Percentage of greater values between two distributions

Importance of raw data in knowing practical or clinical significance as opposed to statistical significance

Sometimes a t test is very useless to achieving real life goals yet it is incredibly popular to publish t tests comparing two sets of data without showing the raw data in medical science.  Although it could make sense to remove some parts of raw data in order to keep patient data confidential someone could simply replace identifying data such as patient name with unidentifiable information such as id numbers from which the patient name can not be determined.  This would not be possible in all cases such as studying if there is a correlation between the latitude or longitude of primary location of residence address and health conditions but would probably have been possible in most cases that occurred in the past.

In many cases it maybe more useful to know what each of two distributions look like rather than if two distributions are statistically significantly different by a certain alpha value or if one distribution is statistically significantly greater than the other by a certain alpha value

In many cases the question of practical significance or clinical significance is more important than statistical significance.  A question of practical significance could be, what is the probability that a single specific choice will result in a value that is acceptable for my goal.  Other times a question of practical significance could be out of all the available choices which choice has the highest probability of resulting in a value acceptable for my goal.

In such a case it might make more sense to simply look at the raw data of a study and count the number of cases in which the desired value was greater than the minimum value acceptable and less than the maximum value acceptable and then divide that number by the total number of cases.

For example if someone is comparing two treatments for blood pressure someone and wants to decrease a patients blood pressure by at least 10 units but by no more than 50 units it would make more sense to figure out the frequency at which each treatment in a study resulted in a decrease of more than 10 units but less than 50 units than to use a t test.  This would be done by counting the number of cases that result in a decrease of at least 10 units but no more than 50 units then dividing by the total number of cases for each distribution.  Next the person choosing the two treatments figures out which treatment has a higher percentage of acceptable results and chooses that treatment.  

This can not be done without the raw data which is commonly removed from scientific journal articles that give the results of a t test comparing two sets of raw data but removes the raw data.  It might be possible however if the raw data was removed but both sets of raw data were given a standard deviation and mean, to use that information, to estimate the percentage or frequency of cases for each of two choices that produced a decrease of at least 10 units but no more than 50 units in the example above, by using the mean and standard deviations published to construct a distribution to represent each choice based on the assumption of a normal distribution for each of the two treatments that were used.  This however would be a very bad idea if no results for tests of having a normal distribution were presented in the scientific journal article.  I would not be surprised if it is very common never to show the results of any tests showing that the two distributions were actually normally distributed before conducting a t test.  I would like to see a study looking at the percentage of studies in which normal distribution was not checked for but a t test was performed.

Sometimes a question of practical significance might not be so delicate as to have both a upper and lower limit in desired outcomes.  For example if someone wanted a fire arm that could shoot at least a distance of 200 units but did not worry if it could shoot too great a distance.  It would make more sense to look at the raw data for each of the different firearm choices and calculate the percent frequency that exceeded a firing distance of 200 units for each firearm choice's distribution than to conduct a t test on the raw data to determine if each of two firing distribution choices available produced statistically significant different results for an alpha value or if one test produced statistically significant greater results for an alpha value

Sometimes a question of practical significance is yet another degree even less delicate than that and someone might not even care if a at least a minimal number of units were produced but only want to know which of two choices produced greater results the most often.  Someone might say, now we can finally use a t test legitimately.  And in such a case you could use a t test but it still would not be the best choice for at least two reasons.

Firstly, a t test would not be the best choice because the question was not which choice had a greater mean but which choice produced greater results than the other available choice most often.  Secondly, a t test would not be the best choice because you can not know if the results of both distributions will be normally distributed before conducting the test.

I am going to write a possibly new statistics test I thought of which may or may not have already been created by someone else under another name.  I will name this test the "Percentage of greater values between two distributions" test.

Test steps

This test works for ordinal, interval or ratio data and it does not matter if the two sets of data being compared have equal sample sizes or not or if they have odd or even sample sizes.

After collecting data from two samples order the values of the data for each sample in a list in ascendimg or descending order.

Pick one of the two lists

For each value on the chosen list count the number of values on the other list that are below that value as 1 point each, the number of data points exactly equal to that value as 0.5 points each and the number of data points higher than that value as 0 points each.  Each value on the chosen list will have a certain score or sum total of points.  If there are more than one data points on the list other than the chosen list that have that value count that data point multiple times.  Add up the total number of points for each value on the chosen list together for all the values on the chosen list and divide it by the product of the sample size of the chosen list and the sample size of the other list.  If a value on the chosen list occurs multiple times then count or use that value multiple times when adding up the points for all of the values.  You can swap lists next and get a score for each list.  The final value for each list should give you the percent of time that a value randomly selected from the population that one list was sampled from should be greater than another value randomly selected from the population the other list was sampled from.  In some cases involving data with discrete quantized values it maybe better to set the 0.5 points each values to 0 points instead as long as you do so consistently for both lists, I choose them to be 0.5 points based on the assumption of non quantized data in which it could have a 50% chance of being above and 50% chance of being below the other data point if a very small measurement difference was made such as one data point being 1.49999 and the other being 1.50001 instead of both being 1.5 when measured to two significant digits each.

Examples

I will use this possibly new test to compare two such sets of random uniform data each with a sample size of five, one between 0 and 1 and the other between 0.1 and 1.1 both to 4 decimal places from

https://www.random.org/decimal-fractions/

https://web.archive.org/web/20220402133005/https://www.random.org/decimal-fractions/ 

I will add 0.1 to the second set of numbers between 0 and 1 to get a range between 0.1 and 1.1  

( I was hoping this would increase the likelihood of the results for the two distributions looking different but unfortunately the distribution which has 0.1 added to it had a lower mean and median and the opposite of intended results were produced)

In order to compare this test with a t-test I will assume the p value for the possibly new data is one minus the percent that a randomly selected value from one list will be greater than a randomly selected value from another list, even though this does not line up the same way as a t test because a t test only compares if one mean is likely to be greater than another.  I will also look at the percent of values from one list that are less than or less than or equal to the mean and median of the other list ( four combinations choice of mean or median and choice of less than or less than or equal to. )

Here are your random numbers:

0.8349
0.3538
0.8425
0.2781
0.5241 

Timestamp: 2022-04-11 00:42:01 UTC

List A (sample A in order from least to greatest)

0.2781

0.3538

0.5241

0.8349

0.8425

Sample B before adding 0.1

Here are your random numbers:

0.1081
0.3479
0.4587
0.7225
0.3923

Timestamp: 2022-04-11 00:46:43 UTC

0.1081

0.3479

0.3923

0.4587

0.7225

List B ( sample B in order after 0.1 is added)

0.2081

0.4479

0.4923

0.5587

0.8225

Start of example for "Percentage of greater values between two distributions" test

Value B, Value A, points ( 1 if A<B, 0.5 if A=B, 0 if A>B)

0.2081, 0.2781, 0

0.2081, 0.3538, 0

0.2081, 0.5241, 0

0.2081, 0.8349, 0

0.2081, 0.8425, 0

0.4479, 0.2781, 1

0.4479, 0.3538, 1

0.4479, 0.5241, 0

0.4479, 0.8349, 0

0.4479, 0.8425, 0

0.4923, 0.2781, 1

0.4923, 0.3538, 1

0.4923, 0.5241, 0

0.4923, 0.8349, 0

0.4923, 0.8425, 0

0.5587, 0.2781, 1

0.5587, 0.3538, 1

0.5587, 0.5241, 1

0.5587, 0.8349, 0

0.5587, 0.8425, 0

0.8225, 0.2781, 1

0.8225, 0.3538, 1

0.8225, 0.5241, 1

0.8225, 0.8349, 0

0.8225, 0.8425, 0

Total points 10

Sample Size A * Sample Size B = 5 * 5 = 25

Total Points / Sample Size = 10 / 25

estimated 40% of random value from population B being greater than value from population A

one tailed P value = 1 - 0.40 = 0.60 used for comparing with one tailed t test if population B > A

100% - 40% = 60% 

estimated 60% of random value from population A being greater than value from population B

one tailed P value = 1 - 0.60 = 0.40 used for comparing with one tailed t test if population A > B

Value A, Value B, points ( 1 if B<A, 0.5 if B=A, 0 if B>A)

