Your data stays private
The list is processed in this page. It is not sent to a server, saved in the URL or retained after you leave.
Descriptive statistics
Statistics calculator
Paste a numerical data set to calculate measures of centre, spread and quartiles. Population and sample results are displayed separately.
Enter your data
Results
Center and frequency
Spread and quartiles
Sorted values
Choose population or sample measures according to how the data set was collected and how the result will be used.
Calculation method
The calculator uses compensated summation and Welford’s online algorithm to reduce floating-point error. Results are rounded to fifteen significant digits.
Arithmetic mean
mean = Σx ÷ nThe sum of all observations divided by their count.
Population variance
σ² = Σ(x − μ)² ÷ NUse when the entered values represent the complete population of interest.
Sample variance
s² = Σ(x − x̄)² ÷ (n − 1)Uses Bessel’s correction when the values are a sample from a larger population.
Standard deviation
standard deviation = √varianceExpresses spread in the same unit as the original values.
Interquartile range
IQR = Q3 − Q1Measures the spread of the middle half of the ordered data.
Quartile convention used
Q1, the median and Q3 use linear interpolation at position (n − 1) × p, commonly called quantile type 7. Other software may use a different convention, so small differences are possible.
Common uses
School and research data
Summarise measurements, scores or experimental observations.
Business metrics
Compare typical values and variability in response times, sales or costs.
Data quality checks
Inspect the range, quartiles and repeated values before deeper analysis.
Important limitations
- Descriptive statistics do not establish causation, significance or a probability model.
- The mode compares the numeric values exactly after parsing; values that differ only because of rounding remain distinct.
- Quartiles can differ between software packages because several accepted conventions exist.
- Large outliers can strongly affect the mean, variance and standard deviation.
Examples
Single mode
1, 2, 2, 3, 4
Mean 2.4; median 2; mode 2; population variance 1.04.
Even-sized data set
1, 2, 3, 4
Median 2.5; Q1 1.75; Q3 3.25 using type 7 interpolation.
No repeated value
5, 10, 15
Mean and median 10; no mode; range 10.
Frequently asked questions
Should I use population or sample standard deviation?
Use population measures when the list is the entire group of interest. Use sample measures when the list is used to estimate variability in a larger population.
Why is sample variance unavailable for one value?
Its denominator is n − 1, which equals zero when n is one.
Can there be more than one mode?
Yes. Every value tied for the highest frequency is displayed. If all values occur once, the result reports no mode.
Why do quartiles differ from another calculator?
There are multiple accepted sample-quantile conventions. This page explicitly uses linear interpolation type 7.
Method references
- NIST — Measures of scale
Reference for variance, standard deviation, range and interquartile range as measures of spread.
- NIST — Percentiles
Defines percentiles and identifies the median as the 50th percentile.
- R documentation — Sample quantiles
Documents the nine sample-quantile algorithms and the type 7 default used by this page.
- OpenStax — Measures of variation
Provides population and sample variance and standard-deviation formulas.