Statistics Calculator
Paste a dataset and this calculator returns the full descriptive picture: count, sum, mean, median, mode, min, max, range, quartiles, IQR, sample and population variance and standard deviation, plus the distribution shape (Fisher-Pearson skewness and excess kurtosis), the standard error of the mean, and the coefficient of variation. Switch between the sample formulas (divide by n – 1, for data drawn from a larger group) and the population formulas (divide by n, for data that is the whole group).
Quartiles use the Tukey hinge method: Q1 is the median of the lower half of the sorted data, Q3 the median of the upper half, and IQR = Q3 – Q1. The skewness and kurtosis figures match Microsoft Excel’s SKEW and KURT functions, so numbers computed elsewhere can be checked here directly.
| Center | |
| Count (n) | 0 |
| Sum | 0 |
| Mean (average) | 0 |
| Median | 0 |
| Mode | – |
| Spread | |
| Minimum | 0 |
| Maximum | 0 |
| Range | 0 |
| Q1 (25th percentile, Tukey hinge) | 0 |
| Q3 (75th percentile, Tukey hinge) | 0 |
| Interquartile range (IQR) | 0 |
| Variance (s²) | 0 |
| Standard deviation (s) | 0 |
| Shape | |
| Skewness (Fisher-Pearson G1) | 0 |
| Excess kurtosis | 0 |
| Precision and relative spread | |
| Standard error of the mean | 0 |
| Coefficient of variation | 0 |
Variance = sum of squared deviations from the mean divided by n – 1 (sample) or n (population); standard deviation is its square root. Quartiles use Tukey hinges (medians of the sorted halves, median excluded from both when n is odd), which can differ slightly from spreadsheet interpolation. Skewness is the bias-corrected Fisher-Pearson G1 (Excel SKEW parity); excess kurtosis follows the Excel KURT convention. Standard error = SD / sqrt(n); coefficient of variation = SD / |mean| x 100%.
Quick Answer
For the classic textbook dataset 2, 4, 4, 4, 5, 5, 7, 9 in sample mode, this calculator returns a mean of 5.0000, a median of 4.5000, a mode of 4, quartiles of 4.0000 and 6.0000 (IQR 2.0000), a sample variance of 4.5714 and a sample standard deviation of 2.1381. Switch to population mode and the variance becomes 32 / 8 = 4.0000 with an SD of 2.0000. Paste the numbers above and every figure matches.
Key takeaways
- Sample formulas divide by n – 1, population formulas by n. The choice is the single most consequential input on the page: on eight values it is the difference between an SD of 2.1381 and 2.0000.
- Skewness here is the bias-corrected Fisher-Pearson G1, the same number Excel’s SKEW returns, and excess kurtosis follows the Excel KURT convention, so spreadsheet figures can be cross-checked directly.
- Quartiles use Tukey hinges (medians of the sorted halves), which can differ slightly from Excel’s interpolation methods. The difference is a convention, not an error, and it is disclosed on every result.
- Standard error and the coefficient of variation answer the two questions that follow every descriptive table: how precisely does the mean estimate the population, and how big is the spread relative to the mean?
- Quick Answer
- How to use the statistics calculator
- Sample vs population: which mode to choose
- The formulas, honestly stated
- Quartile conventions: why your spreadsheet may disagree slightly
- Worked example: 2, 4, 4, 4, 5, 5, 7, 9, checked step by step
- Reading the precision statistics: standard error and coefficient of variation
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the statistics calculator
Paste one dataset, choose sample or population, click Calculate statistics, and read the four sections: center, spread, shape, and precision. The results stay hidden until you calculate.
- Paste your numbers: commas, spaces, tabs, semicolons or line breaks all work, so a copied spreadsheet column pastes straight in. The classic dataset 2, 4, 4, 4, 5, 5, 7, 9 is pre-filled as the worked example.
- Pick sample or population: sample (n – 1) when your data is a subset of something bigger, population (n) when your data is the entire group. The variance and SD symbols flip to s and s² or σ and σ² accordingly.
- Read the sections in order: center tells you the typical value, spread how far values stray, shape whether the distribution leans or has heavy tails, and precision whether the mean is estimated tightly and whether the spread is large relative to the mean.
- Copy or share: the buttons under the results copy a plain-text claim summary with every figure, handy for pasting into a report or message.
Anything that is not a number stops the calculation with the message “Enter numbers separated by commas, spaces, or line breaks.” rather than a silently wrong answer. Sorted data is shown under the results, capped at the first 60 values with a count of the rest, so a few-thousand-value paste stays readable.