0.2781, 0.2081, 1

0.2781, 0.4479, 0

0.2781, 0.4923, 0

0.2781, 0.5587, 0

0.2781, 0.8225, 0

0.3538, 0.2081, 1

0.3538, 0.4479, 0

0.3538, 0.4923, 0

0.3538, 0.5587, 0

0.3538, 0.8225, 0

0.5241, 0.2081, 1

0.5241, 0.4479, 1

0.5241, 0.4923, 1

0.5241, 0.5587, 0

0.5241, 0.8225, 0

0.8349, 0.2081, 1

0.8349, 0.4479, 1

0.8349, 0.4923, 1

0.8349, 0.5587, 1

0.8349, 0.8225, 1

0.8425, 0.2081, 1

0.8425, 0.4479, 1

0.8425, 0.4923, 1

0.8425, 0.5587, 1

0.8425, 0.8225, 1

Total points 15

Sample Size A * Sample Size B = 5 * 5 = 25

Total Points / Sample Size = 15 / 25

estimated 60% of random value from population A being greater than value from population B

one tailed P value = 1 - 0.60 = 0.40 used for comparing with one tailed t test if population A > B

100% - 60% = 40% 

estimated 40% of random value from population B being greater than value from population A

one tailed P value = 1 - 0.40 = 0.60 used for comparing with one tailed t test if population B > A

End of test example for "Percentage of greater values between two distributions" test

Comparing with t - test

One tailed t-test to determine if population A > B

https://www.socscistatistics.com/tests/studentttest/default2.aspx

p-value is .351604

Comparing with percent of values above and below mean and median of data from other list

Percent of values in one list that are less than or greater than mean or median of values from other list

List B ( sample B in order after 0.1 is added)

0.2081

0.4479

0.4923

0.5587

0.8225

median = 0.4923

mean = 0.5059

list of values from sample A that are less than mean of sample B ( 0.2781, 0.3538 )

list of values from sample A that are greater than mean of sample B ( 0.5241, 0.8349, 0.8425 )

no values from sample A that are equal to the mean of sample B

list of values from sample A that are less than median of sample B ( 0.2781, 0.3538 )

list of values from sample A that are greater than median of sample B ( 0.5241, 0.8349, 0.8425 )

no values from sample A that are equal to the median of sample B

60% of values from A are greater than or equal to the mean of B

60% of values from A are greater than the mean of B

60% of values from A are greater than or equal to the median of B

60% of values from A are greater than the median of B

https://www.calculatorsoup.com/calculators/statistics/average.php

https://web.archive.org/web/20220324192741/https://www.calculatorsoup.com/calculators/statistics/average.php

List A (sample A in order from least to greatest)

0.2781

0.3538

0.5241

0.8349

0.8425

median = 0.5241

mean = 0.56668

list of values from sample B that are less than mean of sample A ( 0.2081, 0.4479, 0.4923, 0.5587 )

list of values from sample B that are greater than mean of sample A ( 0.8225 )

no values from sample B that are equal to the mean of sample A

list of values from sample B that are less than median of sample A ( 0.2081, 0.4479, 0.4923 )

list of values from sample B that are greater than median of sample A ( 0.5587, 0.8225 )

no values from sample B that are equal to the median of sample A

20% of values from B are greater than or equal to the mean of A

20% of values from B are greater than the mean of A

40% of values from B are greater than or equal to the median of A

40% of values from B are greater than the median of A

Start of example for "Percentage of greater values between two distributions" test

Sample E

0.1081

0.3479

0.4587

0.7225

0.3923

List E

0.1081

0.3479

0.3923

0.4587

0.7225

Calculating scores for list E

Value B, Value A, points ( 1 if A<B, 0.5 if A=B, 0 if A>B)

0.1081, 0.2781, 0

0.1081, 0.3538, 0

0.1081, 0.5241, 0

0.1081, 0.8349, 0

0.1081, 0.8425, 0

0.3479, 0.2781, 1

0.3479, 0.3538, 0

0.3479, 0.5241, 0

0.3479, 0.8349, 0

0.3479, 0.8425, 0

0.3923, 0.2781, 1

0.3923, 0.3538, 1

0.3923, 0.5241, 0

0.3923, 0.8349, 0

0.3923, 0.8425, 0

0.4587, 0.2781, 1

0.4587, 0.3538, 1

0.4587, 0.5241, 0

0.4587, 0.8349, 0

0.4587, 0.8425, 0

0.7225, 0.2781, 1

0.7225, 0.3538, 1

0.7225, 0.5241, 1

0.7225, 0.8349, 0

0.7225, 0.8425, 0

Total points 8

Sample Size E * Sample Size B = 5 * 5 = 25

Total Points / Sample Size = 8 / 25

estimated 32% of random value from population E being greater than value from population A

one tailed P value = 1 - 0.32 = 0.68 used for comparing with one tailed t test if population E > A

100% - 32% = 68% 

estimated 68% of random value from population A being greater than value from population E

one tailed P value = 1 - 0.68 = 0.32 used for comparing with one tailed t test if population A > E

End of example for "Percentage of greater values between two distributions" test

Comparing with other tests

one tailed t-test for mean of population A > mean of population E

P value = 0.163344

https://www.socscistatistics.com/tests/studentttest/default2.aspx

Copyright Carl Janssen 2022 April 10

Custom Distribution Statistical Hypothesis Test

This might not be mathematically accurate in terms of probability theory I am experimenting with the results of doing different things with numbers.  These assumptions I am using do contradict with reality as do all probability based statistical hypothesis tests but possibly in different ways than some others.

I am literally making up this statistical test and calling it the custom distribution statistical hypothesis test.  I do not know if someone already made a test like this before me and called it a different name

Goal :  The probability distribution model that best represents data collected from the population that corresponds to real life samples is not known before collecting the data therefore it is erroneous to assume it will be a normal distribution before analyzing the sample data.  I am trying to create a model that creates a custom model for frequency distribution to represent the population that corresponds to data collected from a sample then create a possibly new type of statistical test.  I will call this possibly new type of statistical test a, "custom distribution statistical hypothesis test."

A "custom distribution statistical hypothesis test" is similar to a replacement for two sample independent unpaired t tests only with the assumption of a "custom distribution" instead of a normal distribution.

This statistics test includes determining the probability that the median of one sample of  data would be at least the distance it is measured to be in a specific direction from the median of another sample of data based on using the "custom frequency distribution" to represent the population of the sample from the other set of data.  This is done twice once for each sample of data compared to the other one.

By similar I do not mean the mathematical definition of the word similar.  The shape of the "custom frequency distribution" is empirically determined in a custom manner for each sample of data simply by using a cumulative frequency table plus certain additional assumptions and it is assumed that the population has a similar shape to the "custom frequency distribution" determined by a table except that it is modeled as continuous instead of discrete.  

The shape of the normal distribution used in t tests however is not determined empirically by the actual sample data but presumed to represent something "close enough" to the true shape of the population's probability distribution from which a sample is collected when running a t test.  Although the standard deviation or variance and mean of a normal distribution used for a t test is determined empirically the shape is not.  For example the shape of a set of data could appear to be uniform (or any other shape than a normal distribution) based on someone rational looking at a empirical probability plot or empirical probability table for a set of data but would still be treated as a normal distribution shape for a t test.

This test should work with ordinal, interval or ratio data when there is a odd sample size for both samples.  When there is a even sample size for one or both samples one can use a mean to get the median for interval or ratio data but the test results should be less trustworthy than with a odd sample size.  When there is a even sample size for one or both samples and the data is ordinal it does not work to use a mean to estimate the median unless you are taking the mean of two identical values.  However, it might be possible in the case of an even number sample size to create a fictitious rank in between the two values that are closest to where the median would be that you would normally take the mean of to get the median with interval or ratio data and use that fictitious rank in place of the median.  For example if the median would be between "10th street" and "11th street" you could create a fictitious rank of "the street that is less than 11th street but greater than 10th street" or if the median was between "3rd degree" and "4th degree" you could create a fictitious rank of "the degree that is more than 3rd degree but less than 4th degree" but you should not use 14th street as a replacement for the median that is in between 12th street and 16th street because there might be a real 14th street that is not located at the mean of the coordinates or positions of 12th and 16th street and you should not use 7th degree to represent a median that is in between 5th degree and 9th degree because 7th degree might be a real rank that is not the same as the mean of 5th degree and 9th degree.

Comparing the medians of two different samples with the probability distributions of the other sample to estimate the probability that the median of one population is greater than the median of the other population.

Assumption 1 :  The sample which is expected to have a greater median is named sample B and the one expected to have a lower median is named sample A.

Assumption 2 : If two values collected from a sample have no collected values taken from a sample found between them then the probability of obtaining or being above a third value that is in between them when collecting a sample with a sample size of one from that population shall be inclusively between the probability of obtaining or being above each of the two values that were collected from the sample.

Assumption 3 : If two values collected from a sample have no collected values taken from a sample found between them then the probability of obtaining or being below a third value that is in between them when collecting a sample with a sample size of one from that population shall be inclusively between the probability of obtaining or being below each of the two values that were collected from the sample.

