The calculator below implements paired sample t-test (also known as a dependent samples t-test or a t-test for correlated samples). The t-test is also known as Student's t-test, after the pen name of William Sealy Gosset.
Paired samples t-tests typically consist of a sample of matched pairs of similar units or one group of units that has been tested twice (a "repeated measures" t-test).
A typical example of the repeated measures t-test would be where subjects are tested before treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure-lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way, the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random between-patient variation has now been eliminated. Note, however, that an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice.
A paired samples t-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample by using additional variables that were measured along with the variable of interest. The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples. The pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.1
So, the test deals with correlated samples. This means that two sets of measures are arranged in pairs, where each item in one set is somehow linked with a corresponding item in another set. The idea is to replace the difference between the means of two sets with the mean difference between the paired observations. This allows us to remove the extraneous effects of pre-existing individual differences between subjects.
The null hypothesis here assumes that the true mean difference is equal to zero. The two-tailed alternative hypothesis assumes that the mean difference is not equal to zero. The upper-tailed alternative hypothesis assumes that the mean difference is greater than zero. The lower-tailed alternative hypothesis assumes that the mean difference is less than zero.
The test assumptions are:
- the scale of measurement has the properties of an equal-interval scale,
- the values have been randomly drawn from the source population,
- the source population from which the values have been drawn can be reasonably supposed to have a normal distribution.
The procedure for the test is almost the same as the procedure for the Two sample t-Test, with the only difference is that it operates on , which is equal to .
Calculate set of values as
Calculate the mean of the sample as
, where N is number of pairs (or number of values.
Calculate sum of squared deviations of sum of squares as
Estimate the variance of the source population as
Estimate the standard deviation of the sampling distribution as
- Calculate t as
After we get the t-value, we can look up the inverse of CDF of Student's t-distribution with degrees of freedom and estimate the confidence.
For details, I can refer you to the excellent explanation here.