Summary statistics calculator
Paste a list of numbers, or a table and pick a column. Every statistic updates as you type, and where textbooks and software disagree on a definition, each version is shown.
Count (n)
40
Mean
35.425
Median
33
Mode
33
Std. dev. (sample)
10.879
Std. dev. (population)
10.7422
Variance (sample)
118.3532
Std. error of mean
1.7201
Minimum
26
Maximum
95
Range
69
Sum
1,417
Coeff. of variation
30.7%
Skewness
4.405
Quartiles, three ways
| Method | Q1 | Q3 | IQR |
|---|---|---|---|
| Linear, inclusive | 30 | 37.25 | 7.25 |
| Linear, exclusive | 30 | 37.75 | 7.75 |
| Tukey's hinges | 30 | 37.5 | 7.5 |
Linear, inclusive: Excel QUARTILE.INC, Google Sheets QUARTILE, R type 7, NumPy default.Linear, exclusive: Excel QUARTILE.EXC, Minitab, SPSS, R type 6.Tukey's hinges: medians of the lower and upper halves, the middle value included in both when n is odd.
Outliers (1.5 × IQR fences)
Lower fence 19.125, upper fence 48.125
Outside the fences: 95.
How to use it
Paste numbers one per line, separated by commas or tabs, or paste a whole table from a spreadsheet
and choose the column. Blank cells and text are skipped rather than counted as zero, and values such
as 1,250, $40 or (12) are read as numbers. The count at the top
left tells you how many values were actually used; check it against what you expected.
The formulas
- Mean:
x̄ = Σx / n. Summed with compensated (Kahan) summation, so long lists of decimals do not drift. - Median: the middle value when sorted, or the average of the two middle values when n is even.
- Mode: the most frequent value. If every value appears once there is no mode; ties are all listed.
- Sample standard deviation:
s = √(Σ(x − x̄)² / (n − 1)). The population version divides by n. - Standard error of the mean:
s / √n, how much the mean itself would vary between samples. - Coefficient of variation:
s / |x̄|, spread relative to size. Meaningless when the mean is near zero. - Skewness:
n / ((n − 1)(n − 2)) × Σ((x − x̄)/s)³, the adjusted Fisher–Pearson coefficient.
Why quartiles have three answers
A quartile falls between two data points more often than on one, and there are several defensible ways to decide where. Hyndman and Fan's 1996 paper catalogued nine definitions in use in statistical software. Three cover almost everything you will meet:
- Linear, inclusive places the quartile at position
(n − 1)p + 1in the sorted list and interpolates. Excel's QUARTILE.INC, Google Sheets, R's default (type 7) and NumPy use it. - Linear, exclusive uses position
(n + 1)p. Excel's QUARTILE.EXC, Minitab and SPSS use it. It spreads the quartiles slightly further apart. - Tukey's hinges take the median of the lower half and of the upper half, with the middle value included in both halves when n is odd. Many textbooks teach the halves method with the middle value excluded instead, as TI graphing calculators do; for even n the two agree.
Worked example
The sample is 40 invented delivery times. They sum to 1,417 minutes, so the mean is 35.43. The median is 33, and 33 is also the mode, appearing four times. The mean is higher than the median because of a single 95-minute delivery, and the skewness of 4.40 says the same thing more formally.
The first quartile is 30 under all three methods. The third quartile is 37.25 (inclusive), 37.5 (Tukey) or 37.75 (exclusive), giving an IQR between 7.25 and 7.75. Using the inclusive quartiles, the fences are 30 − 1.5 × 7.25 = 19.125 and 37.25 + 1.5 × 7.25 = 48.125, and only the 95 lies outside.
Delete the 95 and the mean drops to 33.90 while the median stays at 33. The sample standard deviation halves, from 10.88 to 5.07. The fences also tighten, to 20.25 and 46.25, and now the 47-minute order is flagged. That is a known property of the rule: removing one outlier and re-running can always reveal another. Decide what to exclude once, for a stated reason, rather than repeating the test until nothing is left.
Limits
These statistics describe the numbers you paste; they do not test hypotheses or give confidence intervals beyond the standard error. With fewer than about ten values, the standard deviation, skewness and quartiles are all unstable, and small changes to one value move them a lot. To see the distribution's shape rather than summarise it, use the histogram maker. For statistics by group (the mean per region, the median per product), use the pivot table.
Common questions
Should I use the sample or the population standard deviation?
Use the sample version (dividing by n − 1) when your numbers are a sample from something larger and you want to describe that larger thing, which is nearly always. Use the population version (dividing by n) only when the list is the complete set you care about, such as every employee in a small company. Excel calls them STDEV.S and STDEV.P.
Why do Excel and my calculator give different quartiles?
Because there is no single agreed definition. Excel's QUARTILE.INC, Google Sheets and R's default interpolate one way, QUARTILE.EXC and Minitab another, and many textbooks and calculators use the medians of the two halves. On large datasets they converge; on small ones they can differ noticeably. This page shows all three so you can match whichever your course or software uses.
When should I report the median instead of the mean?
When the data is skewed or has outliers, as incomes, house prices, waiting times and response times usually do. The mean is pulled towards the long tail; the median is the middle value and barely moves. If the two differ by a lot, report both and say why.
Does an IQR outlier mean the value is a mistake?
No. The 1.5 × IQR rule, from John Tukey, flags values worth looking at, not values to delete. In a normal distribution about 0.7% of values fall outside the fences by chance alone. Check whether an outlier is an error, such as a typo or a wrong unit, before removing it, and say so when you do.
What does the skewness number mean?
It measures lopsidedness. Zero means symmetric, positive means a longer tail to the right (a few large values), negative a longer tail to the left. As a rough guide, values beyond about ±1 indicate strong skew. The figure here is the adjusted Fisher–Pearson coefficient, which matches Excel's SKEW function.
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