How I am defining "inclusively between"

2 <= X <= 3 in this example X is inclusively between 2 and 3

2 < X < 3  in this example X is exclusively between 2 and 3

Assumption 4 : The median of a population shall be estimated to be the median of the sample taken from that population.  Sample A is a sample taken from population A and sample B is a sample taken from population B.

Assumption 5 : If both the median of sample A is lower than the median for the population estimated from sample B and the median of sample B is higher than the median for the population estimated from sample A, both for the same alpha level, then the median of population B is higher than the median of population A for that alpha level.  If exactly one of those two conditions are met but not both then it is unclear if the median of population A is lower than the median of population B for that alpha level.  If neither of those conditions are met then the median of population A is not lower than the median of population B for that alpha level.

Null Hypothesis 1 : Median of sample B <= Median of population A

Alternative Hypothesis : Median of sample B > Median of population A

Null Hypothesis 2 : Median of sample A => Median of population B

Alternative Hypothesis : Median of sample A < Median of population B

Null Hypothesis 3 : Median of population B <= Median of population A

Alternative Hypothesis 3 : Median of population B > Median of population A

If Null Hypotheses 1 and 2 are both rejected then null hypothesis 3 is rejected.  If exactly one of Null Hypotheses 1 and 2 are rejected but not both of them then it is unclear if Null Hypothesis 3 is rejected or not.  If Null Hypothesis 1 and 2 both fail to be rejected then then Null Hypothesis 3 fails to be rejected.

Assumption 6 : The P value of rejecting Null Hypothesis 1 shall be one minus the cumulative frequency distribution as calculated from left to right of achieving the median of Sample B on the probability distribution table for Sample A.  In the case where no value from sample A exists with the same value as the median of Sample B this shall be estimated by using the closest value above and the closest value below the median of sample B from the probability distribution table on sample A as explained in assumption 2.  In the case where a value exists on sample A which is the same as the median of sample B the cumulative frequency distribution for the value already on the chart of the cumulative frequency distribution for  sample A shall be used with the frequency greater than or equal the value instead of the greater than only option

Assumption 7 : The P value of rejecting Null Hypothesis 2 shall be one minus the cumulative frequency distribution as calculated from right to left of achieving the median of Sample A on the probability distribution table for Sample B.  In the case where no value from sample B exists with the same value as the median of Sample A this shall be estimated by using the closest value above and the closest value below the median of sample B from the probability distribution table on sample B as explained in assumption 3.  In the case where a value exists on sample B which is the same as the median of sample A the cumulative frequency distribution for the value already on the chart of the cumulative frequency distribution for  sample B shall be used with the frequency less than or equal to the value instead of the less than only option

Examples 1-4

Comparing results between t tests and custom distribution statistical hypothesis tests for uniform distributions in which both samples have equal variance and independent unpaired samples are used

Example 1

sample A = (1, 2, 3, 4, 5)

sample B = (2.5, 3.5, 4.5, 5.5, 6.5)

custom distribution statistical hypothesis test

sample A = (1, 2, 3, 4, 5), median = 3

value, frequency=value, frequency <= value, frequency < value

0.5, 0, 0, 0, 0

1, 1/5, 1/5, 0

1.5, 0, 1/5, 1/5

2, 1/5, 2/5, 1/5

2.5, 0, 2/5, 2/5

3, 1/5, 3/5, 2/5

3.5, 0, 3/5, 3/5

4, 1/5, 4/5, 3/5

median of sample B 4.5, 0, 4/5, 4/5

5, 1/5, 1, 4/5

5.5, 0, 1, 1 

1-1  <= P <= 1 - 4/5

0 <= P <= 0.2 for the median of sample B being higher than the estimated median of population A  

sample B = (2.5, 3.5, 4.5, 5.5, 6.5) median = 4.5

value, frequency=value, frequency => value, frequency > value

7, 0, 0, 0

6.5, 1/5, 1/5, 0

6, 0, 1/5, 1/5

5.5, 1/5, 2/5, 1/5

5, 0, 2/5, 2/5

4.5, 1/5, 3/5, 2/5

4, 0, 3/5, 3/5

3.5, 1/5, 4/5, 3/5

median of sample A 3, 0, 4/5, 4/5

2.5, 1/5, 1, 4/5

2, 0, 1, 1

1-1  <= P <= 1 - 4/5

0 <= P <= 0.2 for the median of sample A being lower than the estimated median of population B

for alpha > P reject the null hypothesis

for alpha < P fail to reject the null hypothesis

for one tailed alpha > 0.2 the median of sample A can be considered lower than the median of population B

and

for one tailed alpha > 0.2 the median of sample B can be considered higher than the median of population A

therefore

for one tailed alpha > 0.2 the median of population B can be considered greater than the median of population A

0 < one tailed alpha < 0.2 the median of sample A can not be considered conclusively if it is lower than the median of population B

and

0 < one tailed alpha < 0.2 the median of sample B can not be considered conclusively if it is higher than the median of population A

therefore

0 < one tailed alpha < 0.2 the median of population B can not be considered conclusively if it is greater than the median of population A

t - test

Null Hypothesis : mean of Population B <=  mean of Population B

Alternative Hypothesis : mean of Population B > mean of Population A

sample A = (1, 2, 3, 4, 5), mean = 3, n = 5

sample variance = [ ( 5 -3 ) ^ 2 + ( 4-3 ) ^ 2 + ( 3 - 3) ^2 + ( 2 - 3 ) ^2 + (1-3) ^2 ] / ( 5 - 1 )

sample variance = ( 4 + 1 + 1 + 4 ) / 4 = 10 /4 = 2.5

sample std = sample variance ^ 0.5 = 2.5 ^ 0.5 = approx 1.58113883008

sample B = (2.5, 3.5, 4.5, 5.5, 6.5), mean = 4.5, n = 5

sample v =[ ( 6.5 -4.5 ) ^ 2 + ( 5.5-4.5 ) ^ 2 + ( 4.5 - 4.5) ^2 + ( 3.5 - 4.5 ) ^2 + (2.5-4.5) ^2 ] / ( 5 - 1 )

sample variance = ( 4 + 1 + 1 + 4 ) / 4 = 10 /4 = 2.5

sample std = sample variance ^ 0.5 = 2.5 ^ 0.5 = ( 5 / 2 ) ^ 0.5

mean sample B - mean sample A = 4.5 - 3 = 1.5

in this case Sp = sample std A = sample std B because the sample sizes and variance are exactly equal

t = ( mean sample B - mean sample A ) / ( Sp *[ (1/na) +(1/nB) ] ^ 0.5 )

1/na + 1/nB = 1/5 + 1/5 = 2/5

t = 1.5 / [ (5/2) ^ 0.5 * (2/5) ^ 0.5 ] = 1.5

degrees of freedom = df = nA + nB -2 = 5 + 5 -2 = 8

one tailed p value = .086002

https://www.socscistatistics.com/pvalues/tdistribution.aspx

one tailed p value = 0.08600

https://www.statology.org/t-score-p-value-calculator/

for one tailed alpha > 0.086 the mean of population B can be considered greater than the mean of population A

for one tailed alpha < 0.086 the mean of population B can not be considered greater than the mean of population A

Example 2

sample A = (1, 2, 3, 4, 5)

sample B = (2, 3, 4, 5, 6)

sample A = (1, 2, 3, 4, 5), median = 3

value, frequency=value, frequency <= value, frequency < value

1, 1/5, 1/5, 0

2, 1/5, 2/5, 1/5

3, 1/5, 3/5, 2/5

3.5, 0, 3/5, 3/5

4, 1/5, 4/5, 3/5

median of sample B 4, 1/5, 4/5, 3/5

4.5, 0, 4/5, 4/5

5, 1/5, 1, 4/5

5.5, 0, 1, 1

P = 1 - 4/5

P = 0.2 for the median of sample B to be greater than than estimated median of population A

custom distribution statistical hypothesis test

Null Hypothesis : Median of sample B <= Median of population A

Alternative Hypothesis : Median of sample B > Median of population A

Null Hypothesis : Median of sample A => Median of population B

Alternative Hypothesis : Median of sample A < Median of population B

t - test

Null Hypothesis : mean of Population B <=  mean Population B

Alternative Hypothesis : mean of Population B > mean of Population A

t = 1

one tailed P = 0.173297

https://www.socscistatistics.com/tests/studentttest/default2.aspx

one tailed P = 0.17330

https://www.statology.org/t-score-p-value-calculator/

Example 3

sample A = (1, 2, 3, 4, 5, 6)

sample B = (2.5, 3.5, 4.5, 5.5, 6.5, 7.5)

