Mean, Median, Mode & Standard Deviation
Find the mean, median, mode, range, quartiles and standard deviation of a list of numbers, with the working shown.
Results are provided as is, for general information. Check anything important before relying on it. See the Terms of Use.
About this tool
Enter a list of numbers to get the main descriptive statistics in one place: the mean, median and mode for the center of the data; the range, quartiles and interquartile range for its spread; and the variance and standard deviation in both sample and population form. The standard deviation steps are written out, and the values are listed in sorted order. Results are rounded to six decimal places for display.
How it works
Mean x̄ = Σx ÷ n Population σ = √( Σ(x − x̄)² ÷ n ) Sample s = √( Σ(x − x̄)² ÷ (n − 1) ) Range = max − min IQR = Q3 − Q1
The median is the middle of the sorted values, or the mean of the two middle values when the count is even. Q1 and Q3 are the medians of the lower and upper halves, leaving the overall median out of both when the count is odd. Variance is the squared standard deviation.
| Question | Working | Answer |
|---|---|---|
| What is the mean of 2, 4, 4, 4, 5, 5, 7, 9? | 40 ÷ 8 | 5 |
| What is the median of 3, 7, 1, 9, 4? | Sorted: 1, 3, 4, 7, 9; middle value | 4 |
| What is the median of 1, 3, 5, 8? | (3 + 5) ÷ 2 | 4 |
| What is the mode of 1, 2, 2, 3, 3? | 2 and 3 each appear twice | 2 and 3 (bimodal) |
| What is the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9? | Σ(x − 5)² = 32; √(32 ÷ 8) | 2 |
| What is the sample standard deviation of the same values? | √(32 ÷ 7) | 2.13809 |
| What are the quartiles of 1, 3, 5, 7, 9, 11, 13? | Halves 1, 3, 5 and 9, 11, 13 | Q1 = 3, Q3 = 11, IQR = 8 |
| What is the range of 12, 15, 15, 18, 22? | 22 − 12 | 10 |
When to use it
Questions
What is the difference between sample and population standard deviation?
Both start from the squared distances of each value from the mean. Population standard deviation divides their sum by n, the number of values, and describes a complete group. Sample standard deviation divides by n − 1 and estimates the spread of a larger group your values were drawn from; dividing by the smaller number corrects for a sample's tendency to underestimate spread. For 2, 4, 4, 4, 5, 5, 7, 9 the population figure is 2 and the sample figure is about 2.13809.
How is the median found when there is an even number of values?
Sort the values and take the mean of the two in the middle. For 1, 3, 5, 8 the middle values are 3 and 5, so the median is 4. With an odd count the median is simply the middle value.
What if no value repeats, or several values tie for most frequent?
If every value appears once, there is no mode. If two or more values share the highest count, they are all modes: 1, 1, 2, 2, 3 has modes 1 and 2, each appearing twice.
How are the quartiles calculated?
Q1 is the median of the lower half of the sorted values and Q3 is the median of the upper half. With an odd count, the overall median is left out of both halves: for 1, 3, 5, 7, 9, 11, 13 the halves are 1, 3, 5 and 9, 11, 13, so Q1 = 3 and Q3 = 11. Spreadsheet functions such as QUARTILE interpolate between values instead and can give different quartiles for the same data.
How should I enter the numbers?
Separate them with commas, spaces, semicolons or new lines, so a column pasted from a spreadsheet works as is. Because a comma always separates two values, write a thousand as 1000, not 1,000. Negative numbers, decimals and exponent notation such as 2.5e3 are accepted.