$Descriptive Statistics$

Descriptive Statistics

Descriptive statistics of a data set: mean, median, mode, range, quartiles, variance and standard deviation, frequency table and chart.

Numbers separated by spaces, semicolons or new lines; fractions as 1/6, decimals as 2.5.

Test yourself

The descriptive statistics calculator takes a data set and finds the sorted data, mean, median, mode, range, quartiles, variance and standard deviation (population and sample), the coefficient of variation, a frequency table and a chart. The solution is shown step by step.

Main sample characteristics

Characteristic	How to find
Mean x̄	the sum of the numbers divided by their count
Median Me	the middle of the sorted data (for even n, the average of the two middle values)
Mode Mo	the most frequent value
Range R	x_max − x_min
Variance D	Σ(x − x̄)² / n
Sample variance s²	Σ(x − x̄)² / (n − 1)
Standard deviation	σ = √D, s = √s²

Example. 5, 3, 8, 3, 9, 4, 7, 3, 6, 2.
Sorted: 2, 3, 3, 3, 4, 5, 6, 7, 8, 9; x̄ = 50 / 10 = 5; Me = (4 + 5) / 2 = 4.5; Mo = 3; R = 7.
Σ(x − x̄)² = 52, D = 5.2, σ ≈ 2.28.

Population or sample variance?

If the data is the whole population (all students of a class), use D with the divisor n. If it is a sample from a large population used to estimate its spread, use the corrected variance s² with the divisor n − 1 — it gives an unbiased estimate.

Unlike the mean, the median is robust to outliers: for 1, 2, 3, 4, 100 the mean is 22 but the median is 3.