Three kinds of average
In everyday speech "average" usually means the arithmetic mean, but statisticians use three common measures of the centre of a set of numbers, and they can give quite different answers for the same data.
| Measure | How to find it | Best for |
|---|---|---|
| Mean | Add all values, divide by how many there are | Symmetric data without extreme values: test scores, temperatures, measurements |
| Median | Sort the values and take the middle one (or the mean of the two middle ones) | Skewed data with outliers: incomes, house prices, waiting times |
| Mode | The value that appears most often | Categories and counts: shoe sizes, the most common answer in a survey |
Mean = (x₁ + x₂ + … + xₙ) ÷ n
Worked example 1: no repeated values
Take the numbers 4, 8, 15, 16, 23, 42 (the default values above).
- Sum = 4 + 8 + 15 + 16 + 23 + 42 = 108, and there are 6 numbers, so the mean = 108 ÷ 6 = 18.
- With an even count, the median is the mean of the 3rd and 4th values in order: (15 + 16) ÷ 2 = 15.5.
- No value repeats, so there is no mode.
- Range = 42 − 4 = 38.
The mean (18) is above the median (15.5) because the single large value, 42, pulls the mean upwards. That gap is a quick sign that data are skewed.
Worked example 2: a clear mode
Eight students rate a lesson from 1 to 10: 2, 3, 3, 5, 7, 7, 7, 9. The sum is 43, so the mean is 43 ÷ 8 = 5.375. The median is (5 + 7) ÷ 2 = 6, and the mode is 7, which appears three times. Here the mode tells you the most popular rating, the median tells you that half the class rated it 6 or higher, and the mean summarises all the ratings in one number.
Why the median is used for incomes
Suppose six people earn 50,000, 52,000, 55,000, 58,000, 61,000 and 250,000 a year. The mean is 526,000 ÷ 6 ≈ 87,667, higher than what five of the six people actually earn. The median is (55,000 + 58,000) ÷ 2 = 56,500, which describes the typical person far better. This is why statistics offices usually report median household income and median house prices rather than means.
Spread: range and standard deviation
Two data sets can share the same mean but behave very differently, so it helps to measure how spread out the values are.
- Range is simply largest minus smallest. It is easy to understand but depends entirely on the two most extreme values.
- Standard deviation measures the typical distance of values from the mean. For 4, 8, 15, 16, 23, 42 the squared distances from 18 add up to 910. Dividing by n (6) and taking the square root gives the population standard deviation, about 12.32; dividing by n − 1 (5) gives the sample standard deviation, about 13.49.
Use the population figure when your list contains every member of the group you care about (all students in one class), and the sample figure when the list is a sample used to estimate a larger group (50 customers out of thousands). Spreadsheets follow the same split: STDEV.P and STDEV.S.
Geometric mean for growth rates
When values are multiplied together, as with investment returns or population growth, the arithmetic mean overstates the typical rate. If an investment grows 10%, then 20%, then falls 5%, the growth factors are 1.10, 1.20 and 0.95. Their arithmetic mean is 1.0833 (8.33% a year), but the geometric mean is the cube root of 1.10 × 1.20 × 0.95 = 1.254, which is about 1.0784, or 7.84% a year. Growing 7.84% for three years reproduces the actual result; growing 8.33% does not. The calculator shows the geometric mean whenever all numbers are positive. For investments, see also the ROI calculator, which gives the annualised return directly.
Weighted averages
Sometimes values do not count equally. If a course grade is made of homework (20%) at 80, a midterm (30%) at 90 and a final exam (50%) at 70, the weighted average is 0.2 × 80 + 0.3 × 90 + 0.5 × 70 = 16 + 27 + 35 = 78, not the simple mean of 80. Grade point averages work the same way, weighted by credits; use the GPA calculator for that, or the final grade calculator to see what you need on an exam.
Common mistakes
- Forgetting to sort before finding the median. The middle of an unsorted list is meaningless.
- Averaging averages. The mean of two class averages is only correct if the classes are the same size; otherwise weight each average by its count.
- Averaging percentages or rates directly. Averaging two fuel economy figures in miles per gallon, or two speeds over equal distances, needs a harmonic or weighted mean.
- Reporting the mean of skewed data alone. Give the median too, or the reader may get the wrong picture.
Frequently asked questions
How do I calculate an average?
Add up all the numbers and divide by how many there are. For 4, 8, 15, 16, 23 and 42 the sum is 108 and there are 6 numbers, so the average is 18.
What is the difference between mean and median?
The mean adds all values and divides by the count; the median is the middle value once the list is sorted. Extreme values pull the mean but barely move the median.
Can a data set have more than one mode?
Yes. If two or more values share the highest count, each is a mode (the data are bimodal or multimodal). If no value repeats, there is no mode.
Should I use sample or population standard deviation?
Use population when your numbers are the whole group you are describing, and sample when they are a sample used to estimate a larger group. The sample version divides by n − 1 and is slightly larger.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.1 (Measures of location) and 1.3.5.6 (Measures of scale).
- Office for National Statistics (UK), guidance on averages: mean, median and mode.
Last reviewed 8 October 2026. Results are estimates; see our disclaimer.