Measures of Dispersion — Range, Variance & Standard Deviation

Measures of dispersion tell you how spread out a data set is, and you compute them whenever a single center value like the mean is not enough to describe the data. The three basic measures are range, variance, and standard deviation. Range uses only the lowest and highest values, variance measures average squared distance from the mean, and standard deviation is the square root of variance, returning the spread to the original units. Use range for a quick scan, variance for formal statistical work, and standard deviation when you want a spread measure that is easy to interpret.

The Procedure, Step by Step

1. Find the range. Identify the minimum and maximum values, then compute $\text{range} = \text{max} - \text{min}$.
2. Find the mean. Add the values and divide by the number of values to locate the center.
3. Compute squared deviations. Subtract the mean from each value and square the result, so positive and negative deviations do not cancel.
4. Get the variance. Average the squared deviations using $N$ for a full population or $n-1$ for a sample.
5. Get the standard deviation. Take the square root of the variance to return to the original units.

The formulas behind those steps:

$$\text{range} = \text{maximum} - \text{minimum}$$

For a full population,

$$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2$$

For a sample used to estimate a larger population,

$$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$$

The standard deviation is the square root of the variance:

$$\sigma = \sqrt{\sigma^2} \qquad \text{or, for a sample,} \qquad s = \sqrt{s^2}$$

Use $N$ only when your data is the entire population you care about; use $n-1$ when your data is a sample from a larger group.

A Full Example: Same Range, Different Spread

Compare these two data sets:

• Set A: $2, 5, 5, 5, 8$
• Set B: $2, 2, 5, 8, 8$

Both share the same minimum, maximum, and mean, so step 1 and step 2 give identical results:

$$\text{range} = 8 - 2 = 6 \qquad \text{mean} = \frac{25}{5} = 5$$

Range alone calls them equally wide. Now run steps 3 through 5 on each.

Set A

Deviations from the mean:

$$-3,\ 0,\ 0,\ 0,\ 3$$

Squared:

$$9,\ 0,\ 0,\ 0,\ 9$$

The sum of squared deviations is $18$. Treating the data as a population,

$$\sigma^2 = \frac{18}{5} = 3.6 \qquad \sigma = \sqrt{3.6} \approx 1.90$$

Set B

Deviations from the mean:

$$-3,\ -3,\ 0,\ 3,\ 3$$

Squared:

$$9,\ 9,\ 0,\ 9,\ 9$$

The sum of squared deviations is $36$, so

$$\sigma^2 = \frac{36}{5} = 7.2 \qquad \sigma = \sqrt{7.2} \approx 2.68$$

Same range, but Set B has the larger variance and standard deviation. That is the key idea the procedure exposes: range sees only the endpoints, while steps 3 to 5 use the whole data set.

Where Each Step Gets Stuck, and How to Check It

• Step 1 vs. spread. Assuming the same range means the same spread is wrong; the example above proves it.
• Step 4 (units). Variance is not in the original units. If the data is in meters, variance is in square meters; this is exactly why step 5 exists.
• Step 4 (denominator). Mixing up the population and sample formulas. Decide whether you have a full population ($N$) or a sample ($n-1$) before dividing.
• Outliers. Variance and standard deviation are sensitive to outliers because large deviations are squared before averaging.

Practice the Procedure

Make two short data sets with the same mean and the same range, then run all five steps and compare their variance and standard deviation, just as Sets A and B above. After working it out by hand, run the same data in a solver to confirm your variance and standard deviation match.

Use range for a quick first look at how wide the data runs, variance when you need the spread measure inside other statistical methods, and standard deviation when you want a practical description of spread in the same units as the data.