custom distribution statistical hypothesis test

Null Hypothesis : Median of sample B <= Median of population A

Alternative Hypothesis : Median of sample B > Median of population A

Null Hypothesis : Median of sample A => Median of population B

Alternative Hypothesis : Median of sample A < Median of population B

t - test

Null Hypothesis : mean of Population B <=  mean of Population B

Alternative Hypothesis : mean of Population B > mean of Population A

Example 4

sample A = (1, 2, 3, 4, 5, 6) 

sample B = (2, 3, 4, 5, 6, 7) 

custom distribution statistical hypothesis test

Null Hypothesis : Median of sample B <= Median of population A

Alternative Hypothesis : Median of sample B > Median of population A

Null Hypothesis : Median of sample A => Median of population B

Alternative Hypothesis : Median of sample A < Median of population B

t - test

Null Hypothesis : mean of Population B <=  mean of Population B

Alternative Hypothesis : mean of Population B > mean of Population A

Example 5

Comparing custom distribution statistical hypothesis test with unpaired independent t test for the following triangular shaped distributions

sample A = ( 1, 2, 2, 3, 3, 3, 4, 4, 5 )

sample B = ( 2, 3, 3, 4, 4, 4, 5, 5, 6 )

custom distribution statistical hypothesis test

Null Hypothesis : Median of sample B <= Median of population A

Alternative Hypothesis : Median of sample B > Median of population A

Null Hypothesis : Median of sample A => Median of population B

Alternative Hypothesis : Median of sample A < Median of population B

t - test

Null Hypothesis : mean of Population B <=  mean of Population B

Alternative Hypothesis : mean of Population B > mean of Population A

Example 6 

custom distribution statistical hypothesis test with unequal sample sizes

sample A = ( 1, 2, 3, 4, 5, 6, 7, 8)

sample B = ( 3, 4, 5, 6, 7 )

Null Hypothesis : Median of sample B <= Median of population A

Alternative Hypothesis : Median of sample B > Median of population A

Null Hypothesis : Median of sample A => Median of population B

Alternative Hypothesis : Median of sample A < Median of population B

Copyright Carl Janssen 2022

https://en.m.wikipedia.org/wiki/Student%27s_t-test

http://web.archive.org/web/20220307082135/https://en.m.wikipedia.org/wiki/Student's_t-test

https://duckduckgo.com/?q=independent+t+test+equal+variance+and+equal+sample+size&ia=web

https://www.real-statistics.com/students-t-distribution/two-sample-t-test-equal-variances/

http://web.archive.org/web/20200804112708/https://www.real-statistics.com/students-t-distribution/two-sample-t-test-equal-variances/

https://duckduckgo.com/?q=unpaired+t+test+equal+variance+and+equal+sample+size&ia=web

https://tungmphung.com/unpaired-two-sample-t-test-independent-t-test/

http://web.archive.org/web/20210423122739/https://tungmphung.com/unpaired-two-sample-t-test-independent-t-test/

https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/a/population-and-sample-standard-deviation-review

http://web.archive.org/web/20220323223702/https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-sample/a/population-and-sample-standard-deviation-review

https://duckduckgo.com/?q=standard+deviation+sample+vs+population+formula&ia=web

https://www.statology.org/population-vs-sample-standard-deviation/

http://web.archive.org/web/20210824024007/https://www.statology.org/population-vs-sample-standard-deviation/

https://duckduckgo.com/?q=test+statistic+and+critical+value+calculator&ia=web

https://duckduckgo.com/?q=p-value+and+alpha&ia=web

https://duckduckgo.com/?q=+hypothesis+one+tail+t+test+example&ia=web

https://duckduckgo.com/?q=t+test+percentile&ia=web

https://duckduckgo.com/?q=t+test+p+value+calculator&ia=web

https://www.socscistatistics.com/pvalues/tdistribution.aspx

https://web.archive.org/web/20211011165105/https://www.socscistatistics.com/pvalues/tdistribution.aspx

https://www.socscistatistics.com/tests/studentttest/default2.aspx

https://web.archive.org/web/20211011072830/https://www.socscistatistics.com/tests/studentttest/default2.aspx

https://duckduckgo.com/?q=p+value+t+test+chart&ia=web

https://www.statology.org/how-to-calculate-a-p-value-from-a-t-test-by-hand/

https://web.archive.org/web/20211123175421/https://www.statology.org/how-to-calculate-a-p-value-from-a-t-test-by-hand/

https://www.statology.org/t-score-p-value-calculator/

https://web.archive.org/web/20210306170506/https://www.statology.org/t-score-p-value-calculator/

fat runner paradox test for special relativity

There is no acceleration the racer starts running before reaching the startline but both the racer and referee start their stopwatch at zero the exact moment the midpoint or center of the racers body crosses the startline and stop the stopwatch the moment the midpoint of the racers body crosses the finish line.  The runner does not stop running when he crosses the finish line but maintains the same speed unless blocked by a wall eliminating the issue of acceleration from the calculations.  The midpoint is measured in the direction the racer is running.  

There shall be one more line called a "percentage line."   The ( Y * 100 ) percent completion line marked whatever percent between the start line and the finish line one wants to know that fraction or percentage of the race has been complete.  Where if Y = 0 the line would be at the start line, if Y = 0.5 the line would be half way between the start line and the finish line marking 50% completion when crossed and if Y = 1 the line will be at the finish line.

There is an observer who is stationary relative to the startline, finish line, racing track and wall who measures the following 

all the following measurements are in one dimension going in the same direction and it's opposite

H is used to represent speed as a fraction times the speed of light

H and Y are unitless

0 < H < 1

0 <= Y <= 1

L, W and D are used to represent thr distances described below in units of the speed of light * time such as light years or light seconds

L, W and D have units of time

L > 0

W > 0

D > 0

If the racer stood still relative to the referee his body would have a length of 2D*c in the direction he is running, or the front of the runner's body is a distance D*c in front of his midpoint

The distance from the startline to the stopline is L*c

The distance from the finish to a wall is W*c

The racer is running at a velocity of H*c

The distance from the startline to the percentage line is Y*L*c

Is there any combination of these variables such that the runner will hit the wall at the exact moment in time when his midpoint touches the stopline from the point of view of the referee but not from the point of view of the runner or will hit the wall from the point of view of the runner but not from the point of view of the referee?

The distance L*c is not necessary to answer that question but prior to answering that question for the sake of practice the running time and running distance from start to finish from the point of view of both the runner and referee will be calculated and it will be confirmed that the speed of the wall moving toward the runner from the runners point of view is the same as the speed of the runner moving toward the wall from the referee's point of view

Even though both are observer's the referee shall be remembered by a lowercase o as in observer which can be a different symbol than the number 0 zero

The runner shall be remembered by j as in jogger because runner and referee both start with r even though running is usually considered faster than jogging

Tof= time to reach the finish line according to referee's stopwatch

Toy= time to reach the percentage line according to referee's stopwatch

Tjf= time to reach the finish line according to the runner's stopwatch

Tjy= time to reach the percentage line according to the runner's stopwatch

Sof = Distance from startline to finish line from referee's viewpoint, this is a proper or rest length

Soy = Distance from startline to percentage line from referee's viewpoint, this is a proper or rest length

Sjf = Distance from startline to start finish line from runner's viewpoint, this is not a proper nor rest length

Sjy = Distance from startline to percentage line from runner's viewpoint, this is not a proper nor rest length

Fow = Length from finish line to wall from referee's viewpoint, this is a proper or rest length

Fjw = Length from finish line to wall from runner's viewpoint, this is not a proper nor rest length

Do = Length from midpoint to front of runner's body from referee's viewpoint, this is not a proper nor rest length

Dj = Length from midpoint to front of runner's body from runner's viewpoint, this is a proper or rest length

From the point of view of the referee and according to the referee's stopwatch

The runners body would appear thinner being shorter in it's direction of movement because it is moving toward the referee

The running track would not have it's length distorted because it is stationary relative to the referee

Sof = L*c

Tof = Sof / ( H*c ) = L*c / ( H*c ) = L / H

Soy = Y*L*c

Toy = Soy / ( H*c ) = Y*L*c / (H*c) = Y * L / H

Dj = D * c

Do = Dj * (1 - [ ( H*c ) ^ 2 ] / c^2 ) ^ 0.5

Do = D * c * ( 1 - H ^ 2 ) ^ 0.5

Fow = W * c

From the point of view of the runner and according to the runners stopwatch

The runner would be stationary and the wall would be moving toward the runner and the running track, startline and finish line would be moving in the same direction as the wall

The runners body would not appear thinner because the runner is stationary relative to himself

The running track would appear shorter because it would be moving relative to the runner

Sjf = Sof * ( 1 - H ^ 2 ) ^ 0.5

Sjf = L*c * ( 1 - H ^ 2 ) ^ 0.5

Tjf = Sjf / H

Tjf = [ L*c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Sjy = Soy * ( 1 - H ^ 2 ) ^ 0.5

Sjy = Y*L*c * ( 1 - H ^ 2 ) ^ 0.5

Tjy = Sjy / H

Tjy = [ Y*L*c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Dj = W * c

Fjw = Fow * ( 1 - H ^ 2 ) ^ 0.5

Fjw = W * c * ( 1 - H ^ 2 ) ^ 0.5

Proof that special relativity can not change order of events when there is no acceleration

Percent of time complete = 100% * Tjy / TjF = 100% * Toy / Tof

Toy / Tof = ( Y * L / H ) / ( L / H ) 

Toy / Tof = Y

Tjy / Tjf =  ( [ Y*L*c * ( 1 - H ^ 2 ) ^ 0.5 ] / H ) /  ( [ L*c * ( 1 - H ^ 2 ) ^ 0.5 ] / H ) 

Tjy / Tjf = Y

The percentage of time completed is the same as the percentage of the distance completed on the race track from the point of view of both the referee and the runner.  