Sample vs population: which mode to choose
Choose sample when your data is a subset of something bigger: a survey of 200 voters, 30 product measurements from a production line, a class of students standing in for all students. Dividing by n – 1 (Bessel’s correction) compensates for the fact that a sample underestimates the true spread. Choose population only when your data is the entire group of interest: all 50 states, every employee in a 12-person company, the full set of monthly sales for a closed year. When in doubt, sample is the safer default, and it is what most textbooks and statistical software assume.
The two modes affect more than variance and SD. Skewness and kurtosis also switch: sample mode reports the bias-corrected Fisher-Pearson G1 and the Excel KURT-style excess kurtosis, while population mode reports the plain moment skewness and the raw moment excess kurtosis. That is why the same dataset shows skewness 0.8185 in sample mode and 0.6563 in population mode on the worked example.
The formulas, honestly stated
Mean = sum / n. Median is the middle value of the sorted list, or the average of the two middle values when n is even. Mode is every value that occurs most often; if no value repeats there is no mode. Population variance = ∑(x – mean)² / n and sample variance divides by n – 1 instead (Bessel’s correction); standard deviation is the square root of each. These are the standard definitions given in the NIST/SEMATECH e-Handbook of Statistical Methods and in the Rice University Online Statistics Education notes on variability.
Skewness (G1) = n / ((n–1)(n–2)) · ∑((x – x̄) / s)³
Excess kurtosis (sample) = n(n+1)/((n–1)(n–2)(n–3)) · ∑((x – x̄) / s)⁴ – 3(n–1)²/((n–2)(n–3))
The skewness formula is the bias-corrected Fisher-Pearson G1, identical to what Microsoft’s SKEW function returns; the excess kurtosis is the convention of Microsoft’s KURT function, which is 0 for a normal distribution and positive for heavy tails. The NIST e-Handbook’s skewness page and kurtosis page describe the same moment-based reasoning.
Quartile conventions: why your spreadsheet may disagree slightly
There are at least five ways to compute quartiles, and they disagree on small datasets. This calculator uses Tukey hinges: Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half, with the median itself excluded from both halves when n is odd. Excel’s QUARTILE.INC interpolates instead (R type 7). The NIST e-Handbook’s percentiles page documents the family of methods and why results differ.
On the worked example (n = 8, even), the hinges give Q1 = 4.0000 and Q3 = 6.0000. On an odd-sized dataset such as 3, 5, 7, 8, 13, 14, 18, 21, 24, the lower half is 3, 5, 7, 8 and Q1 = 6.0000, the upper half is 14, 18, 21, 24 and Q3 = 19.5000. Excel’s QUARTILE.INC would report 7.0000 and 18.0000 for the same data. Neither is wrong; they answer slightly different questions about the 25% and 75% marks, and this page states its convention on every result so figures can be reproduced by hand.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9, checked step by step
This dataset is pre-filled in the calculator. The run, in sample mode, gives:
- Count n = 8, sum = 40, mean = 40 / 8 = 5.0000
- Sorted: 2, 4, 4, 4, 5, 5, 7, 9. Median = (4 + 5) / 2 = 4.5000
- Mode = 4 (appears 3 times, more than any other value)
- Range = 9 – 2 = 7.0000; Tukey hinges: Q1 = 4.0000, Q3 = 6.0000, IQR = 2.0000
- Sum of squared deviations = 32. Sample variance = 32 / 7 = 4.5714; sample SD = √4.5714 = 2.1381
- Shape: skewness 0.8185 (right-skewed, the 9 drags the tail), excess kurtosis 0.9406
- Precision: standard error = 2.1381 / √8 = 0.7559; CV = 2.1381 / 5 × 100 = 42.76%
Paste the same numbers above with the sample mode selected and every figure matches. Switch to population mode and the variance becomes 32 / 8 = 4.0000 with SD 2.0000.
The same code, run on more datasets, produces the checks below. Every row is a real run of this calculator, so you can paste any of them in and get the same figures back, to the fourth decimal.
| Dataset (sample mode) | Mean | SD | Skewness | Excess kurtosis |
|---|---|---|---|---|
| 1, 2, 3, 4, 5 | 3.0000 | 1.5811 | 0.0000 | –1.2000 |
| 2, 4, 4, 5, 9, 12, 20 | 8.0000 | 6.2981 | 1.3281 | 1.3866 |
| –5, –2, 0, 3, 8 | 0.8000 | 4.9699 | 0.5645 | –0.0005 |
| 1, 1, 2, 2, 3 (bimodal) | 1.8000 | 0.8367 | 0.5122 | –0.6122 |
| 7, 7, 7, 7 (constant) | 7.0000 | 0.0000 | n/a | n/a |
Two edges worth knowing: constant data (no spread at all) shows skewness and kurtosis as n/a, because both divide by the standard deviation; and a dataset whose mean is exactly 0 shows CV as n/a, since a relative spread against a zero mean is meaningless. Single values report Q1/Q3 as NaN, the honest consequence of asking for the median of an empty half.
