Descriptive Statistics › Measures of Spread

While central tendency measures locate the center of a distribution, measures of spread quantify how much data points vary around that center. Two datasets can have identical means but very different spreads, so center and scale should be examined together [5].

Population variance is the mean squared deviation from the population mean. For a random sample, the following corrected sample variance divides the sum of squared deviations by $n-1$:

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

Where:

• $s^2$: Sample variance
• $x_i$: Individual observation
• $\bar{x}$: Sample mean
• $n$: Number of observations

For $n \geq 2$, division by $n-1$ is commonly called Bessel's correction. Under an independent, identically distributed sampling model with finite variance, it makes $s^2$ an unbiased estimator of the population variance; it does not make $s$ an unbiased estimator of the population standard deviation [6]. For independent variables with finite variances, the variance of their sum equals the sum of their variances. Because variance has squared units, such as dollars squared, direct interpretation can be awkward.

Standard deviation addresses this interpretation issue by taking the square root of variance, returning to the original units:

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

Standard deviation expresses dispersion in the data's original units. Its distance interpretation is clearest for distributions that are roughly symmetric and not heavy-tailed. For a normal distribution, about 68% of probability lies within one population standard deviation of the mean, 95% within two, and 99.7% within three [7]. These percentages are approximations for data whose distribution is only approximately normal, not a distribution-free outlier rule.

The range is the simplest measure of spread, calculated as the difference between the maximum and minimum values:

$$\text{Range} = x_{\text{max}} - x_{\text{min}}$$

While easy to compute and interpret, the range uses only two data points and is extremely sensitive to outliers. A single erroneous or extreme value can inflate the range dramatically, making it an unreliable measure of typical spread.

The interquartile range (IQR) is less sensitive to extreme values because it measures the spread of the middle 50% of the data:

$$\text{IQR} = Q_3 - Q_1$$

Where:

• $Q_1$: First quartile (25th percentile)
• $Q_3$: Third quartile (75th percentile)

Because the IQR focuses on the central half of the ordered values, changing an already extreme tail value usually leaves it unchanged. The conventional inner-fence rule flags values below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$ as potential outliers. A flag is a prompt for investigation, not a reason to delete an observation automatically [8].

For example, consider a dataset of daily website visits over a week: 120, 135, 142, 128, 155, 130, 138. We can calculate several descriptive measures of center and spread:

• Mean: (120 + 135 + 142 + 128 + 155 + 130 + 138) / 7 = 135.43
• Median (middle sorted value): 135
• Mode: No unique mode (all values occur once)
• Range: 155 - 120 = 35
• IQR: Q3 - Q1 = 142 - 128 = 14 (exclusive median-of-halves convention, omitting the overall median)
• Standard Deviation: 11.22

The mean and median are close, although that alone does not establish symmetry. The range is 35, the IQR is 14 under the exclusive median-of-halves convention, and the sample standard deviation is 11.22. Whether that spread is large or small depends on the scale and application.