Statistics How To

How to Calculate the Least Significant Difference (LSD)

Hypothesis Testing > How to Calculate the Least significant Difference

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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 you reject the null hypothesis as a result of your hypothesis test results. The LSD calculates the smallest significant between two means as if a test had been run on those two means (as opposed to all of the groups together). This enables you to make direct comparisons between two means from two individual groups. Any difference larger than the LSD is considered a significant result.

Least Significant Difference (LSD): Formula

The formula for the least significant difference is:
fishers lsd
Where:
t = critical value from the t-distribution table
MSw = mean square within, obtained from the results of your 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 on Group 1 and Group 2 with the following test results:
lsd

Step 1: Find the t-critical value. The t-critical value for α=0.05, dfw = 36 is 2.028 (I used the TI-83 calculator to find a t-distribution value). Make sure you are using the Within groups DF from your results!

Step 2: Insert the given values, the MSE from your results (I used 26.65 from Within groups on the above table) the t-distribution value from Step 2 into the least significant difference formula:
LSD = 2.028 √ (26.65 * (2/10)) = 4.68.
Put this number aside for a moment,

Step 3: Calculate ybar1-ybar2 from the results. For this example, we get -8.7.

Step 4: Compare Step 2 and Step 3. If |ybar1-ybar2| ≥ LSD1,2 then you can reject the null hypothesis that the means are the same (H01 = μ2). Our value of 8.7 is larger than 4.68 so we can reject the null hypothesis.

That’s it!

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

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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How to Calculate the Least Significant Difference (LSD) was last modified: October 15th, 2017 by Andale

21 thoughts on “How to Calculate the Least Significant Difference (LSD)

  1. Serah Kiano

    I need more examples for clarification if you could simply write it in
    other way around.

    Thankyou!

  2. Andale Post author

    Hello, Serah,
    Could you clarify what you mean by “write it in
    other way around.”? An example would help.
    Thanks!

  3. Andale Post author

    I’d be happy to add another example when I update the article :)
    What did you need clarified?

  4. Niaba Teme

    I need to know how to calculate LSD for combined analysis for varieties in 3 replication over year 2 and 3 location for variety, interaction variety year and location*year. I am more concerned on what to use as denominator under for each LSD calculation and t values. If you do have reference please provide electronic copy.
    Please help.

  5. Andale Post author

    “I am more concerned on what to use as denominator under for each LSD calculation and t values”
    n = number of scores used to calculate the means. I’m not familiar with your data — what did you use to calculate the means in ANOVA??