Mean, median, range and standard deviation calculator
Five numbers is enough to show the difference between the sample and population standard deviations, which is where most of the confusion lives.
Mean
14.2
- Median
- 14
- Sample standard deviation Divides by n-1. Use this when your five numbers are a sample.
- 3.114 482 3
- Population standard deviation Divides by n. Use this only when the five numbers are the entire population.
- 2.785 677 655
- Sample variance
- 9.7
- Population variance
- 7.76
- Range
- 8
- Minimum
- 11
- Maximum
- 19
- Standard error of the mean
- 1.392 838 828
- Coefficient of variation
- 21.9330 %
- Sum
- 71
Show the working
- Value 1 = 12
- Value 2 = 15
- Value 3 = 11
- Value 4 = 19
- Value 5 = 14
- Sum = 12+15+11+19+14 = 71
- Mean = 71 / 5 = 14.2
- Minimum = min(12,15,11,19,14) = 11
- Maximum = max(12,15,11,19,14) = 19
- Range = 19 - 11 = 8
- lo_pair = max(min(12,15), min(11,19)) = 12
- hi_pair = min(max(12,15), max(11,19)) = 15
- Median = max(min(14, 12), min(max(14, 12), 15)) = 14
- ss = (12-14.2)^2 + (15-14.2)^2 + (11-14.2)^2 + (19-14.2)^2 + (14-14.2)^2 = 38.8
- Population variance = 38.8 / 5 = 7.76
- Sample variance = 38.8 / 4 = 9.7
- Population standard deviation = sqrt(7.76) = 2.785 68
- Sample standard deviation = sqrt(9.7) = 3.114 48
- Coefficient of variation = 14.2 != 0 ? 3.114 48 / abs(14.2) * 100 : 0 = 21.933
- Standard error of the mean = 3.114 48 / sqrt(5) = 1.392 84
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- mean = sum / n
- sample variance = sum of squared deviations / (n - 1)
- population variance = sum of squared deviations / n
- standard error = standard deviation / sqrt(n)
Bessel's correction — dividing by n-1 rather than n — is what makes the sample variance an unbiased estimate of the population variance.
The median comes from a five-element selection network rather than a sort, which keeps it a single expression.