How to Calculate ANOVA by Hand

by Damon Verial Google

After collecting data from two groups, researchers often use ANOVA (analysis of variance) to contrast differences between the groups. Before beginning the ANOVA calculations, you need to have the basic summary statistics of the data you collected. Though you can easily compute ANOVA with statistical software, calculating ANOVA by hand will allow you to understand the individual steps involved and how they contribute to showing the differences between groups.

List your summary statistics for easy reference. Your summary statistics include the individual data points for the first group, "x;" the number of data points for the second group, "y;" and the number of data points for each group, "n." Make these summary statistics easy to see and reference in preparation for the ANOVA calculation.

Add the data points for each group. Add up all of the data points for the first group, and call the result "SX." Do the same for the data points in the second group and call the result "SY."

Calculate the mean correction. Use the formula C = (SX + SY)^2 / (2n).

Calculate the sum of squares between groups, SSB. Use the formula SSB = [(SX^2 + SY^2) / n] -- C.

Square all of the data points. Write these results in an easy-to-reference way.

Sum all of the squared data points. Do not separate the sums for the two groups; the final result should be a single sum. Call this final sum D.

Calculate the sum of squares total, SST. Use the formula SST = D -- C.

Calculate the sum of squares within groups, SSW. Use the formula SST -- SSB.

Compute the degrees of freedom for between and within groups, "dfb" and "dfw," respectively. For between groups, dfb = 1, and for within groups dfw = 2n -- 2.

Compute the mean square for between groups, MSB. Use the formula MSB = SSB / dfb.

Compute the mean square for within groups, MSW. Use the formula, MSW = SSW / dfw.

Calculate the final statistic, the F statistic. Use the formula, F = MSB / MSW.

Things You Will Need

  • A data set


  • "Experimental Designs Using ANOVA"; Barbara Tabachnick and Linda Fidell; 2007

About the Author

Having obtained a Master of Science in psychology in East Asia, Damon Verial has been applying his knowledge to related topics since 2010. Having written professionally since 2001, he has been featured in financial publications such as SafeHaven and the McMillian Portfolio. He also runs a financial newsletter at Stock Barometer.