Mean, Median, Mode & Standard Deviation Calculator
Free statistics calculator. Enter a list of numbers for the mean, median, mode, range, sum, and both sample and population standard deviation and variance.
What standard deviation measures
Standard deviation is the typical distance of a value from the mean of its data set. A small standard deviation means the numbers cluster tightly around the average; a large one means they are spread out. Paste a list of numbers and this tool returns the mean, median, mode, sample and population standard deviation, both variances, the range, and the sum.
How the calculation works
Every statistic comes from the same three steps.
- Add the values and divide by the count to get the mean.
- Subtract the mean from each value, square the result, and add all the squares. That total is the sum of squared deviations.
- Divide by n − 1 for the sample variance, or by n for the population variance, then take the square root.
In formulas, sample SD = √(Σ(x − μ)² / (n − 1)) and population SD = √(Σ(x − μ)² / n). Values may be separated by commas, spaces, or new lines, and anything that is not a number is dropped.
Worked example with the default data
The box is pre-filled with 12, 15, 18, 22, 27, so n = 5 and the sum is 94, giving a mean of 18.8.
| Value | Value − mean | Squared deviation |
|---|---|---|
| 12 | −6.8 | 46.24 |
| 15 | −3.8 | 14.44 |
| 18 | −0.8 | 0.64 |
| 22 | 3.2 | 10.24 |
| 27 | 8.2 | 67.24 |
The squared deviations add to 138.8. Dividing by n − 1 = 4 gives a sample variance of 34.7, and the square root is 5.8907. Dividing by n = 5 gives a population variance of 27.76 and a population SD of 5.2688. The median of the sorted list is the middle value, 18, and the range is 27 − 12 = 15. All of these are the numbers the tool reports.
Reference: interpreting the result
| Situation | Which to report | Reason |
|---|---|---|
| All members of a group (every exam score in one class) | Population SD | You hold the entire set, so divide by n |
| A sample drawn from a larger group | Sample SD | Dividing by n − 1 corrects the low bias |
| SD is zero | Either | Every value is identical |
| Mixed units in the list | Neither | The result is meaningless; convert first |
If the data is roughly bell-shaped, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. That rule of thumb only applies to mound-shaped data, not to every distribution.
Mean, median and mode
The tool reports the three measures of central tendency alongside the spread, so you can judge a data set from one paste. The mean is the sum divided by the count. The median is the middle value once the list is sorted, or the average of the two middle values when the count is even. The mode is the value that appears most often; a set can have several modes, or none when every value is unique.
When the mean and median sit close together the data is roughly symmetric. When the mean is pulled well above the median, a few large values are dragging it up and the median is the more honest summary — which is why incomes and house prices are usually reported as medians.
| Measure | Best for | Weakness |
|---|---|---|
| Mean | Roughly symmetric data; inputs to further statistics | Distorted by outliers |
| Median | Skewed data such as income, prices, house values | Ignores the size of every other value |
| Mode | Repeated or categorical data such as shoe sizes or survey answers | Can be absent or multiple |
Common mistakes
- Using population SD on sample data. It divides by the larger n, so the spread comes out slightly too small.
- Squaring then forgetting to take the root. Variance is in squared units; standard deviation returns to the original units.
- Assuming a small SD means accurate data. SD describes spread, not correctness — a badly calibrated instrument can be very consistent.
- Leaving a stray letter in the list. Non-numeric tokens are silently skipped, which changes n and every result.
- Ignoring outliers. One extreme value inflates the mean and the standard deviation together.
Should I use sample or population standard deviation?
Use the population version when your list is the complete set you care about, such as every score in a class. Use the sample version when your list is a subset used to estimate a larger group, because dividing by n − 1 compensates for the fact that a sample tends to underestimate true spread.
Can standard deviation be negative?
No. It is the square root of a sum of squares, so it is always zero or positive. A negative value would signal an arithmetic error. Variance can never be negative either.
Why is the two-standard-deviation figure important?
In a normal distribution roughly 95% of observations fall within two standard deviations of the mean. Values beyond that are uncommon enough to be worth a second look, which is why two SD is a common threshold in quality control and outlier screening.
How is this different from the mean absolute deviation?
Mean absolute deviation averages the distance from the mean without squaring. Standard deviation squares the distances first, so it weights large deviations more heavily and is the measure used in most inferential statistics.
The percentage calculator expresses differences between values as relative change, and the basic calculator is handy for checking a sum by hand.