Average Calculator
Separated by commas, spaces or new lines.
- 8.5714286
- Mean
- 7
- Median
- 7
- Mode
- 12
- Range
- 7
- Count
- 60
- Sum
- 3.90969
- SD (sample)
- 3.61967
- SD (population)
The average calculator reports the mean, median and mode together, because "average" means all three and they disagree whenever a distribution is skewed. It also gives both standard deviations, the sample and population forms differ, and using the wrong one is a common error.
How it works
- Mean: the sum divided by the count. Sensitive to outliers.
- Median: the middle value once sorted, or the average of the two middle values for an even count. Unaffected by outliers.
- Mode: the most frequent value. If every value appears once, there is no mode, the tool says so rather than listing everything.
- Range: largest minus smallest.
sample SD divides by (n - 1); population SD divides by n
- n - 1
- Bessel’s correction, used when your numbers are a sample and you are estimating a wider population
- n
- used when your numbers are the entire population you care about
The correction exists because a sample tends to cluster more tightly than the population it came from, so dividing by n underestimates the spread. With large samples the two figures converge; with small ones the gap is substantial.
Examples
A set with a repeated value
Numbers
12, 7, 3, 7, 9, 15, 7
Result
Mean 8.571 · Median 7 · Mode 7 · Range 12 · Sample SD 3.910 · Population SD 3.620
Seven appears three times, so it is the mode and also the median. The two standard deviations differ by about 8 percent, because dividing by 6 rather than 7 inflates the sample figure at this size.
Where mean and median disagree
Numbers
20, 22, 24, 25, 200
Result
Mean 58.2 · Median 24
One outlier pulls the mean far above every value except itself. The median of 24 describes the data far better, which is why incomes are usually reported as medians.
No mode at all
Numbers
1, 2, 3, 4, 5
Result
Mode: none
Every value appears once, so no value is more frequent than the others. Reporting all five as modes, which some calculators do, is wrong.
Frequently asked questions
Which standard deviation should I use?
The sample form, dividing by n − 1, if your numbers are a sample and you want to say something about a wider population. The population form, dividing by n, if your numbers are the complete set you care about. Most real analysis uses the sample form, and spreadsheets default to it as STDEV.S.
When should I report the median instead of the mean?
Whenever the data is skewed or contains outliers. Five values of around 20 plus one of 200 produce a mean of 58, which describes none of them. This is exactly why house prices and salaries are reported as medians.
Why does it say there is no mode?
Because no value repeats. A set where every value appears once has no mode, and listing all of them as modes, which several calculators do, is mathematically wrong. A set can also have two or more modes, and all of them are shown.
What separators can I use?
Commas, spaces, semicolons and new lines, in any combination. Anything that is not a number is skipped, so pasting a spreadsheet column with a header still works.