A% will never be completed before B% for both the referee and the runner when A is less than B

B% will never be completed before A% for both the referee and the runner when B is less than A

This proves that the order of events never changes in special relativity without acceleration, the time that the same event occurs would sometimes be different for both the referee and the runner but the order that events occur in does not change.  The order of events is the same for both the referee and the runner.

Example to help understand the proof

For both the referee and the runner these 11 events happen in the same order

First 0% completion Y = 0

Toy = 0

Tjy = 0

Second 10% completion Y = 0.1

Toy = 0.1 * L / H 

Tjy = [ 0.1 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Third 20% completion Y = 0.2

Toy = 0.2 * L / H 

Tjy = [ 0.2 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Fourth 30% completion Y = 0.3

Toy = 0.3 * L / H 

Tjy = [ 0.3 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Fifth 40% completion Y = 0.4

Toy = 0.4 * L / H 

Tjy = [ 0.4 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Sixth 50% completion Y = 0.5

Toy = 0.5 * L / H 

Tjy = [ 0.5 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Seventh 60% completion Y = 0.6

Toy = 0.6 * L / H 

Tjy = [ 0.6 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Eighth 70% completion Y = 0.7

Toy = 0.7 * L / H 

Tjy = [ 0.7 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Ninth 80% completion Y = 0.8

Toy = 0.8 * L / H 

Tjy = [ 0.8 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Tenth 90% completion Y = 0.9

Toy = 0.9 * L / H 

Tjy = [ 0.9 * L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Eleventh 100% completion Y = 1

Toy = L / H 

Tjy = [ L * c * ( 1 - H ^ 2 ) ^ 0.5 ] / H

Disproof of special relativity

Proof of contradiction with reality inherant in special relativity

If someone can hit the wall when their midpoint crosses the finish line from the point of view of one observer but not hit the wall when the midpoint crosses the finish line from the point of view of the other observer then special relativity can not be a true representation of reality and no excuses about diffetent orders of events from different reference frames can justify the theory of special relativity because it has already been proven above that the order of events is the same in all reference frames in cases such as this one where no acceleration is occuring

If Dj = Fjw but Do does not equal Fow for at least one positive combination of values for the variables in which travel occurs slower than the speed of light then special relativity is in contradiction with physical reality

or

If Do = Fow but Dj does not equal  Fjw for at least one positive combination of values for the variables in which travel occurs slower than the speed of light then special relativity is in contradiction with physical reality

or

If Do < Fow and simultaneously Dj > Fjw for at least one positive combination of values for the variables in which travel occurs slower than the speed of light then special relativity is in contradiction with physical reality

or

If Do > Fow and simultaneously Dj < Fjw for at least one positive combination of values for the variables in which travel occurs slower than the speed of light then special relativity is in contradiction with physical reality

Example of disproof

In the case where the proper distance between the front of the runner's body and the center of the runner's body when the runner is at rest is equal to the proper distance from the finish line to the wall when the wall is at rest

If Dj = Fow 

then

Do = Dj * ( 1 - H ^ 2 ) ^ 0.5

Do = Fow * ( 1 - H ^ 2 ) ^ 0.5 < Fow

Do < Fow      

meaning collision with wall does not occur from referee's viewpoint

Fjw = Fow * ( 1 - H ^ 2 ) ^ 0.5

Dj = Fow > Fow * ( 1 - H ^ 2 ) ^ 0.5 = Fjw

Dj > Fjw

meaning collision with wall does occur from runner's viewpoint

Copyright Carl Janssen 2022

https://wikimedia.org/api/rest_v1/media/math/render/svg/37ae2718b3d30ba8c8f9019bedde2e289f1f3b28

http://web.archive.org/web/20220312022749/https://wikimedia.org/api/rest_v1/media/math/render/svg/37ae2718b3d30ba8c8f9019bedde2e289f1f3b28

https://en.m.wikipedia.org/wiki/Length_contraction

http://web.archive.org/web/20211130084048/https://en.m.wikipedia.org/wiki/Length_contraction

The proper length[1] or rest length[2] of an object is the length of the object measured by an observer which is at rest relative to it, by applying standard measuring rods on the object. 

https://en.m.wikipedia.org/wiki/Proper_length

http://web.archive.org/web/20211126104939/https://en.m.wikipedia.org/wiki/Proper_length

Stars Red shift can be from a medium instead of a movement

1 The intensity of light should be inversely proportional to the distance between the source and the observer before considering how much intensity is lost by traveling through a medium

2 The intensity of light is lost as it travels through a medium by a different amount for each frequency or wavelength

3 The farther a light source is from an observer the more mass it will need to have as a source if made of the same material at the same temperature and density in order to have sufficient intensity of brightness to be seen by the observer and the more mass the light source has the greater gravitational red shift will occur and the greater the gravitational red shift that occurs the less energy per area per time the observer will experience from the light emitted by the source.  Light from some sources might not experience a red shift but the percent of those light sources that do not have enough mass to be visible would increase as their distance from the earth increases.  As the distance increases the minimum mass required and thus the minimal amount of red shift required for a stationary light source (of the same material, temperature, pressure and density) to be seen increases.  This is  assuming that gravitational red shifting and blue shifting really does occur in real life.

4 From a probabilistic viewpoint, the further the distance between the source and the observer the more likely at least one opaque object of a given size or greater will be between the source and the observer.  This might be similar to the relationship of the number of cracks on a section of road and the length of the segment of a road or the Poisson distribution.

Points 1 through 4 could account for red shifts from distant stars and a lack of unlimitted or infinite brightness experienced on earth by an individual in a universe that might be of unlimited or infinite volume and might have never experienced a big bang and in which some stars that were thought to be moving away from an observer due to a doppler shift might not be moving away from the observer in all cases that big bang believers have claimed them to be doing so.

Alleged shifts in light spectrums may occur from light sources that are stationary relative to the earth as the light from a stationary source traveles through the medium of outer space not necessarily through a doppler effect caused by a light source moving away from or towards the earth.

The intensity of light at one frequency or wavelength may be decreased by a different percent than the intensity at another frequency or wavelength for the same medium at the same thickness

Outer space is supposedly an imperfect vacuum which would mean it is a physical material.  If the physical material outer space is made out of reduces the intensity of blue light more strongly than it reduces the intensity of red light or absorbs blue light better than red light traveling through the same distance or thickness of material at the same temperature, pressure, chemical concentration and density then the further the distance between a stationary light source and an observer the more red shifted the light would appear to be to that observer.

By red shift in the context of the effects of light traveling through a medium, I mean a shift in the frequency and wavelength distributions of the light.  In this specific type of red shift the weighted average of frequency with respect to intensity would decrease and the weighted average of wavelength with respect to intensity would increase as the distance the light has to travel from the source through the medium to reach the observer increases.  A blue shift in this context would mean the weighted average of frequency with respect to intensity would increase and the weighted average of wavelength with respect to intensity would decrease.  The frequency of the individual photons would not change as the distance traveled through the same medium increases but the intensity would change for a given frequency as the distance traveled through the same medium increases.  The percent decrease in intensity would be different for blue light than red light since the medium the light is traveling through blocks or absorbs one wavelength or frequency a different amount than it does with another wavelength or frequency.  I mention traveling a further distance through the same medium because changing what medium a wave is in can also change it's wavelength intensity distribution and or frequency intensity distribution but that is not the source of the red shift I am talking about.  

