Analysing a single scale variable
Part 3b: Effect size
In the previous part we found out that the average age in the population is significantly different from 50, but is it a big difference based on the sample? The problem is that if you have a very large sample size, almost every statistical test will be significant, even if it is just a very small difference.
To determine the size of the difference, we can use a so-called effect size measure and the one that goes well with the one-sample t-test is known as Cohen's d (Cohen, 1988). The calculation is fairly easy, it is the difference between the sample mean and the expected population mean (the test value or hypothesized mean), divided by the standard deviation. Note that there is also a Cohen's ds, but that is used for an independent samples t-test.
In the example the sample mean was 48.19, the expected age in the population was 50, so the difference would be 48.19 - 50 = -1.81. The standard deviation was 17.69, so Cohen's d becomes: d = -1.81 / 17.69 = 0.10. Cohen suggests to multiply this with the square root of 2, to get a generic d, for which he gives a rule of thumb for the classification as shown in the Table 1 (Cohen, 1988, p. 40).
0.00 < 0.20
0.20 < 0.50
0.50 < 0.80
0.80 or more
The 0.10 from the example would then indicate a negligible effect size. We can add this to our report:
The mean age of customers was 48.19 years, 95% CI [47.4, 49.0]. The claim that the average age is 50 years old can be rejected, t(1968) = -4.53, p < .001, with a negligible effect size (d = .10).
Click here to see how to determine Cohen's D with SPSS, R (studio), Excel, Python, an Online calculator, or Manually.
This has become available in SPSS 27. As mentioned on this site, there is no option in earlier versions of SPSS to determine Cohen's D, but it can easily be calculated using by using the output from the t-test and entering the results below in the 'Online calculator' option.
with R (Studio)
Enter the sample mean, the hypothesized population mean, and the sample standard deviation:
Manually (formula and example)
The formula for Cohen's d for a one-sample t-test is:
Where x̄ is the sample mean, μH0 the expected mean in the population (the mean according to the null hypothesis), and s the sample standard deviation.
The formula for the sample standard deviation is:
In this formula xi is the i-th score, x̄ is the sample mean, and n is the sample size.
The sample mean can be calculated using:
Example (different example)
We are given the ages of five students, and have an hypothesized population mean of 24. The ages of the students are:
Since there are five students, we can also set n = 5, and the hypothesized population of 24 gives
For the standard deviation, we first need to determine the sample mean:
Then we can determine the standard deviation:
Now for Cohen's d:
Remember that this manual calculation was done using a different example.
We've done enough analysing, so let's combine all the reports bit on the next section.
Single scale variable