Descriptive Statistics Calculator
Paste your numbers — separated by commas, spaces or new lines — and get the mean, median, mode, sample and population standard deviation, variance, range, quartiles and IQR in one table, ready to copy. It shows both versions of the standard deviation so you can pick the right one. Free, no sign-up.
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How to Use the Statistics Calculator
- 1. Paste your data. Put the numbers in the box separated by commas, spaces, tabs or new lines, in any mix. Decimals and negatives work; anything that is not a number is skipped, and a counter shows how many numbers were read.
- 2. Read the results. Count, sum, minimum, maximum and range come first, then mean, median and mode(s), then variance and standard deviation in both versions, then Q1, Q3 and the IQR. Values are shown to up to four decimals.
- 3. Choose sample or population. Use the sample figures if your numbers are a sample of a larger group, and the population figures if they are every value in the group (see below).
- 4. Copy or clear. Copy Results puts all fifteen lines on the clipboard as plain text with a decimal point; Clear empties the box.
Sample or Population Standard Deviation?
Both measure the typical distance of the values from the mean, and they differ only in the divisor. Population: σ = √(Σ(x − μ)² ÷ N). Sample: s = √(Σ(x − x̄)² ÷ (n − 1)). The n − 1 (Bessel's correction) compensates for the fact that a sample sits closer to its own mean than to the true population mean, so dividing by n would underestimate the spread.
Eight scores: 12, 15, 9, 22, 18, 14, 15, 20
Sum = 125 and n = 8, so the mean = 125 ÷ 8 = 15.625
Sum of squared deviations Σ(x − 15.625)² = 125.875
Population variance = 125.875 ÷ 8 = 15.7344; population SD = 3.9667
Sample variance = 125.875 ÷ 7 = 17.9821; sample SD = 4.2405
Use the sample standard deviation when your numbers are a sample drawn to learn about a larger group, which is the usual case in research, surveys and lab work, and what Excel's STDEV.S, R's sd and most statistics courses mean by default. Use the population version when the data are every member of the group you care about, such as all the scores in one class that you are only describing. With one number the sample statistics are undefined, and the calculator shows a dash. As n grows the two converge.
Mean, Median and Mode
- Mean — the sum divided by the count (15.625 above). It uses every value, so one extreme score pulls it.
- Median — the middle value of the sorted list, or the average of the two middle values when the count is even. For the eight scores it is the average of 15 and 15, which is 15. It resists outliers: replace 22 with 220 and the mean jumps to 40.4 while the median stays 15.
- Mode — the most frequent value (15 here). If several values tie, all are listed; if no value repeats, the calculator reports none, because a list of all-different numbers has no mode.
Quartiles: One Method of Several
Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half, leaving out the middle value when the count is odd. For the scores that gives Q1 = 13, Q3 = 19 and IQR = 6. Spreadsheets and other software often interpolate instead: Excel's QUARTILE.INC returns Q1 = 13.5 and Q3 = 18.5 for the same data. Neither is wrong, but they disagree on small data sets, so say which method you used if your course or journal cares.
Which Tool Do You Need?
| You want | Use |
|---|---|
| Mean, median, mode, SD and quartiles of a list | This Statistics Calculator |
| Skewness, kurtosis and tests of shape | Skewness and Kurtosis Calculator |
| The same statistics from a frequency table or class intervals | Grouped Data Calculator |
| How many standard deviations a value is from the mean | Z-Score Calculator |
| The standard error of a mean | Standard Error Calculator |
| The percentile of one value in a list | Percentile Calculator |
Frequently Asked Questions
- How do I calculate standard deviation?
- Find the mean, subtract it from each value, square those differences, add them up, divide by n − 1 for a sample (or n for a population) to get the variance, then take the square root. For 12, 15, 9, 22, 18, 14, 15, 20 the sample SD is 4.2405.
- What is the difference between sample and population standard deviation?
- The population version divides by n and treats your numbers as the whole group. The sample version divides by n − 1 to correct for estimating the spread of a larger population from a sample. For research data use the sample version.
- How is the median different from the mean?
- The mean is the arithmetic average of all values; the median is the middle value of the sorted list. The median is more robust to outliers, so reports of skewed data such as incomes often give the median.
- Can a data set have more than one mode?
- Yes. If two or more values tie for the highest frequency, the data are multimodal and every tied value is a mode. If no value repeats, there is no mode.
- Why do my quartiles differ from Excel's?
- This calculator uses the median-of-halves method. Excel's QUARTILE.INC interpolates between data points and QUARTILE.EXC uses a third rule, so small data sets give different Q1 and Q3. All are accepted conventions.
- What does a standard deviation of zero mean?
- Every value is the same, so nothing deviates from the mean. Any spread at all makes the standard deviation positive.