Computing Statistics - Test for Paired Samples

Sample size

Number of valid samples

The number of valid data samples for both attributes n is displayed.

This is particularly useful when there is a heavily unbalanced distribution of missing data among the two attributes, which might cause the analysis to be based on an unacceptably small sample size.

Student t-test

Student t-value and P-value for Student t-test

This test is used to assess the hypothesis that the difference between each couple under observation is zero.

The corresponding test statistic is based on the ratio between the average of all differences and the corresponding standard error:

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where  is the average of all differences (X_i−Y_i), S_d is the corresponding standard error and n is the number of couples. μ₀ is the expected value under the null hypothesis and is set to 0. Under the hypothesis of Normal distribution of the differences X_d follows a Student t distribution with n−1 degrees of freedom.

Wilcoxon and Mann-Whitney test

• W-value for Wilcoxon test 

• P-value for Wilcoxon test 

• Number of unequal pairs

This test is generally used instead of the Student t test when X_d is not Normally distributed.

The test is distribution-free (“non-parametric”) being based on the ranks of X_d observed distribution. Ranks are the progressive numbers of the ordered values of a variable:

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where N_r is the number of couples whose difference is not zero, and R_i is the rank of the i-th couple. Inference (i.e. the p-value calculation) is made by the following test:

z asymptotically follows a standard Normal distribution with standard error σ_W given by the following equation: