Stats, Stop Making us Cry: Simplifying the T-Test


The main purpose of statistics in psychology is to quantify it as a science. To be a science, you need to be able to back up theories with data. This series of articles are designed to try and help make statistics a little bit easier. This article explores T-Tests; when they are used, what they show, how to report them in APA format, and how to conduct one in SPSS and by hand.


T-Tests are a type of statistical test, used to see if there is a difference between two values (so two participants for example, or two scores by the same participant in differing conditions). It is a parametric test, which means the data it is used on fits two assumptions: its distribution of values is normal, and the data from the two populations used do not significantly differ from each other either.  There are two types of multi sample T-Tests ( there is another single sample t-test, but this article does not cover it), an independent T-Test, and a dependent T-Test. The first type is used to compare two groups of independent participants, the second is used to compare the same participants in two different conditions. An example of when you would use an Independent T-Test would be determining whether music helps or hinders ability on a memory test. One group would do the test with music, and one without, then the independent T-Test would compare the means of both groups to see if there are any significant differences. An example of when you would use a dependent T-Test would be doing the same experiment as above, but having participants do similar tests both with and without music, then comparing the means from the with and without conditions. If your data is non-parametric, you would use a Wilcoxon test instead of an Independent T-Test, and you would use a Mann-Whitney test instead of a Dependent T-Test. These are not covered in this article.

As briefly mentioned above, what T-Tests use in their analysis are the means, also known as μ, for a group of participants. The mean is the sum of all the scores, divided by the amount of people there where in the group. Therefore comparing the two means can compare performance of the two groups. Generally, a significance level if p<0.05 shows a significant difference.

Reporting in APA

Another important aspect of statistics in research is knowing how to report your results in APA format. To do this, we need certain bits of data. We need the t-test value, the significance level, the degrees of freedom (df)( this just means your number of samples minus one) and the mean and standard deviation(SD) for both groups. Firstly, you would explain what the test was, e.g. “an independent/dependent T-Test was run, looking at the effect of the IV on the DV”. You would then give a comparison of the two groups and the descriptive statistics associated with each (the mean and the standard deviation), and state if there was a difference or not. An example would read “the no music group (M= meanvalue, SD = SDvalue) performed significantly better than the music group (M=meanvalue, SD=SDvalue).” Finally, you report the T-Test value and the significance level after the ), so “t(dfvalue)=tvalue, p=signicance level.”

How to Perform in SPSS

Most psychologists do not do these calculations by hand, but instead use SPSS to organise their data, and to run their analyses. First, I will show you how to run an Independent T-Test on SPSS, next I will show you how to run a Dependent T-Test.

Independent T-Test

  1. Input and organise your data, paying particular attention to the group variable that you are running the analysis on (if you need help with this see a more basic SPSS guide).
  2. Choose Analyse -> Compare Means -> Independent Samples T-Test and the box pictured below will appear. T-Test article screenshot 1
  3. For the testing variable, click on your DV (dependent variable) then press the arrow to choose it. For the grouping variable, click the IV/grouping variable, and click the arrow.
  4. You’ll then need to click “define group” and input the group variable values (in this case 1 and 2), see image below for visual help. Click continue and an output will appear. T-Test article screenshot 2

Reading the Output

T-Test article independent output

Above is what the output will look like. The mean and SD for each group can be found in the group statistics. The df can be found under the column headed “df”. The significance level can be found under the column labelled “significance”. And the t-value can be found in the column labelled “t”. If the Levene’s Test is not significant you read from the “equal variance assumed”, if it is you read from the “equal variance not assumed”.

Dependent T-Test

  1. Input and organise your data (if you need help with this see a more basic SPSS guide).
  2. Choose Analyse -> Compare Means – > Paired Sample T-Test and the box pictured below will appear.T-Test article screenshot 3
  3. You will need to put condition one into the variable one box, by clicking it and clicking the arrow, do the same for condition two. Click ok and the output will appear.

Reading the Output

T-Test article dependent output

Above is what the output will look like. The mean and SD for each condition can be found in the Paired Samples statistics box. The df can be found under the column headed “df”. The significance level can be found under the column labelled “significance”. And the t-value can be found in the column labelled “t”.

T-Tests by Hand

The first thing you need to do when you are calculating statistics by hand is to list out all your sample numbers. This gives a better visual of the data you are working with. The easiest way to do this is to put a number on each line, and line them up one above the other.  At the bottom of the list, calculate the sum of the numbers by adding all of the numbers together.

The first thing you will want to calculate is μ. μ stands for “the mean”. This line by line method comes in handy here as it makes it much easier to count the samples you have. Once you have done the division, congratulations! You have found μ. This is an extremely important symbol in statistics because it is contained in almost every formula.

The next thing you need to calculate is SS (the “sum of the squares”). In order to begin this calculation, go back to the top where you started and subtract the μ  you solved for from each number. When you have gotten an answer for each of these, square them all. Keeping them in neat lines aids in your visualization. When you have completed squaring all of your results, add the squared numbers together. The reason that we square the numbers is because they become normally distributed when we subtract the mean. This results in the new numbers adding up to zero. And you cannot accomplished statistical calculations with zero.

Once you have obtained your sum, you now need to calculate your standard deviation. To find this, you get the square root of your SS divided by the number of samples minus 1. The formula for this looks like this:  Image

That may look big and scary, but you just calculated SS and you can just count the numbers for the n!  So you already have all the numbers you need to get your standard deviation, also known as sigma.

Two Sample Independent t-tests

An independent design is known as a between subjects design (a heads up about keywords here). The formula that I was taught is t= (M1-M2)/s(M1-M2)  the numbers by the M’s in reality are subscripts.

This is known as “Welch’s t-test” and it is used to test the hypothesis that the two samples have equal means. M1 is representative of group 1, while M2 is group two. s still represents sigma, or the standard deviation. This is a two tailed test, so when finding your p-values, you must do both sides. The only thing left to do now is to compare your p-values and decide whether or not you reject the null hypothesis!


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