Descriptive Statistics Calculator
Paste a list of numbers and get a complete summary in one table: center (mean, median, mode), spread (range, IQR, variance, standard deviation), shape (skewness, kurtosis), outliers and a histogram. Choose sample or population and the quartile method your class or software uses.
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You can paste a column straight from Excel or Google Sheets. Text and blank cells are ignored.
How to use the Descriptive Statistics Calculator
- Paste your data into the box: one column from a spreadsheet, or numbers separated by commas, spaces or new lines.
- Choose Sample or Population. Pick Sample when the numbers are a subset of a larger group (the usual case).
- Pick the quartile method that matches your textbook, calculator or Excel, if it matters for your work.
- Press Calculate to see the key results, the worked formulas, the histogram and the full statistics table.
- Download the CSV or copy the results for a report.
Formulas used
| Statistic | Formula | What it tells you |
|---|---|---|
| Mean | x̄ = Σx ÷ n | The balance point of the data; pulled by extreme values |
| Median | Middle value of the sorted data (average of the two middle values if n is even) | The typical value; resistant to outliers |
| Sample variance | s² = Σ(x − x̄)² ÷ (n − 1) | Average squared distance from the mean, for a sample |
| Population variance | σ² = Σ(x − μ)² ÷ N | The same, when you have the whole population |
| Standard deviation | s = √s² or σ = √σ² | Typical distance from the mean, in the data's own units |
| Interquartile range | IQR = Q3 − Q1 | Spread of the middle 50% of values |
| Coefficient of variation | CV = SD ÷ |mean| × 100% | Spread relative to the mean, for comparing data in different units |
| Sample skewness (G1) | n ÷ ((n − 1)(n − 2)) × Σ((x − x̄) ÷ s)³ | Asymmetry: positive = long right tail |
| Sample excess kurtosis (G2) | n(n + 1) ÷ ((n − 1)(n − 2)(n − 3)) × Σ((x − x̄) ÷ s)⁴ − 3(n − 1)² ÷ ((n − 2)(n − 3)) | Tail heaviness compared with a normal distribution (0) |
The sample skewness and kurtosis formulas are the bias-adjusted versions used by Excel (SKEW, KURT), Google Sheets and SPSS. When you choose Population, the calculator uses the plain moment ratios g1 = m3 ÷ m2^1.5 and g2 = m4 ÷ m2² − 3.
Reading the results together
- Mean vs median. If the mean is well above the median, a few large values are pulling it up (right skew). For skewed data such as incomes, home prices or response times, the median and IQR describe a typical value better than the mean and standard deviation.
- Standard deviation vs IQR. For roughly normal data, the IQR is about 1.35 standard deviations. A much larger ratio suggests outliers or heavy tails.
- The 68–95–99.7 rule only applies when the histogram looks bell-shaped: then about 68% of values fall within one SD of the mean and 95% within two.
- Five-number summary. Minimum, Q1, median, Q3 and maximum are what a box plot draws; the fences in the table mark where its whiskers stop.
- Histogram bins follow Sturges' rule (about log₂ n + 1 bins) rounded to a tidy width. Each bin includes its lower edge and excludes its upper edge, except the last.
Frequently asked questions
What are descriptive statistics?
Descriptive statistics summarize a data set with a few numbers: measures of center (mean, median, mode), measures of spread (range, interquartile range, variance, standard deviation) and measures of shape (skewness, kurtosis). They describe the data you have, unlike inferential statistics, which draw conclusions about a larger population.
Should I choose sample or population?
Choose population only when your data includes every member of the group you care about, such as the scores of all 25 students in one class. If the data is a subset used to learn about a bigger group, choose sample. The sample variance divides by n − 1 instead of n (Bessel's correction), which makes it slightly larger.
Why do my quartiles differ from Excel or my calculator?
There is no single standard formula for quartiles. TI-84 calculators and most intro textbooks take the median of each half of the data, Excel QUARTILE.INC and R interpolate at position (n − 1)p, and QUARTILE.EXC uses (n + 1)p. With small data sets the results can differ, so choose the method your course or software uses.
How are outliers identified?
This calculator uses Tukey's 1.5 × IQR rule: a value is flagged as an outlier if it is below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. It is the same rule box plots use to draw separate points.
How do I interpret skewness?
Skewness near 0 means the data is roughly symmetric. A positive value means a longer tail on the right (a few unusually large values, common for incomes or house prices), and a negative value means a longer tail on the left. As a rule of thumb, values beyond ±1 indicate strong skew. When data is right-skewed, the mean is usually larger than the median.
What does excess kurtosis tell me?
Kurtosis describes how heavy the tails are compared with a normal distribution. This tool reports excess kurtosis, so a normal distribution scores about 0, positive values mean more extreme values than normal (heavy tails) and negative values mean fewer (light tails). The sample version needs at least 4 values.
Is my data sent anywhere?
No. All calculations run in your browser, and the numbers you paste are not uploaded or stored on a server.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.