One should also consider that if the path of a light source orbited around a observer in a perfect circular path at constant absolute value of acceleration and absolute value of velocity and maintained a constant distance from that observer what sort of doppler effect ( if any ) would happen even though the  light source is neither moving towards or away from the observer in terms of the absolute value of the distance between the source and the observer

Supposedly gravitation could also cause a shift in frequency that is separate from or in addition to the one allegedly caused by velocity.  The farther a star is from earth the more massive the star would usually have to be to be seen from earth because the intensity of light is inversely proportional to the square of the distance between the observer and the light source before intensity lost due to traveling through a medium is even considered.  This means a greater gravitational red shift would usually be experienced the farther the distance from the earth a star is observed from even if it was not moving away from the earth.

In the case of gravitational red shift I believe the claim would be that each photon actually individually increases it's wavelength and or decreases it's frequency.  It is possible for a wave to change it's frequency without changing it's wavelength or change it's wavelength without changing it's ftequency if the velocity or speed of the wave changes which can happen if it changes it's medium or if the medium changes some factor like temperature.

A universe of infinite or unlimited volume with an unlimited number of stars would not necessarily be unlimitedly or infinitely bright as viewed from the earth because objects between some of the stars and the earth would be opaque or partly opaque and block some of the light from other stars or objects in addition to light becoming more dim before it reaches or fails to reach earth through the inverse square law and the effect of traveling through a medium as well as allegedly shifting in frequency and or wavelength and reducing in energy per photon through gravitational red shifting it experiences on the way to earth from some objects near it's path exceeding any gravitational blue shifting it experiences on the way to earth from other objects.

Someone might argue that although I listed three different sources of red shifts ( something similar to Beer-Lambert Law, Doppler, Gravitation) we can still know the universe is expanding because each source of red shift effects things in a mathematically different way and we can know which one of each type contributes to how much of each and from there calculate the doppler shift and know how fast each of the stars are moving away from the earth based on the frequency distribution a star should emit based on it's chemical composition.  The problem with that is we do not know the chemical composition of stars.  Supposedly every element we have tested on earth emits a unique distribution of light frequencies or wavelengths and the distribution emitted by a star is the same as the distribution emitted by the elements or perhaps chemical compounds that make it up only shifted in a precise way such that there is only one possible combination of elements or compounds that could have emitted that light frequency or wavelength distribution with not a single one to possibly be added or removed from the list mathematically determinable for that star.  That however is simply untrue because we can only compare the distributions with distributions for materials found on earth and we have never flew a rocket to the sun and collected a sample of the matetial to know it is not made from something different, perhaps an undiscovered element that does not even fit on the periodic table with it's integer atomic numbers the way we currently understand it to work and maybe this element emits what looks like a shifted distribution of another element.  It sounds ridiculus but theoritical astronomers claim there is dark matter that is not like any material discovered on earth but somehow eliminates their massive calculation errors if you put the right amount of this dark matter in if I am being ridiculus for claiming the stars might be made of a material not discovered on earth then they are being ridiculus for claiming outer space has a material not discovered on earth that just so happens to exist in the quantities to reduce their calculation errors from the amount that disproves their theories to an amount that excuses their theories.  I am not saying the stars are made of a undiscovered or hypothetical material, I am saying they might be but to some people it is a definitive fact that 85% of the universe is made out of a hypothetical material.  Here on earth I have never known a construction worker to build a building out of a hypothetical matetial, they might build something out of a material they do not know the name of but it is not hypothetical they can actually touch it, unlike the sun which has never been touched by human hands if it is so hot you will burn up before you reach it as has been claimed and more importantly if it is so high in the sky I can not build a ladder or skyscraper or climb a mountain to reach it.  There is another problem than not knowing the material the stars are made out of and that is not knowing how much gravity they have which would mean not knowing how much of the red shift is from gravity vs how much is from doppler.  We can know how much acceleration is caused by local gravity on earth but we can not personally know how much acceleration is caused by local gravity on the moon unless we personally go to the moon which most people can not do and therefore can not personally confirm the moon data is not fake.  If the most people could go to the moon then most people could travel to the moon and take the acceleration measutements from there personally instead of going by the word of people who already have a track record of fabricating stories about dark matter.  We can not personally measure the Universal gravitational constant from the earth only the local gravitational constant because the universal gravitational constant is calculated by manipulating the position of objects in a physics laboratory and carefully measuring their masses but the person who takes the observation's mass and position is not measured but worse yet the mass and position of the walls, ceilings and floors of the building next to the objects is not measured.

We do not know

1 How much of the red shift is from gravity - because we do not know the universal gravitational constant - because they ignore the mass of the building but measure the mass of the equipment inside the building) - you can take a video camera and drop an object in front of a ruler with a stopwatch also running on video and record it and measure local acceleration due to gravity but that is different than proving a Universal Gravitational Constant even exists - And no you probably have not tested it on the moon if you are reading this

2 How much of the red shift is from traveling through a medium -  Because we do not know how outer space effects light over long distances involving something similar to the Beer-Lambert law - Because the common person can not fly up in a satelite and shoot several different types of monochromatic lasers through the medium outer space is made up of and take measurements.  Even if we could we have not taken those measurements to make sure outer space between the star and wherever the observers atmosphere becomes outer space is the same material

3 What the starting spectrum that the light frequency distribution was shifted from - Because we do not know what the spectrum of the star would have been based on it's material chemical composition - Since we do not know what material the stars are made out of since we can not assume they are made from chemicals that exist on earth because for all we know they could be 85% dark matter, why not 85% of everything else is except things we can observe on earth

4 Once we know those three things we do not know we can calculate the doppler shift of a star based on a extrapolation of other things that have never been tested in that data range  - If you do not believe in doppler you could set up a race track and take a video camera and could measure the location and time positions of the car and compare it with the doppler speed data for a wide variety of speeds less than 120 miles per hour that a commoner can afford a car to drive at but you can not drive your car at 3/4 the speed of light and take measurements.

 "The primary evidence for dark matter comes from calculations showing that many galaxies would behave quite differently if they did not contain a large amount of unseen matter"

That is not the primary evidence for dark matter that is the primary evidence of at least one of five things 

1 the "physics" assumptions that led to you doing those calculations are wrong

2 You made an expertimental error collecting the data

3 You made an error doing the calculations with the data you collected

4  It's not the theoritical astrophysisicts fault some other physicist calculated G wrong they just plugged the wrong number someone else calculated into the correct equations

Although a novice  would come to the instinctual conclusion that astronomers made a 85% error in how much mass is in the universe after further examination they would reslise it is only a negative 46 % error.  But a true expert would say there was no calculation error at all for the mass except we put in the wrong value for G which was clearly wrong because we ignored those massive walls every time we did a calculation to figure out G

We really need to multiply or perhaps divide what we thought G was by 3.4225 to get the true value of G.  Or perhaps G is another value altogether calculated a different way then how I suggested but it would not surprise me at all if G was pure fiction since g is measurable but no experiment to this day has used correct methodology to measure g that I know of taking into account the mass and position of the human being that looked into the equipment and the mass and position of any walls structures or buildings nearby.  Perhaps it could be measured out doors far away from massive buildings in a region where local g measurements are consistent at several locations near the test site to make sure there is not extra density mass in a section of the ground nearby and with no wind to shift the results.

1.85 * 1.85 = 3.4225

F = G*m1*m2 / (R^2)

3.4225*G*m1*m2  = 1.85*m1*1.85*m2

(G / 3.4225) * m1* m2= G* ( m1 / 1.85 ) * ( m2 / 1.85 )

It is theoritical that there is 85% more dark matter than expetimentally observed.

Traditional Percent Error formula based on assumption that the theory is correct and the experimental data is wrong

  experimental - theoritical / theoritical

Theoritical amount of mass in universe = 1.85 * Experimental

( E - 1.85E ) / 1.85 E= - 0.85 / 1.85 = - 0.459459459459

So I made up a theory and my observed data does not match my theory, I know reality is wrong not my theory the error is in the data not the theory

Keeping it real percent error based on that the experimental observations are correct and the theory that dark matter exists is wrong

theoritical - experimental / experimental

( 1.85E - E ) / E = 0.85

5 There was never any data and this is a practical joke and the common people can not ever look into one of these expensive telescopes to find out there never were any observations just fabricated observations used as an excuse to request tax funding for research and to brag about how smart you are but humbly still not educated enough until you get more money because there is a 85% unknown missing matter mystery that needs more funding.

