In this post, we will learn about performing paired samples t test in SPSS. The paired samples t-test is a statistical hypothesis testing procedure used to determine whether the mean differences between two sets of observations are zero. In paired samples t-tests (also known as dependent samples) t-tests, each observation in one set is paired with the corresponding observation in another. In this test means/averages of two related groups are compared. By related we mean that the observations in the two groups are paired or matched in some way.

## Table of Contents

### Points to Remember

The following are points that need to be remembered:

A Paired samples t-test can be used when two measurements are taken from the same individuals/objects/respondents or related units. The paired measurements can be:

**Before and After Comparisons:**A comparison of before and after situations, such as measuring blood pressure before and after taking medication.**Matched Pairs:**Used when comparing the test scores of twins or blood relations.**Repeated Measures:**When measuring a person’s happiness level at different points in time.

A paired samples t-test is also known as a dependent samples t-test, paired samples t-test, or repeated measures t-test.

### Paired Samples t-test Cannot be used

Note that a paired samples t-test can only be used to compare the means for two related (paired) units having a continuous outcome that is normally distributed. This test is not appropriate when

- The data is unpaired
- There are more than two units/ groups
- The continuous outcome is not normally distribution
- The outcome is ordinal or ranked

### Hypothesis for Paired Samples t test

The hypotheses for a paired/ dependent samples t-test can be stated as

$H_0:\mu_d = 0$ (the difference between the mean of pairs is zero (or equal) )

$H_1: \mu_d \ne$ (the difference between the mean of pairs is not zero (or different) )

$H_1: \mu_d < 0$ (upper tailed test)

$H_1: \mu_d > 0$ (lower-tailed test)

The test statistics for a paired samples t-test are as follows

$$t=\frac{\overline{d} }{\frac{s_d}{\sqrt{n}} }$$

where

- $\overline{d}$ is the sample mean of the differences
- $n$ is the sample size
- $s_d$ is the sample standard deviation of the differences

### Performing Paired Samples t test in SPSS

To run a paired samples t test in SPSS, click **Analyze > Compare Means > Paired Samples t-test**.

Paired samples t-test dialog box, the user needs to specify the variables to be used in the analysis. The variables from the left side need to be moved from the paired variables box. A blue button in between both boxes may be used to shift the variables from left to right or right to left side. Note that the variables you specify in paired variables pan need to be in pair form.

In the above dialog box, the following are important points to follow:

**Pair:**The pair row (on the right side pane) represents the number of paired samples t-tests to run. More than one paired samples t-test may be run simultaneously by selecting multiple sets of matched variables.**Variables 1:**The first variable represents the first match group (such as the before situation).**Variables 2:**The second variable represents the second match group (such as the after situation).**Options:**The options button can be used to specify the**confidence interval percentage**and how the analysis will deal with the missing values.

Note that setting the confidence interval percentage does not have any impact on the calculation of the p-value.

### Paired Samples t test Data Example

Consider the following example about 20 students’ academic performance by taking an examination before and after a particular teaching methodology.

Student Number | Marks before Teaching Methodology | Marks After Teaching Methodology |
---|---|---|

1 | 18 | 22 |

2 | 21 | 25 |

3 | 16 | 17 |

4 | 22 | 24 |

5 | 19 | 16 |

6 | 24 | 29 |

7 | 17 | 20 |

8 | 21 | 23 |

9 | 23 | 19 |

10 | 18 | 20 |

### Testing the Assumptions of Paired Samples t-test

Before performing the Paired Samples t-test, it is better to test the assumptions (or requirements) of the paired samples t-test.

- The dependent variable should be continuous (that is measured on interval or ratio level).
- The dependent observations (related samples) should have the same subject/ objects, that is, the subjects in the first group are also in the second group.
- Sampled data should be random from the respective population.
- The differences between the paired values should follow the normal (or approximately) normal distribution
- There should be no outliers in the differences between the two related groups.

Note that when testing the assumptions (such as normality, and outliers detection) related to paired samples t-test, one must use a variable that represents the differences between the paired values, not the original variables themselves.

Also note that when one or more assumptions for a paired samples t-test are not met, you may run the non-parametric test, Wilcoxon Signed Ranks Test.

### Output: Paired Samples T test

The SPSS will result in four tables:

**Paired Samples Statistics**

The paired samples statistics table gives univariate descriptive statistics (such as mean, sample, size, standard deviation, and standard error) for each variable entered as paired variables.**Paired Samples Correlations**

The paired samples correlation table gives the bivariate Person correlation coefficient for each pair of variables entered.**Paired Samples Test**

The paired samples test table gives the hypothesis test results with p-value and confidence interval of difference.**Paired Samples Effect Sizes**

The paired sample Effect sizes tables give Cohen’s d and Hedges’ Correction values with confidence interval

### Interpreting the Paired Samples t test Output

From the “Paired Samples Test” the two-tailed p-value (0.121) is greater than 0.05 (level of significance), which means that the null hypothesis is accepted which means that there is no difference between marks before and after the teaching methodology. It means that improvement in marks is due to chance or random variation marely. The “Paired Samples Correlations” Table shows that the paired variables are correlated/ related to each other as the p-value for Pearson’s Correlation is less than 0.05.

The marks related to before and after teaching methodology are statistically and significantly related to each other, however, the average difference of marks between before and after teaching methodology is not statistically significant. The differences are due to change or random variation.

### How to report the Paired Samples t-test Results

One might report the statistics in the following format: *t*(degrees of freedom) = *t*-value, *p* = significance level.

From the above example, this would be: *t*(9) = -1.714, *p* > 0.05. Due to the averages of the two situations and the direction of the *t*-value, one can conclude that there was a statistically non-significant improvement in marks due to the teaching methodology from 19.9 Â± 2.685 marks to 21.50 Â± 3.922 marks (*p* > 0.05). So, there is no improvement due to the teaching methodology.

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