Statistics How To

How to Calculate the Least Significant Difference (LSD)

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How to Calculate the Least Significant Difference (LSD): Overview

When you run an ANOVA (Analysis of Variance) test and get a significant result, that means at least one of the groups tested differs from the other groups. However, you can’t tell from the ANOVA test which group differs. In order to address this, Fisher developed the least significant difference test in 1935, which is only used when the ANOVA F omnibus returns a significant result. The LSD calculates the smallest significant between two means as if a t-test had been run on those two means. This enables you to make direct comparisons between two means from two groups. Any difference larger than the LSD is considered a significant result.

How to Calculate the Least Significant Difference (LSD): Formula

The formula for the least significant difference is:

How to Calculate the Least Significant Difference (LSD)

The least significant difference formula.


Where:
t = critical value from the t-distribution table with a df from the denominator of the f statistic
MSE = mean square error, obtained from the ANOVA test
n = number of scores used to calculate the means

How to Calculate the Least Significant Difference (LSD): Sample Problem

Sample problem: Calculate the Least Significant Difference for the difference between two means with the following statistics:
Significance level (α)=0.05, two-tailed test, df = 32, an MSE of 0.975 and 10 scores per mean.

Step 1: Run an ANOVA test (i.e. a two way ANOVA in Excel). This is a prerequisite for calculating the LSD (in fact, if you don’t run an ANOVA test, the LSD will make no sense!). You’ll need the Mean Square from the test (circled below) in Step 3.

Step 2: Find the t-critical value in the t-table. The t-critical value for α=0.05, two-tailed test, df = 32 is 2.0369 (I used the TI-83 calculator to find a t-distribution value).

Step 3: Insert the given values, the MSE from Step 1 and the t-distribution value from Step 2 into the least significant difference formula:
fisher's least significant difference example

That’s it!

Tip: The LSD will only make sense if you have a significant result from ANOVA. Therefore, you shouldn’t run the test if you do not get a significant result from ANOVA.

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