Descriptive Statistics Calculator

StatisticsLast updated: August 22, 2026

Descriptive statistics summarize a data set with a handful of numbers that describe its center, its spread, and its shape. It is the first step to take before any analysis: a test chosen without first getting to know the data is usually the wrong test.

From a single data entry, this calculator computes measures of central tendency (mean, median, mode), spread (variance, standard deviation, range, IQR, coefficient of variation), position (quartiles, percentiles), and shape (skewness, kurtosis) together. It also flags outliers with Tukey's rule and draws a histogram, a box plot, and a normal Q-Q plot.

You can separate the numbers with commas, spaces, or line breaks. A decimal comma is also accepted (3,14).
Determines the denominator of the variance and the standard deviation.
If you leave it empty, the number is set automatically by k = 1 + 3.322·log₁₀(n).

Descriptive Statistics Formulas

Mean:                x̄ = Σxᵢ / n
Variance (sample):   s² = Σ(xᵢ − x̄)² / (n − 1)
Standard deviation:  s = √s²
Standard error:      SE = s / √n
Interquartile range: IQR = Q3 − Q1
Coefficient of var.: CV = s / |x̄|
Skewness:            g₁ = [n / ((n−1)(n−2))] · Σ((xᵢ − x̄) / s)³
Excess kurtosis:     g₂ = {[n(n+1)] / [(n−1)(n−2)(n−3)]} · Σ((xᵢ − x̄)/s)⁴ − 3(n−1)² / [(n−2)(n−3)]
Outlier fences:      [Q1 − 1.5·IQR , Q3 + 1.5·IQR]

Skewness and kurtosis are computed with the sample-corrected formulas, the same ones behind Excel's SKEW/KURT functions and the SPSS Descriptives output.

How to Calculate

  1. Paste your data values into the box; commas, spaces, and line breaks are all accepted.
  2. Choose whether the data is a sample or the entire population — this sets the denominator of the variance (n−1 or n).
  3. Optionally set the number of histogram classes by hand; if you leave it empty, Sturges' rule is applied.
  4. Read the measures of center, spread, and shape together: the gap between the mean and the median is the first clue about skewness.
  5. Check the outlier warning and the box plot; extreme observations noticeably affect both the mean and the standard deviation.
  6. Look at the Q-Q plot to decide whether the normality assumption is reasonable.

Worked Examples

A measurement series with an outlier

The mean of the 15 measurements is 27.73 and the median is 27. With Q1 = 25, Q3 = 28.5, and IQR = 3.5, Tukey's upper fence is 33.75; the value 45 lies above it and is flagged as an outlier. The mean exceeding the median shows that the distribution is right-skewed.

Örneklem büyüklüğü (n): 15 · Aritmetik ortalama (x̄): 27.73 · Standart sapma (s): 5.39

A small symmetric data set

The mean and the median both land on 14 and the skewness is 0. The sample standard deviation works out to √(40/4) ≈ 3.16; had the same data been treated as a population, the denominator would be 5 instead of 4 and the deviation would be a little smaller. In a symmetric distribution the mean can safely serve as the measure of center.

Örneklem büyüklüğü (n): 5 · Aritmetik ortalama (x̄): 14.00 · Standart sapma (s): 3.16

Treating the data as a population

Because all eight values were measured, the population formula is selected: the mean is 5.5, the population variance is 2.25, and the population standard deviation is 1.5. Had the same data been treated as a sample, the denominator would be 7 instead of 8 and the standard deviation would be a little larger.

Örneklem büyüklüğü (n): 8 · Aritmetik ortalama (x̄): 5.50 · Population standard deviation (σ): 1.50

Frequently Asked Questions

What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize the data in front of you and generalize nothing. Inferential statistics use hypothesis tests and confidence intervals to carry conclusions from a sample to a population. A descriptive summary is the first step of every analysis; the choice of test is usually made by looking at that summary.
Should I use the mean or the median?
If the distribution is roughly symmetric and free of outliers, the mean is preferred because it carries more information. In skewed distributions or in the presence of outliers the median is more reliable, since it is unaffected by extreme observations. For right-skewed variables such as income, rent, or duration, the median is almost always the more accurate measure of center.
How should I interpret skewness and kurtosis?
A skewness close to 0 means a symmetric distribution; the range ±0.5 counts as approximately symmetric and values beyond ±1 as clearly skewed. An excess kurtosis of 0 indicates a normal distribution; positive values point to a peaked, heavy-tailed shape and negative values to a flat one. Many sources accept the range ±2 as compatible with normality.
Should I remove outliers from my data?
Not automatically. An outlier must first be explained: if it is a measurement or data-entry error it is corrected or removed; if it is a genuine extreme observation it should stay in the data. Rather than deleting extreme observations, it is better practice to use robust measures such as the median and the IQR, or to report the results both with and without them.
What is the coefficient of variation good for?
By expressing the standard deviation relative to the mean, it gives a unitless measure of variability. That makes it possible to compare series measured in different units or on very different scales: you cannot compare monthly income and body height through their standard deviations, but you can through their coefficients of variation. It is meaningless for variables whose mean can be near zero or negative.
How should I choose the number of histogram classes?
By default Sturges' rule (k = 1 + 3.322·log₁₀n) is applied, and it is a reasonable starting point for most data sets. Too few classes hide the shape of the distribution, while too many amplify the noise. Changing the number of classes to check whether the distribution really is bimodal or unimodal is a good habit.
Do the results match SPSS and Excel?
Yes. The mean, standard deviation, variance, skewness, and kurtosis come from the same formulas as SPSS Descriptives and Excel's AVERAGE, STDEV.S, VAR.S, SKEW, and KURT functions. The quartiles are found by linear interpolation (Excel's PERCENTILE.INC method); small differences may appear against some of the quartile methods used by SPSS.