Copyright Carl Janssen 2022

https://en.m.wikipedia.org/wiki/Beer%E2%80%93Lambert_law

http://web.archive.org/web/20220314215558/https://en.m.wikipedia.org/wiki/Beer%E2%80%93Lambert_law

https://en.m.wikipedia.org/wiki/Gravitational_redshift

http://web.archive.org/web/20211126104210/https://en.m.wikipedia.org/wiki/Gravitational_redshift

https://en.m.wikipedia.org/wiki/Inverse-square_law

http://web.archive.org/web/20220320133858/https://en.m.wikipedia.org/wiki/Inverse-square_law

https://en.m.wikipedia.org/wiki/Poisson_distribution

https://hubpages.com/@fatfist

http://web.archive.org/web/20220313060928/https://hubpages.com/@fatfist

https://discover.hubpages.com/education/OLBERS-PARADOX-A-Physical-Explanation-For-The-DARK-Night-Sky

http://web.archive.org/web/20210518201220/https://discover.hubpages.com/education/OLBERS-PARADOX-A-Physical-Explanation-For-The-DARK-Night-Sky

https://duckduckgo.com/?q=red+shift+not+caused+by+movement&ia=about

https://duckduckgo.com/?q=red+shift+beer+lambert&ia=web

The most absorbance is obtained when beam color is complementary to the color of solution.

https://colors-newyork.com/what-is-red-shift-what-does-it-indicate/

http://web.archive.org/web/20220323062447/https://colors-newyork.com/what-is-red-shift-what-does-it-indicate/

Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe

https://en.m.wikipedia.org/wiki/Dark_matter

http://web.archive.org/web/20220321143913/https://en.m.wikipedia.org/wiki/Dark_matter

 The primary evidence for dark matter comes from calculations showing that many galaxies would behave quite differently if they did not contain a large amount of unseen matter

http://web.archive.org/web/20220321143913/https://en.m.wikipedia.org/wiki/Dark_matter

14 out of 54 mentions of walls

That is to say, we now accept apples as having easily measurable and verifiable gravitational attractions, but we ignore the gravitational attractions of walls weighing thousands of pounds. 

http://milesmathis.com/caven.html

http://web.archive.org/web/20220130225907/http://milesmathis.com/caven.html

some scientists continue to work on models that might not require dark energy. Inhomogeneous cosmology falls into this class.

https://en.m.wikipedia.org/wiki/Inhomogeneous_cosmology

http://web.archive.org/web/20220322060158/https://en.m.wikipedia.org/wiki/Inhomogeneous_cosmology

https://en.m.wikipedia.org/wiki/Exotic_matter

http://web.archive.org/web/20220321143900/https://en.m.wikipedia.org/wiki/Exotic_matter

https://en.m.wikipedia.org/wiki/Exotic_atom

http://web.archive.org/web/20220319073945/https://en.m.wikipedia.org/wiki/Exotic_atom

Historical Origins Science is not real experimental science

 The only model consistent with science is an eternal universe not big bang or finite amount of time ago earth creation.  The assumption for the big bang is we can use input  data to extrapolate what happened in the past as an output and then put that output data in as a new input to figure out what happened even further in the past.  The only way to be consistent about it is to allow someone to extrapolate as far back in the past as they want, if you limit yourself to billions of years ago when the special event of the big bang occured instead of trillions or an unlimmitted amount of years big bangers are no better than the creationists who limit extrapolation to thousands of years in the past when the special event of creation occured or when the special event of a worldwide flood occured.  Both secular evolutionary big bang cosmology origin science and creationist origin science are not real experimental science.

Copyright Carl Janssen 2022 March 22

sine and cosine of sum of two angles

http://web.archive.org/web/20220314121223/https://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/AngleAdditionDiagramSine.svg/338px-AngleAdditionDiagramSine.svg.png

http://web.archive.org/web/20201111211702/https://en.m.wikipedia.org/wiki/File:AngleAdditionDiagramSine.svg

http://web.archive.org/web/20150208110925/https://upload.wikimedia.org/wikipedia/commons/f/f5/AngleAdditionDiagramSine.svg

http://web.archive.org/web/20220319192920/https://en.m.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2013_November_24

http://web.archive.org/web/20220311205122/https://en.m.wikipedia.org/wiki/List_of_trigonometric_identities

https://en.m.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities

https://en.m.wikipedia.org/wiki/Small-angle_approximation#Angle_sum_and_difference

http://web.archive.org/web/20210827113612/https://en.m.wikipedia.org/wiki/Small-angle_approximation

https://en.m.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Angle_sum_identities

http://web.archive.org/web/20210506193820/https://en.m.wikipedia.org/wiki/Proofs_of_trigonometric_identities

https://ccssmathanswers.com/trigonometrical-ratios-of-90-degree-minus-theta/

http://web.archive.org/web/20210411120031/https://ccssmathanswers.com/trigonometrical-ratios-of-90-degree-minus-theta/

https://imgflip.com/i/69a0vj

https://i.imgflip.com/69a0vj.jpg

http://web.archive.org/web/20220319200030/https://imgflip.com/i/69a0vj

http://web.archive.org/web/20220319200030im_/https://i.imgflip.com/69a0vj.jpg

Square Roots of Complex Numbers without trigonometry or exponential functions

 

M, N, D and E do not equal 0 

D and E are real numbers

i * i = - 1

- i * - i = - 1

D + E i = ( M + N i ) ^ 2 = M ^ 2 - N ^ 2 + 2 M N i

D = M ^ 2 - N ^ 2

E = 2 M N

N = E / 2 M

D = M ^ 2 - ( E / 2 M ) ^ 2

You should not multiply or divide both sides of an equation by a variable unless you exclude cases of division by zero but that has already been done in the preconditions already set

D M ^ 2 = M ^ 4 - ( E ^ 2 ) / 4

0 = 4 M ^ 4 - 4 D M ^ 2 - E ^ 2

if a x ^ 2 + b x + c = 0 then x = ( - b +- [ b ^ 2 - 4 a c ] ^ 0.5 ) / ( 2 a )

x = M ^ 2

a  = 4

b = - 4 D

c = - E ^ 2

M ^ 2 = ( 4 D +- [ ( - 4 D ) ^ 2 - 4 * 4 * (- E ^ 2) ] ^ 0.5 ) / ( 2 * 4 )

M ^ 2 = ( 4 D +- [ 16 D ^ 2 + 16 E ^ 2 ] ^ 0.5 ) / 8

M = + - [ ( 4 D +- [ 16 D ^ 2 + 16 E ^ 2 ] ^ 0.5 ) / 8 ] ^ 0.5

N = +- 0.5 E / [ ( 4 D +- [ 16 D ^ 2 + 16 E ^ 2 ] ^ 0.5 ) / 8 ] ^ 0.5

Although there are eight possible solutions for N and four possible solutions for M based on what sign is chosen based on the plus or minus signs, only certain combinations of M and N values work and there should theoritically be exactly two solutions for the square root of D + Ei and not all combinations of solutions for M and N will work so you should check your solutions

if M is a pure imaginary number then N will also be a pure imaginary number and this would not prevent a solution

if M is a real number then N will also be a real number

Example

D + E i = 1 +  i

M =  + - [ ( 4 * 1 +- [ 16 + 16 ] ^ 0.5 ) / 8 ] ^ 0.5

Real solutions for M and N

In this case when M is positive then N is positive and when M is negative then N is negative because both D and E are positive

https://duckduckgo.com/?q=(+(+4+*+1+%2B+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5&ia=calculator

https://duckduckgo.com/?q=0.5+%2F+(+(+(+4+*+1+%2B+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5+)&ia=calculator

M = approximately +- 1.09868411347

N = +- 0.5 / 1.09868411347 approximately

N = approximately +- 0.455089860562

M ^ 2 - N ^ 2 = 1.09868411347 ^ 2 - 0.455089860562 ^ 2 = 1.00000000001 = D = = 1 approximately

2MN = 2*1.09868411347*0.455089860562 = 1 = E approximately

https://duckduckgo.com/?q=2*1.09868411347*0.455089860562&ia=calculator

https://duckduckgo.com/?q=(+1.09868411347+%5E+2+)+-+(+0.455089860562+%5E+2+)&ia=calculator

Pure imaginary solutions for M and N

Flip Flopped Solutions for the absolute values of M and N compared to real solutions

These have the same absolute values as before except M and N are switched and M and N are pure imaginary numbers instead of real numbers.  Only two out of these four solutions work based on choosing the sign correctly.  I recommend simply solving for the real solutions for M and N because figuring out the proper sign may be easier for most people that way.

M = +- 0.455089860562 i

N = +- 1.09868411347 i

https://duckduckgo.com/?q=(+(+4+*+1+-+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5&ia=calculator

https://duckduckgo.com/?q=0.5+%2F+(+(+(+4+*+1+-+(+16+%2B+16+)+%5E+0.5+)+%2F+8+)+%5E+0.5+)&ia=calculator

Copyright Carl Janssen 2022 March 18

Skills, majors and careers to prepare you for medical school to become a doctor, physical therapist or pharmacist

The entry requirements for different career fields and college degrees may vary by time and region and the material in this article may not be correct or may no longer be correct at a future date 

Normally to attend college to get a doctoral degree to become a medical doctor or physical therapist or pharmacist one can get a Bachelors degree in anything as long as someone completes a minimum number of courses and gets a high enough grade point average.  

