Enter a dataset to compute comprehensive descriptive statistics: mean, median, mode, standard deviation, variance, quartiles, IQR, skewness, kurtosis, and outliers.
| Statistic | Formula |
|---|---|
| Mean | Σxᵢ / n |
| Variance (pop.) | Σ(xᵢ−μ)² / n |
| Variance (sample) | Σ(xᵢ−x̄)² / (n−1) |
| Std Dev (σ) | √(population variance) |
| IQR | Q3 − Q1 |
| Outlier fences | Q1 − 1.5·IQR, Q3 + 1.5·IQR |
| Statistic | Value |
|---|---|
| Count (n) | 8 |
| Sum | 40 |
| Minimum | 2 |
| Maximum | 9 |
| Mean (x̄) | 5 |
| Median | 4.5 |
| Mode | 4 |
| Range | 7 |
| Sum of Squares | 32 |
| Variance (pop. σ²) | 4 |
| Variance (sample s²) | 4.57142857 |
| Std Dev (pop. σ) | 2 |
| Std Dev (sample s) | 2.13808994 |
| Coeff. of Variation | 42.76179871% |
| Q1 (25th pct) | 4 |
| Q2 / Median | 4.5 |
| Q3 (75th pct) | 6 |
| IQR | 2 |
| Lower Fence | 1 |
| Upper Fence | 9 |
| Skewness | 0.81848755 |
| Excess Kurtosis | 0.940625 |
| Outliers | None |
Karl Pearson began using the term standard deviation in lectures in 1893 and printed it the next year, and the habit of flagging outliers with a 1.5 × IQR fence comes from John Tukey's 1977 book Exploratory Data Analysis. This page puts both traditions in one place. Paste a list of numbers and it returns the mean, median, mode, both versions of variance and standard deviation, quartiles, skewness, excess kurtosis and any values outside Tukey's fences.
With n values: mean x̄ = Σx ÷ n. Sum of squares SS = Σ(x − x̄)². Population variance σ² = SS ÷ n and sample variance s² = SS ÷ (n − 1); each standard deviation is the square root of its variance. The median is the middle value of the sorted list, or the average of the two middle ones. Quartiles are the medians of the lower and upper halves, leaving the middle value out when n is odd, so other software with a different quartile convention can report slightly different Q1 and Q3. IQR = Q3 − Q1, and the outlier fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. The coefficient of variation is s ÷ |mean| × 100, using the sample standard deviation.
What changes if one value moves? Start from 2, 4, 4, 4, 5, 5, 7, 9 and replace the 9 with 19. The mean climbs from 5 to 6.25 and the sample standard deviation from about 2.14 to about 5.34, yet the median stays at 4.5 and the quartiles stay at 4 and 6, so the IQR is still 2. The upper fence is still 9, so 19 is now flagged as an outlier. Replace the 9 with 12 instead and the mean only rises to 5.375, but 12 is still beyond the fence. That is why the median and IQR cope better with messy data, while the mean and standard deviation react to every point.
Worked example with the default dataset 2, 4, 4, 4, 5, 5, 7, 9. n = 8, the sum is 40 and the mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, so SS = 32. Population variance = 32 ÷ 8 = 4 and σ = 2; sample variance = 32 ÷ 7 = 4.5714 and s = 2.1381. The median is (4 + 5) ÷ 2 = 4.5, the mode is 4 and the range is 9 − 2 = 7. Q1 = 4, Q3 = 6, IQR = 2, and the fences are 1 and 9, so nothing is flagged: the 9 sits exactly on the upper fence and only values strictly beyond it count. The coefficient of variation is 2.1381 ÷ 5 × 100 = 42.76%. Output is rounded to 8 decimal places and shown with up to 10 significant digits.
Use the sample version (s) when your numbers are a sample from a larger group, which is the usual case, and the population version (σ) only when you hold every member. The big tile shows s. For the default data that is 2.1381 against 2, about 7% higher.
A trailing comma or a leading space leaves an empty piece of text, and the parser turns it into 0. Type 1, 2, 3, with a trailing comma and the page analyses four values: 1, 2, 3 and 0. Delete the stray separator.
With very small datasets the fences are tight. Try 2, 4, 4, 4, 5, 5, 7: Q1 = 4, Q3 = 5, IQR = 1, so the fences are 2.5 and 6.5 and both the 2 and the 7 are flagged. A flag means unusual relative to the middle of this list, not wrong.
When every value appears exactly once there is no mode. If two or more values tie for the highest count, all of them are listed.
Skewness needs more than 2 values and excess kurtosis more than 3. If every value is identical the standard deviation is zero and both measures are undefined. With a single value the sample variance is shown as 0, although strictly it is undefined.