Statistics Calculator
Mean, median, mode, range, quartiles, IQR, outliers and skewness from any list of numbers — with sample and population standard deviation shown side by side.
Statistics Calculator: with the default inputs, mean (average) is 22.4.
Separate with commas, spaces, tabs or semicolons. Decimals, negatives and 1.2e3 all work. The box holds 2,000 characters.
These three conventions disagree on the same data. The table below shows all three.
- Count (n)
- 10
- Sum
- 224
- Median
- 15
- Mode
- 15 (appears 2×)
- Minimum
- 11
- Maximum
- 90
- Range (max − min)
- 79
- Standard deviation — sample (s)
- 23.90351Divides by n − 1. This is Excel's STDEV.S and the usual default.
- Standard deviation — population (σ)
- 22.67686Divides by n. Use it only when your list is the entire group.
- Variance — sample (s²)
- 571.377778
- Variance — population (σ²)
- 514.24
- Standard error of the mean
- 7.558953s ÷ √n — how far the sample mean is expected to sit from the true mean.
- Q1 (25th percentile)
- 13.25
- Q3 (75th percentile)
- 17.5
- Interquartile range (Q3 − Q1)
- 4.25
- Lower outlier fence
- 6.875
- Upper outlier fence
- 23.875
- Outliers
- 90
- Skewness — sample (G₁)
- 3.088503Excel's SKEW. Positive means a long right tail.
Assumptions
- Values are an unweighted list; repeat a value to weight it.
- Sample variance and standard deviation use Bessel's correction (n − 1) and need at least two values.
- Sample skewness is the bias-adjusted Fisher–Pearson coefficient G₁ (Excel's SKEW) and needs at least three distinct-enough values; the population coefficient g₁ appears in the comparison table.
- Quartiles follow the method you select; all three common conventions are listed in the table.
- Histogram bins follow Sturges' rule, clamped to between 3 and 12 bins.
| Measure | Population (÷ n) | Sample (÷ n − 1) |
|---|---|---|
| Divisor used | 10 | 9 |
| Variance | 514.24 | 571.377778 |
| Standard deviation | 22.67686 | 23.90351 |
| Skewness | 2.604454 | 3.088503 |
Sample figures are always the larger of the pair, because dividing by n − 1 rather than n corrects the tendency of a sample to look tighter than the population it came from.
| Method | Q1 | Q3 | IQR |
|---|---|---|---|
| Inclusive (Excel QUARTILE.INC) | 13.25 | 17.5 | 4.25 |
| Exclusive (Excel QUARTILE.EXC) | 12.75 | 18.5 | 5.75 |
| Tukey hinges (box plot) | 13 | 18 | 5 |
If your answer key disagrees with the headline figure, it almost certainly used one of the other two rows.
| Range | Count | Share | Cumulative |
|---|---|---|---|
| 11–26.8 | 9 | 90% | 90% |
| 26.8–42.6 | 0 | 0% | 90% |
| 42.6–58.4 | 0 | 0% | 90% |
| 58.4–74.2 | 0 | 0% | 90% |
| 74.2–90 | 1 | 10% | 100% |
5 equal-width bins by Sturges' rule. The top of each range is excluded except the last, which includes the maximum.
How this is worked out
The formula
n = how many values, Σx = their total Mean: x̄ = Σx ÷ n Median: middle of the sorted list (mean of the middle two when n is even) Range: max − min Population variance: σ² = Σ(x − x̄)² ÷ n σ = √σ² Sample variance: s² = Σ(x − x̄)² ÷ (n − 1) s = √s² Standard error: SE = s ÷ √n IQR: Q3 − Q1 Outlier fences: Q1 − 1.5·IQR and Q3 + 1.5·IQR Sample skewness: G₁ = n ÷ ((n−1)(n−2)) × Σ(x − x̄)³ ÷ s³ Population skewness: g₁ = [Σ(x − x̄)³ ÷ n] ÷ [Σ(x − x̄)² ÷ n]^(3/2)
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Your numbers
- Separate with commas, spaces, tabs or semicolons. Decimals, negatives and 1.2e3 all work. The box holds 2,000 characters.up to 2000 characters · defaults to "12, 15, 11, 18, 14, 20, 13, 16, 15, 90"
- Quartile method
- These three conventions disagree on the same data. The table below shows all three.Inclusive — Excel QUARTILE.INC, R type 7 (default) · Exclusive — Excel QUARTILE.EXC, R type 6 · Tukey hinges — the classic box plot
- Outlier fence multiplier(under More options)
- Tukey's rule flags anything beyond 1.5 × IQR from a quartile. Use 3 for extreme outliers only.from 0 to 10 · defaults to 1.5
What you get back
- Mean (average)main answer
- Count (n)
- Sum
- Median
- Mode
- Minimum
- Maximum
- Range (max − min)
- Standard deviation — sample (s)
- Divides by n − 1. This is Excel's STDEV.S and the usual default.
