Descriptive Statistics Calculator
Calculate mean, standard deviation, variance, range, min, and max for 8 data values. Shows both population and sample statistics.
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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: 4.04
Sample std dev: 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()).
Frequently Asked Questions (FAQ)
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by N; sample standard deviation (s) divides by N − 1. Use σ when your dataset is the entire population. Use s when it is a sample — the N − 1 denominator (Bessel's correction) makes s an unbiased estimator of the true population standard deviation.
How do you calculate the median of a dataset?
Sort the values in ascending order. If N is odd, the median is the middle value. If N is even (as with 8 values), the median is the average of the two middle values: (x₄ + x₅) ÷ 2 after sorting. This calculator does not compute median because it requires sorting — use MEDIAN() in Excel or Google Sheets instead.
What does standard deviation tell you about your data?
Standard deviation measures how spread out values are around the mean. A small value means the data clusters tightly; a large value means it is widely spread. For a normal distribution, roughly 68% of values fall within one standard deviation of the mean and 95% within two.
When should I use variance instead of standard deviation?
Standard deviation (in the same units as the data) is more intuitive for reporting. Variance is used in mathematical derivations and ANOVA — because variances from independent datasets add together, whereas standard deviations do not.
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