CALCSLIP

Average and standard deviation calculator

Mean, median, mode, range and standard deviation for any list of numbers.

Separate with commas, spaces or new lines. Do not use thousands separators.

Result

Mean (average)
80.6
Median
83
Mode
85 (×3)
Range
28
Standard deviation (sample)
9.021
Standard deviation (population)
8.558
Variance (sample)
81.3778
Count
10
Sum
806
Minimum
66
Maximum
94

How this was calculated

  1. Mean

    x̄ = Σx ÷ n

    806 ÷ 10 = 80.6

  2. Median

    Average of the two middle values

    Sorted: 66, 70, 72, 78, 81, 85, 85, 85, 90, 94 → 83

  3. Sample standard deviation

    s = √( Σ(x − x̄)² ÷ (n − 1) )

    √(732.4 ÷ 9) = 9.021

What this calculation assumes

  • Use the sample standard deviation when your numbers are a sample from a larger group (most surveys and experiments) and the population figure when you have every member of the group.
  • When several values tie for most frequent, all of them are modes.

Mean, median and mode

The mean is the sum divided by the count. The median is the middle value once the numbers are sorted, or the average of the two middle values when the count is even. The mode is the value that appears most often; a list can have several modes or none.

For the scores 72, 85, 90, 66, 85, 78, 94, 85, 70 and 81, the mean is 80.6, the median is 83 and the mode is 85. When a few extreme values pull the mean away from the middle, as with incomes or house prices, the median describes a typical value better.

Standard deviation

s = √( Σ(x − x̄)² ÷ (n − 1) ) σ = √( Σ(x − μ)² ÷ n )

Standard deviation measures how far values typically sit from the mean, in the same units as the data. The sample version divides by n − 1 (Bessel's correction) because a sample's own mean sits closer to its values than the true population mean would, which would otherwise understate the spread.

For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean and about 95% within two.

Questions people ask

Should I use sample or population standard deviation?

Population when your numbers are the whole group you care about (every student in one class). Sample when they stand for a larger group (a survey of 500 voters). Spreadsheet functions STDEV.S and STDEV.P are the sample and population versions.

Can I paste from Excel or Google Sheets?

Yes. Copy a column or a row and paste it into the box. Numbers can be separated by new lines, spaces, tabs or commas.

What is variance?

The square of the standard deviation. It is used inside many statistical tests, but the standard deviation is easier to read because it has the same units as the data.