Measures of Dispersion — Range, Variance & Standard Deviation

Measures of dispersion tell you how spread out a data set is. 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, so it returns the spread to the original units.

If you want the fast takeaway, use range for a quick scan, variance for formal statistical work, and standard deviation when you want a spread measure that is easier to interpret.

Range, variance, and standard deviation at a glance

The range is the distance from the minimum to the maximum:

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

It is fast to compute, but it only uses two values. One extreme value can change it a lot.

The variance measures how far values tend to sit from the mean after those distances are squared.

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,

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

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.

The standard deviation is the square root of variance:

$$\sigma = \sqrt{\sigma^2}$$

or, for a sample,

$$s = \sqrt{s^2}$$

Because it is in the original units, standard deviation is usually easier to read than variance.

Worked 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 have the same minimum, the same maximum, and the same mean.

For each set,

$$\text{range} = 8 - 2 = 6$$

and

$$\text{mean} = \frac{25}{5} = 5$$

So range alone says they are equally wide. But the values are arranged differently around the mean.

Set A

The deviations from the mean are

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

Squaring them gives

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

The sum of squared deviations is $18$. If we treat the data as a population,

$$\sigma^2 = \frac{18}{5} = 3.6$$

and

$$\sigma = \sqrt{3.6} \approx 1.90$$

Set B

The deviations from the mean are

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

Squaring them gives

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

The sum of squared deviations is $36$, so

$$\sigma^2 = \frac{36}{5} = 7.2$$

and

$$\sigma = \sqrt{7.2} \approx 2.68$$

Both sets have the same range, but Set B has the larger variance and standard deviation. That is the key idea: range sees only the endpoints, while variance and standard deviation use the whole data set.

Common mistakes with measures of dispersion

One common mistake is assuming that the same range means the same spread. The example above shows why that is false.

Another mistake is treating variance as if it were in the original units. It is not. If the data is in meters, variance is in square meters.

A third mistake is mixing up the population and sample formulas. The correct denominator depends on the situation: use $N$ for a full population and $n-1$ for a sample.

It also helps to remember that variance and standard deviation are sensitive to outliers because large deviations are squared before averaging.

When each measure is useful

Use range when you want a quick first look at how wide the data runs.

Use variance when you need the spread measure inside other statistical methods. Many formulas in probability and statistics are built around variance, even when reports later show standard deviation instead.

Use standard deviation when you want a practical description of spread in the same units as the data. In many classroom and real-world summaries, it is the most readable choice.

Try a similar problem

Make two short data sets with the same mean and the same range, then compare their variance and standard deviation. If you want a next step, try your own version in a solver after you work it out by hand.