The short answer
For the ten numbers 3, 7, 7, 2, 9, 12, 5, 7, 4 and 10:
The mean
- 3 + 7 + 7 + 2 + 9 + 12 + 5 + 7 + 4 + 1066
- 66 ÷ 106.6
The mean is what most people mean by “the average”. It uses every number, which is its strength and its weakness.
The median
Sort the numbers: 2, 3, 4, 5, 7, 7, 7, 9, 10, 12. With ten numbers, the middle falls between the 5th and 6th, which are both 7, so the median is 7. With an odd count, the median is simply the middle number.
The mode
The mode is the most common value. Here 7 appears three times. A list where two values tie has two modes; a list where nothing repeats has no mode. The mode is the only average that works for categories, such as the most popular colour.
The range
The range is the highest number minus the lowest: 12 − 2 = 10. It is quick but depends on just two numbers, so one unusual value changes it completely.
Which average to use
- Best for
- Even data, totals, scores
- Weakness
- Pulled by extremes
- Best for
- Pay, house prices, skewed data
- Weakness
- Ignores how far out values are
- Best for
- Categories, sizes
- Weakness
- May not exist or be unique
Outliers and skewed data
Six salaries: 24,000, 26,000, 27,000, 29,000, 31,000 and 250,000.
| Measure | Value |
|---|---|
| Mean | 64,500 |
| Median | 28,000 |
| Geometric mean | about 39,480 |
Five of the six people earn far less than the mean. This is why the Office for National Statistics reports median household income alongside the mean.
The weighted mean
- 65 × 30 + 72 × 50 + 58 × 206,710
- ÷ (30 + 50 + 20)÷ 100
Grade point averages are weighted means too, with course credits as the weights: see the GPA and grade calculator.
The geometric mean
For growth rates, multiply rather than add. An investment that grows 10%, falls 5% and grows 8% has growth factors 1.10, 0.95 and 1.08. Their geometric mean is about 1.0412, so the steady yearly rate that gives the same result is about 4.1%, not the simple average of about 4.33%. The compound interest calculator uses the same idea. The geometric mean needs every number to be above zero.
Standard deviation
Standard deviation measures how far numbers typically sit from the mean. Two lists with the same mean of 50:
| Numbers | Mean | Population SD | Sample SD |
|---|---|---|---|
| 48, 49, 50, 51, 52 | 50 | 1.41 | 1.58 |
| 30, 40, 50, 60, 70 | 50 | 14.14 | 15.81 |
To work it out: take each number’s distance from the mean, square it, add the squares, divide by n (or n − 1), and take the square root. For the ten numbers at the top, the sample standard deviation is about 3.17.
Sample or population?
If your numbers are the whole group you care about, such as every score in one class, use the population figure (divide by n). If they are a sample used to estimate a bigger group, such as a survey, use the sample figure (divide by n − 1). The sample figure is a little larger, which corrects for a sample usually looking less spread out than the whole group.
Entering your numbers
Commas separate numbers
The calculator reads commas, spaces, semicolons, tabs and new lines as separators. Type 1200, not 1,200, or it will read 1 and 200. Text and symbols are skipped and listed in a warning.
Averages in a spreadsheet
In Excel or Google Sheets: =AVERAGE(A1:A10) for the mean, =MEDIAN(), =MODE.MULT() for every mode, =STDEV.S() for the sample standard deviation, =STDEV.P() for the population one, =GEOMEAN() and =SUMPRODUCT(A1:A3,B1:B3)/SUM(B1:B3) for a weighted mean. For percentages of a total, the percentage calculator shows the working.
Averaging averages
You cannot simply average two averages if the groups are different sizes. A class of 20 with a mean mark of 70 and a class of 30 with a mean of 60 do not average 65 together. Weight each mean by its class size: (70 × 20 + 60 × 30) ÷ 50 = 64. Enter 70 and 60 as the numbers and 20 and 30 as the weights to check it.
Zeros, blanks and missing values
A blank is not the same as a zero. If a pupil missed a test, leaving the mark out gives the average of the tests taken; entering 0 pulls the mean down as if they scored nothing. Decide which you mean before you paste your data. The calculator skips blanks and words such as “absent”, and tells you what it left out.