Statistics Calculator
Enter a list of numbers to see a full descriptive summary: count, sum, mean, median, mode, minimum, maximum, range, quartiles, interquartile range, variance, standard deviation, standard error, coefficient of variation and any outliers.
Last updated November 3, 2026
Separate numbers with commas, spaces or new lines.
Result
Mean = 5, median = 4.5
Sample (n − 1)
| Count | 8 |
|---|---|
| Sum | 40 |
| Mean | 5 |
| Median | 4.5 |
| Mode | 4 |
| Minimum | 2 |
| Maximum | 9 |
| Range | 7 |
| First quartile (Q1) | 4 |
| Third quartile (Q3) | 6 |
| Interquartile range (IQR) | 2 |
| Sample variance | 4.571429 |
| Sample standard deviation | 2.13809 |
| Standard error of the mean | 0.755929 |
| Coefficient of variation | 42.76% |
| Outlier fences (1.5 × IQR) | 1 to 9 |
| Outliers | None |
Sorted data: 2, 4, 4, 4, 5, 5, 7, 9
Show the work
Formula: mean = sum / n; Q1 and Q3 = medians of the lower and upper halves; IQR = Q3 − Q1; fences = Q1 − 1.5 × IQR and Q3 + 1.5 × IQR; SD = √(sum of squared distances / divisor)
- Sort the numbers: 2, 4, 4, 4, 5, 5, 7, 9
- Sum and mean: 40 / 8 = 5
- Median: n = 8 is even, so average the middle pair (4 and 5): 4.5
- Mode: the most frequent value is 4
- Range: 9 − 2 = 7
- Quartiles: lower half 2, 4, 4, 4 gives Q1 = 4; upper half 5, 5, 7, 9 gives Q3 = 6
- IQR: 6 − 4 = 2. Fences: 4 − 1.5 × 2 = 1 and 6 + 1.5 × 2 = 9. Outliers: None
- Squared distances from the mean: (2 − 5)² = 9; (4 − 5)² = 1; (4 − 5)² = 1; (4 − 5)² = 1; (5 − 5)² = 0; (5 − 5)² = 0; (7 − 5)² = 4; (9 − 5)² = 16. Total 32
- Sample variance: 32 / 7 = 4.571429; standard deviation √4.571429 = 2.13809
- Standard error: 2.13809 / √8 = 0.755929
- Coefficient of variation: 2.13809 / |5| = 42.76%
Result: Mean = 5, median = 4.5
How each statistic is calculated
mean = sum ÷ n
variance = sum of (x − mean)² ÷ (n − 1 for a sample, n for a population)
standard deviation = √variance
standard error = s ÷ √n
IQR = Q3 − Q1
fences = Q1 − 1.5 × IQR and Q3 + 1.5 × IQR
The calculator sorts your numbers first. The mean is the sum divided by the count, and the median is the middle value, or the average of the two middle values when the count is even. The mode is the most frequent value; ties are all listed, and there is no mode if every value occurs once.
Variance measures spread. Subtract the mean from each number, square the result, add the squares, and divide. A sample divides by n − 1 and a population divides by n. The standard deviation is the square root of the variance.
The standard error of the mean is the sample standard deviation divided by the square root of n, so it needs a sample of at least two numbers. The coefficient of variation is the standard deviation divided by the absolute mean, shown as a percentage; it is undefined when the mean is 0.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Percentage Calculator, Z-Score Calculator and Probability Calculator.
Quartiles, IQR and outliers
Quartiles use the median of the halves method. Sort the data and split it at the median into a lower half and an upper half. When the count is odd, the middle value is left out of both halves. Q1 is the median of the lower half, Q3 is the median of the upper half, and the interquartile range is Q3 − Q1.
Spreadsheets and statistics packages often use other quartile methods, such as interpolated percentiles, and can return slightly different Q1 and Q3 values on the same list. A mismatch does not mean either tool is wrong.
A value is flagged as an outlier when it falls below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. A value exactly on a fence is not flagged. This is a rule of thumb, not proof that a value is a mistake.
Sample or population
Choose Population when your list is every value you care about. Choose Sample when the list is a subset used to learn about a larger group. Dividing by n − 1 corrects for a sample understating the spread of the whole group.
The choice changes the variance, standard deviation and coefficient of variation, and the standard error is shown for samples only. Everything else is unchanged.
Small lists and special cases
A sample of one number has no variance, because n − 1 is zero, so the calculator asks for at least two numbers or a switch to Population.
Negative numbers work normally. The calculator accepts up to 1,000 numbers and reports the first token that is not a number.
How to use the Statistics Calculator
- Type or paste your numbers, separated by commas, spaces or new lines.
- Choose Sample if the list is part of a larger group, or Population if it is the whole group.
- Read the summary table, the outlier fences and the sorted data.
- Open Show the work to follow each step for your list.
- Use Copy, Print or Share to keep the result.
Worked example
Take 2, 4, 4, 4, 5, 5, 7, 9. The sum is 40, so the mean is 40 ÷ 8 = 5. The middle pair is 4 and 5, so the median is 4.5. The squared distances from the mean add up to 32. The halves are 2, 4, 4, 4 and 5, 5, 7, 9, with medians 4 and 6. The fences are 1 and 9, so nothing is an outlier.
| Statistic | Value |
|---|---|
| Sum | 40 |
| Mean | 5 |
| Median | 4.5 |
| Q1 and Q3 | 4 and 6 |
| Population standard deviation | 2 |
| Sample standard deviation | 2.13809 |
| Coefficient of variation (sample) | 42.76% |
Frequently asked questions
What is the difference between sample and population standard deviation?
The population version divides the squared distances by n and the sample version divides by n − 1. For 2, 4, 4, 4, 5, 5, 7, 9 they are 2 and about 2.13809.
How does the calculator find outliers?
It flags values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. In the list 1 to 9 plus 30, Q1 is 3, Q3 is 8 and the IQR is 5, so the fences are −4.5 and 15.5 and only 30 is flagged.
Why do my quartiles differ from another tool?
There is no single quartile method. This page uses the median of the lower and upper halves, leaving out the middle value for an odd count. Tools that interpolate percentiles can give different results.
Why is the coefficient of variation undefined?
It divides the standard deviation by the mean, so a mean of 0 makes it undefined. For −3, −1, 1, 3 the mean is 0, so the calculator shows Undefined rather than a number.
Why does a sample of one number show no standard deviation?
A sample divides by n − 1, which is zero for a single number, so the spread is not defined. Add a second number or switch to Population.