Two-Sample Paired t Test

Statistics > Basic Statistics > 2-Sample Paired t


Tests the null hypothesis that the population mean of the paired differences of two samples is zero.

Paired-Sample t-Test is used to test if two related samples come from populations with the same mean.

For a paired t-test:

H0: µd = µ0     versus     H1: µd ≠ µ0

where µd is the population mean of the differences and µ0 is the hypothesized mean of the differences.

Pairing involves matching up individuals in two samples so as to minimize their dissimilarity except in the factor under study. Paired samples often occur in pre-test/post-test studies in which subjects are measured before and after an intervention. They also occur in matched-pairs (for example, matching on age and sex), cross-over trials, and sequential observational samples. Paired samples are also called matched samples and dependent samples.


Dialog box items

Samples in one column:
Choose if the sample data are in a single column, differentiated by factor levels in a second column.

oSamples: Enter the columns containing the sample data.

oFactor and Levels: Enter the columns containing the sample factor, and select the levels.

Samples in different columns:
Choose if you have entered raw data in two separate columns.

oFirst Sample: Enter the column containing one sample.

oSecond Sample: Enter the column containing the other sample

Summarized data:
Choose if you have summary values for the sample size, mean, and standard deviation of the difference.


-Sample size: Enter the value for the sample size.

-Sample Mean: Enter the value for the sample mean.

-Sample Std Dev: Enter the value for the sample standard deviation.

The display of outputs of VisualStat.



Data can be entered in one of two ways:

Both samples in a single numeric column with another grouping column (called factor) to identify the population. The grouping column may be categorical, numeric or text.

Each sample in a separate numeric column.

Each row contains the paired measurements for an observation. Paired observations where one of values is missing are ignored.



Displays a histogram, a histogram with a normal curve, and a boxplot.

Choose to display a histogram for each variable

Histogram with Normal Curve:
Choose to display a histogram with a normal curve for each variable

Boxplot of data:
Choose to display a boxplot for each variable

Options: Choose the options you want.

oExclude missing values: Check to excludes rows that have missing values.

oInverted Bar: Check to reverse the axes.



Alternative Hypothesis:
Enter Two-sided, Upper One-sided, or Lower One-sided. If you choose an One-sided hypothesis test, an upper or lower confidence bound will be constructed, respectively, rather than a confidence interval.

Confidence Level:
Enter the level of confidence desired. Enter any number between 0 and 100. Entering 90 will result in a 90% confidence interval. The default is 95%.



Suppose we measure the thickness of plaque (mm) in the carotid artery of 10 randomly selected patients with mild atherosclerotic disease. Two measurements are taken, thickness before treatment with Vitamin E (baseline) and after two years of taking Vitamin E daily.

1.Open the DataBook compare.vstz

2.Select the sheet plaque

3.Choose the tab Statistics, the group Basic Statistics and the command 2-Sample Paired t

4.Select group Samples in different columns

5.In First Sample, select Before. In Second Sample, select After.

6.Click Options page. In Alternative Hypothesis, select Upper One-sided.

7.Click OK


Report window output


Two-Sample Paired t Test


Paired T-Test of mean difference = 0 vs > 0

alternative hypothesis: true mean of differences is greater than 0

                N    Mean   StDev  SE Mean    Corr   95% Conf Interval       t  df  p-value  Alt Hypothesis

Before - After  10  0.0450  0.0264   0.0083  0.9350  [0.0297; Infinity]  5.4000   9   0.0002          Accept




Interpreting the results




Null Hypothesis: Hobefore - µafter = 0

(One-sided) Alternative Research Hypotheses: H1before - µafter > 0.

Test statistics: We can use the sample summary statistics to compute the t-Statistic: t = 5.4

p-value = P(T(df = 9) > t = 5.4) = 0.0002 for this (one-sided) test.

Therefore, we can reject the null hypothesis at a = 0.05!


These data show that the true mean thickness of plaque after two years of treatment with Vitamin E is statistically significantly different than before the treatment (p =0.0002). In other words, vitamin E appears to be an effective in changing carotid artery plaque after treatment.




See Also:

Paired-Samples Sign Test | Wilcoxon Signed Rank Test for Paired-Samples | t Distribution | Report | Numeric Formats

Web Resource: Weisstein, Eric W., Wolfram MathWorld. | Probability and statistics EBook