Reading the precision statistics: standard error and coefficient of variation
After the center and spread, two derived figures answer the follow-up questions. The standard error of the mean (SD / √n) tells you how precisely the sample mean estimates the true population mean: on the worked example it is 0.7559, so the population mean is probably within about ±0.76 of 5. The true mean almost certainly lies inside a wider band, which is the subject of the confidence interval calculator.
The coefficient of variation (SD as a percentage of |mean|) makes spreads comparable across different units or scales: 42.76% on the worked example is moderate, while 621.24% on the –5, –2, 0, 3, 8 dataset flags a spread that dwarfs the mean. The dedicated coefficient of variation calculator goes deeper for head-to-head comparisons of two datasets. When the data pairs up with a second variable instead, the linear regression calculator and the correlation coefficient calculator take over from single-column description to the relationship between two columns.
What competing calculators miss
Most online statistics tools stop at the easy half:
- Basics-only tools (calculator.net and many clones): mean, median, mode, range, both SDs and the geometric mean, but no quartiles, no skewness, no kurtosis, no standard error, no coefficient of variation, and no copy or share. Their strength is the on-page keypad; their ceiling is descriptive completeness.
- Distribution-shape tools (StatCalcPro): a strong suite with skewness, excess kurtosis, SE, CV and outlier flags. The gap is method transparency: quartile conventions and skew/kurt definitions are not stated where the numbers appear, so a figure that disagrees with a spreadsheet cannot be reconciled on the page.
- Outlier-first tools (HeroYears average calculator): quartile methods documented, outliers by the 1.5 × IQR rule, frequency tables and MAD. The gap is shape: no skewness and no kurtosis at all.
- The gap this page fills: Excel-parity skewness and excess kurtosis next to both variance modes, Tukey hinges stated explicitly with the reconciliation math against Excel interpolation, standard error and CV wired into the results, tied modes listed in full rather than truncated, and a copyable claim summary so the numbers can leave the page. No account, nothing installed, and every figure reproducible by hand from the formulas section.
Frequently asked questions
Use sample when your data is drawn from a larger group you want to describe, and population when your data is the entire group. Sample divides the sum of squared deviations by n – 1 (Bessel’s correction) to compensate for the underestimation a sample makes; population divides by n. When in doubt, sample is the safer default and matches most textbooks.
Yes. Numbers separated by commas, spaces, tabs, semicolons or line breaks all parse, so a copied column pastes directly. Anything that is not a number stops the calculation with a clear message instead of a silently wrong answer.
With Tukey hinges: Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half, excluding the median itself from both halves when n is odd. Excel’s QUARTILE.INC interpolates instead, so small differences against a spreadsheet are a convention difference, not an error.
The dataset is bimodal: two values tie for most frequent, which often signals two subgroups mixed together, such as test scores from two different classes. The calculator lists all tied modes, not just the first.
Variance is the average of the squared deviations from the mean, so it is in squared units (dollars-squared, for instance), which is hard to interpret. Standard deviation is its square root, back in the original units, so it can be compared directly with the mean.
Skewness and kurtosis both divide by the standard deviation, so they are undefined when all values are identical (SD = 0). Sample skewness also needs at least 3 values and sample kurtosis at least 4; smaller datasets show n/a rather than a misleading number.
Skewness measures asymmetry: positive means a long right tail (mean above median), negative a long left tail. The calculator uses the bias-corrected Fisher-Pearson G1, the same figure Excel’s SKEW returns. Excess kurtosis measures tail weight relative to a normal distribution: near 0 means normal-like tails, positive means heavy tails and more outliers, negative means light tails.
A few thousand is comfortable. Very large pastes still compute, but the sorted-data display is capped at the first 60 values with a count of the rest, to keep the page readable.
Add This Calculator to Your Website
Embed this calculator on your own site with the code below. Pick a size to preview it live, then copy the code.
Powered by Calculator Zilo
References
- National Institute of Standards and Technology / SEMATECH. Engineering Statistics Handbook: Measures of Location. Accessed October 23, 2026.
- National Institute of Standards and Technology / SEMATECH. Engineering Statistics Handbook: Measures of Skewness and Kurtosis. Accessed October 23, 2026.
- National Institute of Standards and Technology / SEMATECH. Engineering Statistics Handbook: Percentiles. Accessed October 23, 2026.
- Microsoft Support. SKEW function. Accessed October 23, 2026.
- Microsoft Support. KURT function. Accessed October 23, 2026.
- Rice University, Online Statistics Education. Measures of Variability: sample vs population variance. Accessed October 23, 2026.