This however may not be best practice because there are certain hands on skills related to medicine someone can practice before completing their bachelors degree which may make them better equipped than other people who make it into the same medical school without learning these skills first. 

It is very common for people to major in biology or chemistry to try to become a medical doctor but I would suggest they could consider earning a bachelors degree where they actually learn how to practice medical skills first and use that with some biology and chemistry electives to get into medical school instead.  One could also get associate degrees or certificates that teach skills related to practicing medicine at a technical college and transfer those credits towards a bachelors degree to get in upon completing a Bachelors degree.  

Human dissection maybe traumatic for some individuals but maybe required as part of the coursework for Physical Therapy School or Medical School, more over someone who can not deal with human dissection of a dead individual might have even more difficulties as a surgeon in terms of either dexterity or psychologically handling gore and or fear of blood and germs involved with human dissection.  Some people physically or psychologically respond with pain upon seeing other people injured and this may also be a barrier to completing human dissection or doing surgery.  There is a high correlation between a career in surgery and the type of psychopathy or sociopathy defined as not experiencing pain upon witnessing other people experience pain, this type of psychopathy or sociopathy is not necessarily bad unless someone also enjoys other people being harmed or wants to harm other people.

Butcher - Better prepares someone for surgery and medical school human dissection labs.  Not all butchers kill animals some butchers simply dissect animals other people have killed, other butchers kill animals.  I am not endorsing killing animals by posting this.  Good preparation for becoming a Physical Therapist or Medical Doctor

Taxidermist - Better prepares someone for surgery and medical school human dissection labs.   Good preparation for becoming a Physical Therapist or Medical Doctor

Mortician - Better prepares someone for surgery and medical school human dissection labs.   Good preparation for becoming a Physical Therapist or Medical Doctor

Sewing - Practicing this skill as a hobby or career will make you better equipped to do surgery, not all medical students have the property dexterity to stitch patients and practicing sewing may help.   Good preparation for becoming a Medical Doctor

Nursing - You can get a nursing degree before going to medical school or there are some types of nursing that do not require a degree.   Good preparation for becoming a Physical Therapist, Pharmacist or Medical Doctor

Dietitian - A dietitian bachelors degree can meet the requirements to go to medical school if the right courses are taken.  Good preparation for becoming a Physical Therapist, Pharmacist or Medical Doctor

Athletic training submajor for a Kinesiology Bachelors degree - Some schools offer multiple Kinesiology Bachelor degree sub majors.  Although any major can get you in with the correct courses not all sub majors provide hands on skills.  Students who choose the correct sub major which may specifically be athletic training in the case of some Universities may be taught how to assess patients for certain injuries as a hands on skill where as if they choose the wrong submjaor for that specific university no hands on skills will be taught and they will simply only learn academic course material related to exercise.  Some of these skills would be the same skills in some cases as some of the same hands on assessment skills taught to complete a physical therapy doctoral degree in graduate school except that someone with this sub major can learn some of these skills ahead of time without completing a Bachelors degree first.  Good preparation for becoming a Physical Therapist or Medical Doctor 

Anatomy -  Anatomy classes often have been required to attend physical therapy graduate school but not medical school to become a medical doctor.  If I had to guess most physical therapists know skeletal muscle anatomy better than general practitioner medical doctors, but medical doctor specialists usually know anatomy in their specialty area better than physical therapists.  Even though Anatomy classes might not be required to enter medical school, taking anatomy in advance might make you better prepared.  Good preparation for becoming a Physical Therapist or Medical Doctor

Map Reading - Practicing using maps as a skill may help you learn anatomy easier.

Memory - There are memory championships and many world memory Champions wrote books on how to improve your memory but I recommend understanding the methods in the book "Use your perfect memory" by Tony Buzan.  Developing a good memory is good preparation for becoming a Physical Therapist, Pharmacist or Medical Doctor

Pharmacist Technician - You can get a pharmacist technician career or take pharmacy technician coursework without having to complete a bachelors degree first, to help you understand the pharmacological aspects of medicine before going to graduate school to become a medical doctor, physical therapist or pharmacist.  Good preparation for becoming a Physical Therapist, Pharmacist or Medical Doctor

Dental Hygenist - You can get a undergraduate degree and career as a dental hygenist before going to grafuate school to become a dentist

Physical Therapy/Therapist Assistant - You can sometimes take classes to become a physical therapist assistant at a technical college and transfer credits towards an associate degree rowards the bachelors degree required to go into medical school or Physical Therapy School.  Good preparation for becoming a Physical Therapist

Personal Trainer - This career can help you gain experience working with exercise hands on before enrolling in Physical Therapy school.  Do not call yourself a PT on your Physical Therapist college admission application but use the full term personal trainer if you are a personal trainer but not a physical therapist.  Some Physical Therapists get extremely offended by personal trainers calling themselves PT.  I have heard some people claim Physical Therapists have sued Personal Trainers for using the initials PT which Physical Therapists use claiming the initials PT are trademarked.   Personal Training is not a college degree but a career.  Good preparation for becoming a Physical Therapist

Massage Therapist / Massage Therapy School - Good preparation for becoming a Physical Therapist or Medical Doctor

Fire Fighter - Fire fighters often learn certain hands on medical skills 

Emergency Medical Technician

Paramedic

Bio-engineer Bachelors - You can use a bio engineering degree to transfer to medical school if you take any extra courses in Biochemistry or biology that maybe required to enter the school.  This maybe difficult however as engineering degrees usually do not have enough elective courses left over to meet course requirements for anything else in addition to the minimum degree requirements without attending college longer than it would take to complete an engineering degree alone.

Reading books ahead of time - There is nothing to stop you from reading textbooks like "Harrison's principles of internal medicine" before you ever attend medical school or even if you never attend medical school

Practicing math and or statistics ahead of time - You can get math and or statistics textbooks with practice problems and do those problems before taking any math courses at college that will later be required to get into other courses required to attend a medical school or required in and of them self to attend a medical school or that are required for a Bachelors degree major you are planning on working on to get into medical school.  Testing out of some math courses may save you a lot of time and money not taking as many classes at college.  And when you can not test out of a class but must take it the class will be much easier if you practiced the math ahead of time.  Unfortunately a large proportion of medical research is done with an emphasis on statistics  which is not predictable but "random" instead of math such as Algebra and Calculus without statistics that is better suited to real science which is testable, observable and predictable.  So someone in any medical field should be familiar with statistics enough in order to use valid reasoning and the correct choice of premises to show the flaws in any statistical study.

Advanced Placement courses and other means of testing out of college classes without taking them - This will give you more time to take elective college classes that teach hands on skills instead or to simply save time and money skipping classes covering material you already learned.

Actually learn concepts do not just memorize answers to complete tests - There was a Physics professor who was very angry that students wanting to become medical doctors who viewed a physics requirement as a barrier to admission into medical school would try to memorize test answers.  I am not talking about students who try to find out what problems were on a test ahead of time so they can know how to solve them, but actually memorizing a number with some units that was an answer to a past test question in a previous semester.  One of the teaching assistants might have tried to emphasize the level of stupidity involved by saying that they were not memorizing problems which would have been bad enough but instead memorizing answers to problems which was worse.  Some people could actually know how to solve a problem but cheat by trying to find out what the problems are going to be on the test before they are supposed to know and solve the specific problems on the test in advance and still put down work for solving the problem while doing the test pretending as if they never saw it before and were solving it live but these students just tried to memorize the answers whether it was as an attempt to cheat or it was where they genuinely thought it was not cheating and they simply thought the problem would occur again and the way to study was to memorize what the answer was to that problem and put it down again I do not know.  Someone can memorize problems they had done in the past without cheating in some circumstances such as memorizing the numbers for a practice problem assigned by the professor before the test and memorizing the steps to solve that practice problem and memorizing the solutions to that problem and understanding how the solution would change if the numbers change.  It might have been ok under certain circumstances of the non cheating variety if they knew how to answer the question if it was the same problem but with different numbers but if one number in the initial problem was changed the entire answer would be changed and they would not know how to get it by a methodology of problem solving, they would simply put down the number they memorized from what the correct answer was in a previous semester.  This might not be an exact description of the situation the Professor was experiencing with these students but it is a close enough description to convey the point I am trying to express.  Do not be like those bad students learn the methodology for how to get a proper answer instead of just memorizing answers.  Memorizing is important if used correctly but should not be misused that way.

Copyright Carl Janssen 2022 

https://duckduckgo.com/?q=surgery+sewing+and+dexterity&ia=web