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Paired TTest
In Statistics, a ttest can be represented as a statistical hypothesis test where the test statistic supports a student’s t distribution if the null hypothesis is established. In Paired TTest, they compare the means of two groups of observations. The observations must be randomly assigned to each of the two groups so that the difference in response seen is due to the treatment and not because of any other factors. If two samples are given, then the observation of one sample can be paired with the observation of the other sample, by using the paired ttest.
Further, this test can be used in making observations on the same sample before and after an event. Now, let us discuss what is paired ttest, its formula, table, and the procedure to perform the paired ttest in detail.
Paired TTest Definition
The paired ttest gives a hypothesis examination of the difference between population means for a set of random samples whose variations are almost normally distributed. Subjects are often tested in a beforeafter situation or with subjects as alike as possible. The paired ttest is a test that the differences between the two observations are zero.
Let us assume two paired sets, such as Xi and Yi for i = 1, 2, …, n such that their paired difference is independent which is identically and normally distributed. Then the paired ttest concludes whether they notably vary from each other.
Paired TTest Formula
Paired Ttest is a test that is based on the differences between the values of a single pair, that is one deducted from the other. In the formula for a paired ttest, this difference is notated as d. The formula of the paired ttest is defined as the sum of the differences of each pair divided by the square root of n times the sum of the differences squared minus the sum of the squared differences, overall n1.
The formula for the paired ttest is given by
\(t = \frac{\sum d}{\sqrt{\frac{n(\sum d^{2})(\sum d)^{2}}{n1}}}\)
Where, Σd is the sum of the differences.
Paired TTest Table
Paired Ttest table enables the tvalue from a ttest to be converted to a statement about significance. The table is given below:
\(\begin{array}{l}
\text { Paired T Test Table }\\
\begin{array}{ccccccc}
\hline {\text { Two Tailed Significance }} \\
\hline \begin{array}{c}
\text { Degrees of } \\
\text { fredom }(n1)
\end{array} & \alpha=0.20 & 0.10 & 0.05 & 0.02 & 0.01 & 0.002 \\
\hline & & & & & & \\
\hline 1 & 3.078 & 6.314 & 12.706 & 31.821 & 63.657 & 318.300 \\
\hline 2 & 1.886 & 2.920 & 4.303 & 6.965 & 9.925 & 22.327 \\
\hline 3 & 1.638 & 2.353 & 3.182 & 4.541 & 5.841 & 10.214 \\
\hline 4 & 1.533 & 2.132 & 2.776 & 3.747 & 4.604 & 7.173 \\
\hline 5 & 1.476 & 2.015 & 2.571 & 3.305 & 4.032 & 5.893 \\
\hline 6 & 1.440 & 1.943 & 2.447 & 3.143 & 3.707 & 5.208 \\
\hline 7 & 1.415 & 1.895 & 2.365 & 2.998 & 3.499 & 4.785 \\
\hline 8 & 1.397 & 1.860 & 2.306 & 2.896 & 3.355 & 4.501 \\
\hline 9 & 1.383 & 1.833 & 2.262 & 2.821 & 3.250 & 4.297 \\
\hline 10 & 1.372 & 1.812 & 2.228 & 2.764 & 3.169 & 4.144 \\
\hline 11 & 1.363 & 1.796 & 2.201 & 2.718 & 3.106 & 4.025 \\
\hline 12 & 1.356 & 1.782 & 2.179 & 2.681 & 3.055 & 3.930 \\
\hline 13 & 1.350 & 1.771 & 2.160 & 2.650 & 3.012 & 3.852 \\
\hline 14 & 1.345 & 1.761 & 2.145 & 2.624 & 2.977 & 3.787 \\
\hline 15 & 1.341 & 1.753 & 2.131 & 2.602 & 2.947 & 3.733 \\
\hline
\end{array}
\end{array}\)
Paired Vs Unpaired TTest
The similarity between paired and unpaired ttest is that both assume data from the normal distribution.
Characteristics of Unpaired TTest:
 The two groups taken should be independent.
 The sample size of the two groups need not be equal.
 It compares the mean of the data of the two groups.
 95% confidence interval for the mean difference is calculated.
Characteristics of Paired TTest:
 The data is taken from subjects who have been measured twice.
 95% confidence interval is obtained from the difference between the two sets of joined observations.
How to Find the Paired TTest?
Let us take two sets of data that are related to each other, say X and Y with xi ∈ X, yi ∈ Y. where i = 1, 2,……., n. Follow the steps given below to find the paired ttest.
 Assume the null hypothesis that the actual mean difference is zero.
 Determine the difference di = yi – xi between the set of observations.
 Compute the mean difference.
 Calculate the standard error of the mean difference, which is equal to Sd /√n, where n is the total number, and Sd is the standard deviation of the difference.
 Determine the tstatistic value.
 Refer to the Tdistribution table and compare it with the t_{n1} distribution. It gives the pvalue.
Important Notes on Paired TTest
Here are a few important notes on the Paired TTest:
 The data is taken from subjects who have been measured twice.
 95% confidence interval is obtained from the difference between the two sets of joined observations.
Related Topics
Solved Examples on Paired TTest

Example 1:
What conclusion should be made with respect to an experiment when the significance level is 0.068?
Solution:
Since the pvalue of 0.068 is greater than α = 0.05, it would fail to reject the null hypothesis.
Answer: As the value of p < 0.05, the null hypothesis is rejected.

Example 2:
In which of the following cases could you use a pairedsamples ttest?
(a) When comparing the same participant's performance before and after training
(b) When comparing two separate groups of peopleSolution:
A Ttest can be used in making observations on the same sample before and after an event. In option (b) the data does not involve observations before and after an event for the same set of people.
Answer: a. When comparing the same participant's performance before and after training
FAQs on Paired TTest
How Do You Know if Paired or Unpaired TTest?
A paired ttest is designed to compare the means of the same group or item under two separate scenarios. An unpaired ttest compares the means of two independent or unrelated groups. In an unpaired ttest, the variance between groups is assumed to be equal. In a paired ttest, the variance is not assumed to be equal.
Should I Use Paired or Unpaired TTest?
If the data are paired or matched, then you should choose a paired ttest instead. If the pairing is effective in controlling for experimental variability, the paired ttest will be more powerful than the unpaired test.
What Is the Difference Between a Paired TTest and a 2 Sample TTest?
A twosample ttest is used when the data of two samples are statistically independent, while the paired ttest is used when data is in the form of matched pairs. ... To use the twosample ttest, we need to assume that the data from both samples are normally distributed and have the same variances.
Why Would You Use a Paired TTest?
A paired ttest is used when we are interested in the difference between two variables for the same subject. Often the two variables are separated by time. ... Since we are ultimately concerned with the difference between two measures in one sample, the paired ttest reduces to the onesample ttest.
How Do You Interpret a Paired TTest?
Complete the following steps to interpret a paired ttest.
...
 Step 1: Determine a confidence interval for the population mean difference. First, consider the mean difference, and then examine the confidence interval. ...
 Step 2: Determine whether the difference is statistically significant. ...
 Step 3: Check your data for problems.
How Do You Know if Data Is Paired?
Two data sets are "paired" when the following onetoone relationship exists between values in the two data sets.
 Each data set has the same number of data points.
 Each data point in one data set is related to one, and only one, data point in the other data set.
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