What is the difference between sample and population standard deviation?

Population standard deviation divides the sum of squared deviations by n and is used when your data covers the entire group you care about. Sample standard deviation divides by n − 1 (Bessel's correction) and is used when your data is a sample drawn from a larger population — dividing by n − 1 corrects the bias that would otherwise underestimate the true spread. This tool reports both so you can use whichever your task requires.

How is the median calculated?

The numbers are sorted in ascending order. If there is an odd count, the median is the middle value; if there is an even count, it is the average of the two middle values. The median is also reported as the second quartile (Q2).

What if my data has no mode or several modes?

If every value appears exactly once the data set has no mode, and the calculator says so. If two or more values tie for the highest frequency, the data is multimodal and all of those values are listed.

Which quartile method should I choose?

The Inclusive method uses linear interpolation between data points and matches Excel's QUARTILE.INC, Google Sheets, NumPy's default percentile and most modern courses. The Exclusive method uses Tukey's hinges — it splits the sorted data at the median and takes the median of each half. Different textbooks and tools use different conventions, so pick the one your reference expects; the two agree on Q2 (the median) and usually differ only slightly on Q1 and Q3.

What is the interquartile range (IQR) used for?

The IQR is Q3 minus Q1 — the spread of the middle 50% of the data. It is a robust measure of dispersion that is not distorted by extreme outliers, and it is the basis of the common outlier rule: a point is a suspected outlier if it lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Does the calculator round my results?

Values are computed in full double precision and then displayed rounded to six decimal places to avoid floating-point noise. The underlying calculation is not rounded until display, so chained statistics stay accurate.

Yes. Every calculation runs entirely in your browser using JavaScript. Nothing you type is ever uploaded, logged or stored on a server.

Statistics Calculator

A complete descriptive statistics calculator: paste a list of numbers and instantly get the mean, median, mode, range, sum, variance, standard deviation (both sample and population), quartiles (Q1, Q2, Q3) and the interquartile range. It is built for students checking homework, analysts summarising a column of data, researchers writing up results, and anyone who wants a fast, trustworthy summary of a data set without opening a spreadsheet.

How it works

You can separate your numbers with commas, spaces, semicolons, tabs or new lines — mix and match freely, and any non-numeric text is ignored. The tool sorts the data, then computes each statistic from first principles:

Mean is the sum of all values divided by the count, mean = Σx ÷ n.
Median is the middle value of the sorted list (or the average of the two middle values when the count is even). It is also reported as the second quartile, Q2.
Mode is the most frequent value. If several values tie, all are shown; if every value is unique, there is no mode.
Range is max − min.
Variance is the average of the squared deviations from the mean. The population version divides by n; the sample version divides by n − 1 (Bessel’s correction). The standard deviation is the square root of the variance.
Quartiles split the data into four parts. You can choose the inclusive (linear-interpolation) method used by Excel and NumPy, or the exclusive Tukey-hinge method used by many textbooks. The IQR is Q3 − Q1.

The variance formula used is σ² = Σ(x − mean)² ÷ n for the population and s² = Σ(x − mean)² ÷ (n − 1) for the sample. Sample variance and sample standard deviation require at least two values, since dividing by n − 1 is undefined for a single point.

Worked example

Take the data set 11, 12, 12, 12, 14, 15, 16, 17, 18, 19 (ten values).

Sum = 146, so the mean = 146 ÷ 10 = 14.6.
The two middle values are 14 and 15, so the median (Q2) = (14 + 15) ÷ 2 = 14.5.
The value 12 appears three times, more than any other, so the mode = 12.
Min = 11, max = 19, so the range = 8.
The squared deviations from the mean sum to about 72.4, giving a population variance of 72.4 ÷ 10 = 7.24 and a population standard deviation of about 2.69.
Using n − 1, the sample variance is 72.4 ÷ 9 ≈ 8.04 and the sample standard deviation ≈ 2.84.

With the inclusive quartile method, Q1 ≈ 12 and Q3 ≈ 16.75, so the IQR ≈ 4.75.

Formula note: standard deviation is the square root of variance. Always confirm whether your assignment or tool wants the sample (n − 1) or population (n) figure — they differ, and using the wrong one is the single most common mistake in introductory statistics.

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