Why Your Variables Need to Vary

Quantitative Methodology

A very basic (but often overlooked) consideration in selecting variables for a statistical analysis is their variance. At bare minimum, your data need to contain at least two unique values for a variable in order to analyze it. For example, you cannot conduct a statistical analysis of sex if there are only females in your sample. This may seem pretty obvious, but I have seen a surprising number of students propose things such as sex comparisons on all female (or all male) samples. This is an extreme example, but it highlights the importance of variance in your data. If your variable does not vary, it is by definition no longer a variable (instead, it would be considered a constant).

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Variance in your data can have an important impact on the statistical power of your analysis. Let’s use an independent samples t-test as an example. Say that you want to compare the standardized test scores of students from two classrooms (Class A and Class B). Ideally, each classroom would have the same number of students (e.g., 50 in Class A and 50 in class B). The independent samples t-test will have the maximum statistical power when group sizes are exactly equal. If your group sizes are unequal (e.g., 75 in Class A and 25 in Class B), the analysis will have less power, meaning it is less likely that you will find a significant result. In other words, if you reduce the variance of your independent variable (classroom membership), your statistical power is reduced. A few hypothetical power calculations in G*Power show just how much your power can be reduced by unequal group sizes. The power values below were calculated for a two-tailed independent samples t-test with a medium effect size and significance level of .05:

  • Equal groups (Group A = 50, Group B = 50): Power = .70
  • Moderately unequal groups (Group A = 75, Group B = 25): Power = .57
  • Extremely unequal groups (Group A = 90, Group B = 10): Power = .32
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You can see that having extremely unequal group sizes can cut the power of the analysis by more than half, even when the overall sample size remains the same. Now, this does not necessarily mean that your group sizes always need to be equal. This also does not necessarily mean that more variance is always better. Rather, this example simply demonstrates one of the consequences of having too little variance in your data.