Mean, Median & Standard Deviation

Paste a list of numbers and get the average, the middle value, how spread out they are, the quartiles and any outliers — for a sample or for a whole population.

StatisticsDescriptiveSpread

Inputs

Input examples

Type or paste your numbers, separated by commas, spaces or new lines (use ; between numbers when you write decimal commas). Choose Sample when your data is a selection from a larger group, and Whole population when you have every value.

Data — separated by commas, spaces or new lines

Mean, Median and Spread

The mean is the ordinary average: add everything up and divide by how many values there are. The median is the middle value once the data is sorted. One extreme value pulls the mean towards it but hardly moves the median — which is why average house prices and incomes are usually given as medians.

The standard deviation measures how far the values typically lie from the mean. For data that is roughly bell-shaped, about two thirds of the values lie within one standard deviation of the mean, and about 95% within two.

meansensitive to outliers

medianrobust to outliers

±1 s≈ 68% of bell-shaped data

±2 s≈ 95%

Histogram & Box Plot

number of values per bin

mean

median

outlier

Result

Enter the values above

Your Data, Sorted

Worked Techniques

Sample or Population?

A sample is usually a little less spread out than the group it came from. Dividing by n − 1 instead of n corrects for that. With every value of the group, divide by n.

Example: 2, 4, 4, 4, 5, 5, 7, 9: σ = 2, s = 2.14.

Quartiles and the Box Plot

Q1 and Q3 cut off the lowest and highest quarter of the data; the box spans the middle half. This calculator interpolates between neighbouring sorted values (the same method as Excel’s QUARTILE.INC); textbooks and other programs sometimes use slightly different rules.

Example: 1, 2, 3, 4, 5 → Q1 = 2, Q3 = 4, IQR = 2.

Spotting Outliers

Values far outside the box are flagged as possible outliers: a typing error, a broken measurement, or simply an unusual case worth a closer look.

Example: commute times 12–19 min and one of 45 min: 45 is an outlier.

Sources & Verification

References

D. C. Montgomery & G. C. Runger, Applied Statistics and Probability for Engineers, 6th ed., Wiley, 2013 — Ch. 6 (Descriptive Statistics).