- Standard deviation — population (σ)
- Divides by n. Use it only when your list is the entire group.
- Variance — sample (s²)
- Variance — population (σ²)
- Standard error of the mean
- s ÷ √n — how far the sample mean is expected to sit from the true mean.
- Q1 (25th percentile)
- Q3 (75th percentile)
- Interquartile range (Q3 − Q1)
- Lower outlier fence
- Upper outlier fence
- Outliers
- Skewness — sample (G₁)
- Excel's SKEW. Positive means a long right tail.
What this assumes
- Values are an unweighted list; repeat a value to weight it.
- Sample variance and standard deviation use Bessel's correction (n − 1) and need at least two values.
- Sample skewness is the bias-adjusted Fisher–Pearson coefficient G₁ (Excel's SKEW) and needs at least three distinct-enough values; the population coefficient g₁ appears in the comparison table.
- Quartiles follow the method you select; all three common conventions are listed in the table.
- Histogram bins follow Sturges' rule, clamped to between 3 and 12 bins.
About this calculator
Paste a column out of a spreadsheet, type a row of test scores, or drop in the readings from a lab notebook, and get the whole descriptive summary at once: centre, spread, shape and the values that don't belong. There's no limit of ten boxes — commas, spaces, tabs and line breaks all separate values, so copy and paste usually just works.
How to use it
Put your numbers in the box. Anything that isn't a number is skipped and reported, so a stray label or a units column won't break the calculation. One thing to watch: a comma is a separator, not a thousands mark, so write 1250 rather than 1,250 — the calculator warns you if it spots that pattern.
Sample or population — the question everyone gets wrong
Most calculators make you choose, then quietly give you one number. This one shows both, because the choice changes the answer and the rule is simple:
- Your numbers are the whole group you want to describe — every student in this class, all twelve months of last year. Divide by n. That's the population standard deviation, σ.
- Your numbers are a sample standing in for something bigger — 40 customers out of thousands, this week's readings from a process that keeps running. Divide by n − 1. That's the sample standard deviation, s.
Why n − 1? A sample's values sit closer to their own mean than to the true population mean, so dividing by n systematically understates the spread. Subtracting one from the divisor (Bessel's correction) removes that bias on average. The gap matters most when n is small: with 5 values the sample SD is about 12% larger than the population SD of the same numbers; with 500 it's a tenth of a percent. Excel's STDEV.S and Google Sheets' STDEV are the sample version, STDEV.P and STDEVP the population one.
Quartiles: three answers, all correct
There is no single agreed definition of a quartile, and the three in common use genuinely disagree on the same data. The table under the results gives all three side by side:
- Inclusive (Excel QUARTILE.INC, Google Sheets, R type 7, NumPy's default) interpolates at position (n − 1)p. This is the default here because it's what most software will hand you.
- Exclusive (Excel QUARTILE.EXC, R type 6, Minitab) interpolates at position p(n + 1) and needs at least three values.
- Tukey's hinges split the sorted list in half and take the median of each half. This is the version most textbooks teach for box plots.
If your answer key disagrees with the headline number, check the other two rows before assuming anything is wrong.
Reading the shape
Skewness measures lopsidedness. Zero is symmetric; positive means a long right tail (a few unusually large values, the classic income distribution); negative means a long left tail. Anything beyond about ±1 is strongly skewed, and it's a signal that the median describes your data better than the mean does.
Outliers are flagged by Tukey's rule: more than 1.5 × IQR beyond a quartile. That's a prompt to look, not a licence to delete — an outlier is often the most interesting value in the set, and sometimes it's just a typo.
Frequently asked questions
▸Should I use the sample or the population standard deviation?
Sample (divide by n − 1) if your numbers are a subset used to estimate a bigger group; population (divide by n) if they are the entire group. When in doubt use the sample version — it is what Excel's STDEV.S gives and it is the conservative choice.
▸Why does my calculator give a different Q1 than my textbook?
Because quartiles have no single agreed definition. Excel's QUARTILE.INC, Excel's QUARTILE.EXC and Tukey's hinges can all give different answers for the same data. This calculator shows all three so you can match whichever your course uses.
▸How do you decide a value is an outlier?
Tukey's rule: anything below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Multiplying by 3 instead of 1.5 marks 'extreme' outliers. It flags values worth checking; it does not mean they are wrong.
▸What does a skewness of 1.5 mean?
The data has a long right tail — a few values much larger than the rest — so the mean sits above the median. Beyond roughly ±1 the distribution is strongly skewed and the median is usually the fairer summary.
▸How many numbers can I paste in?
As many as fit in the box, which holds 2,000 characters — several hundred values, separated by commas, spaces, tabs or line breaks. Entries that are not numbers are skipped and listed in a warning rather than silently dropped.
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