Last updated: 2026-05-18

What descriptive statistics are

Descriptive statistics are summary measures that describe the center, spread, and extent of a dataset without making inferences beyond the data at hand. This calculator takes 8 numerical values and returns the core measures: the arithmetic mean, population and sample variance, population and sample standard deviation, the minimum, the maximum, and the range. Together they answer two basic questions about any quantitative dataset — where it is centered, and how widely it is spread.

The four measures

Mean (arithmetic average): the sum of all values divided by the count. The mean is the balance point of the dataset and is sensitive to outliers — a single extreme value shifts it noticeably.

Variance: the average squared distance from the mean. Squaring the deviations keeps every term positive and gives larger deviations disproportionate weight. It comes in two forms — population variance (σ², dividing by N) and sample variance (s², dividing by N − 1).

Standard deviation: the square root of variance, expressed in the original units of the data (meters, dollars, seconds). It is the most commonly reported measure of spread because it shares the data's units.

Range: the maximum minus the minimum. It is the simplest spread measure but depends on only two values, which makes it highly sensitive to outliers.

Population variance versus sample variance

The choice between dividing by N and dividing by N − 1 is the distinction that separates the two variance formulas.

Population variance (σ², divide by N) applies when the 8 values are the entire population — there is no larger group from which they were drawn. An example is the annual rainfall totals for exactly the 8 cities under study.

Sample variance (s², divide by N − 1) applies when the 8 values are a random sample drawn from a larger population, and the goal is to estimate the true population variance. The N − 1 denominator is Bessel's correction, which makes s² an unbiased estimator. Without it, the sample variance would systematically underestimate the true variance.

The reason is that a sample tends to cluster closer to its own mean than to the true population mean, so the raw sum of squared deviations divided by N is too small on average. Dividing by N − 1 compensates for this bias.

As a guide: data points collected from a much larger group (survey responses, measurements, test scores) call for the sample standard deviation, while a set of 8 values that defines the entire universe of interest calls for the population standard deviation.

Worked example — exam scores

Scores: 12, 15, 11, 19, 14, 22, 9, 17

Mean: (12 + 15 + 11 + 19 + 14 + 22 + 9 + 17) ÷ 8 = 119 ÷ 8 = 14.875

Sum of squared deviations (SS):

• (12 − 14.875)² = 8.27
• (15 − 14.875)² = 0.02
• (11 − 14.875)² = 15.02
• (19 − 14.875)² = 17.02
• (14 − 14.875)² = 0.77
• (22 − 14.875)² = 50.77
• (9 − 14.875)² = 34.52
• (17 − 14.875)² = 4.52

SS = 130.875

Population std dev: $\sigma = \sqrt{130.875 \div 8} = \sqrt{16.36} \approx$ 4.04

Sample std dev: $s = \sqrt{130.875 \div 7} = \sqrt{18.70} \approx$ 4.32

The score of 22 sits well above the rest of the data and accounts for about 39% of the total SS on its own.

Why the median is not computed

The median is the middle value after ordering the data from smallest to largest. For 8 values, an even count, the median is the average of the 4th and 5th values once sorted. Computing it requires sorting, and this calculator's engine evaluates formulas algebraically rather than ordering arbitrary user inputs. A spreadsheet covers this case directly (Excel: MEDIAN(), Google Sheets: MEDIAN()).