Index of dispersion (Variance to Mean Ratio)

Measures of Dispersion > Index of dispersion (Variance to Mean Ratio)

What is the Index of dispersion?

The index of dispersion is a measure of dispersion for nominal variables and partially ordered nominal variables. It is usually defined as the ratio of the variance to the mean. As a formula, that’s:
D = σ2 / μ.

More complicated data sets can be calculated with the following formula (Walker, 1999):
index of dispersion 1


  • k = number of categories in the data set (including categories with zero items),
  • N = number of items in the set,
  • f = number of frequencies or ratings,
  • Σf2 = sum of squared frequencies/ratings.

If D = 0, all ratings fall into the same category.
IF D = 1, all ratings are equally divided between k categories.


Question: 100 people what their least favorite high school class was. Their responses were: Math(25), Economics(42), Chemistry(13), Physical Education (8), Religious Studies (13). What is the Index of Dispersion?

Step 1: Find k, the number of categories. For this example, that’s 5 (economics, chemistry, chemistry, P.E., and religious studies).

Step 2: Square the number of items in each category to get f2:

  • Math (25) = 625
  • Economics (42) = 1764
  • Chemistry (13) = 169
  • Physical Education (8) = 64
  • Religious Studies (13) = 169.

Step 3: Add up all of the results in Step 2 to get Σf2:
625 + 1764 + 169 + 64 + 169 = 2791

Step 4: Insert the numbers from steps 1 – 3 into the formula and solve:
index of dispersion 2

Solution: A D of 0.90 indicates there is a very even spread across all categories.

Index of Dispersion vs. Coefficient of Variation

The index of dispersion is very similar to the Coefficient of Variation, but they are not the same:

  • The Coefficient of Variation is the ratio of the standard deviation to the mean. It is always dimensionless (i.e. it is a plain number without a unit) and is scale invariant (in other words, you can make changes to the length, height or other scale factors without it affecting the result).
  • The Index of Dispersion is the ratio of the variance to the mean (hence the alternate name variance to mean ratio). I usually has a dimension (unless being applied to dimensionless counts) and it is not scale invariant.

Walker, J. (1999). Statistics in Criminal Justice: Analysis and Interpretation. Jones & Bartlett Learning.

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