Enter a numerical dataset to calculate its mean, median, modes, range, quartiles, variance, standard deviation, robust spread measures, distribution shape and potential outliers. Every result uses the same validated, unrounded values.
Descriptive Statistics Calculator
Paste up to 50,000 values. The calculator checks every token before producing a descriptive summary.
Dataset summary
Calculated from unrounded values
Your descriptive statistics will appear here.
Box plot
Calculated measures
| Measure | Value | Method or meaning |
|---|
Frequency table
| Value | Frequency | Relative frequency | Cumulative frequency |
|---|
Sorted data preview
The ordered values will appear here.
Calculation steps
- Enter and calculate a valid numerical dataset.
Recent calculations
How to Use This Statistics Calculator
- Enter one numerical dataset in the text box. Separate values with commas, spaces, semicolons, tabs or new lines.
- Select whether the main spread result should use the sample or population formula. Both versions still appear in the detailed table.
- Choose the inclusive Type 7 quartile method or the median-of-halves method.
- Select the display precision. This changes only the displayed rounding, never the stored values or calculations.
- Press Calculate statistics. Review the measures, box plot, frequency table, sorted preview and calculation steps.
You can enter integers, negative values, decimals, scientific notation and simple fractions. For example, 1/2; -0.75; 2e1 becomes 0.5, -0.75 and 20. Do not attach units or labels to the numbers. A comma always separates values, so write 2500 instead of 2,500.
The calculator keeps processing inside your browser. It validates complete tokens rather than accepting partial text. An entry such as 12abc is rejected instead of being silently treated as 12.
What This Descriptive Statistics Calculator Finds
Descriptive statistics organize and summarize the values you entered. They do not prove a cause, test a hypothesis or show that your dataset represents a larger population. The tool reports several complementary measures because no single statistic describes every important feature.
The frequency table groups numerically equal entries. Therefore 1, 1.0, 1e0 and 2/2 count as the same value. The sorted preview helps you check order, duplicates and possible entry mistakes before relying on the summary.
Mean, Median and Mode Formulas
The arithmetic mean adds all values and divides by their count. It uses every observation, which makes it useful but sensitive to extreme values.
The median is the middle ordered value. With an even number of observations, it is the mean of the two middle values. The mode is the value that occurs most often. A dataset can have one mode, several tied modes or no mode when every value occurs once.
Worked center example
For 2, 4, 4, 4, 5, 5, 7 and 9, the count is 8 and the sum is 40. The mean is 40 ÷ 8 = 5. The middle values are 4 and 5, so the median is 4.5. The value 4 occurs three times, more than any other value, so the mode is 4.
Mean and median answer different questions. If one unusually large value pulls the mean upward, the median often gives a more stable description of the typical position. The frequency table shows whether a reported mode represents a strong concentration or only a small difference in counts.
Sample vs Population Variance and Standard Deviation
Variance measures average squared distance from the mean. Standard deviation is the square root of variance, so it returns to the original unit of the data. The denominator depends on what your dataset represents.
Sample variance: s² = Σ(xi − x̄)² / (n − 1)
Standard deviation: σ = √σ² or s = √s²
Use the population formula when the entered values are the complete group you want to describe. Use the sample formula when the values are a sample used to estimate variability in a larger population. Dividing by n − 1 applies Bessel's correction and compensates for the tendency of a sample to underestimate population variance.
For the worked dataset, the population variance is 4 and population standard deviation is 2. The sample variance is 4.571428571 and sample standard deviation is about 2.13809. Sample variance and sample standard deviation are not defined for a single observation, so the calculator shows Not available instead of a misleading zero.
The engine uses centered, scaled calculations rather than subtracting two large squared totals. This reduces avoidable cancellation, overflow and underflow when values are large, small or close together.
Quartiles, IQR, Outlier Fences and the Box Plot
Quartiles split ordered data into lower, middle and upper regions. Q1 marks the 25th percentile, Q2 is the median and Q3 marks the 75th percentile. Their difference is the interquartile range.
Lower fence = Q1 − 1.5(IQR)
Upper fence = Q3 + 1.5(IQR)
Values strictly outside these fences are flagged as potential outliers. The rule is a screening convention, not proof that a value is wrong. An unusual observation may be valid and important. Check the measurement, collection process and subject context before excluding it.
Why quartiles differ between calculators
There is more than one accepted quartile convention. The default inclusive Type 7 method uses position (n − 1)p and linearly interpolates between neighboring sorted values. This matches a common spreadsheet and statistical-software convention. The alternative median-of-halves method finds the median of the lower and upper halves and excludes the overall median when n is odd.
Changing the method updates Q1, Q3, IQR, fences, potential outliers and the box plot together. The plot's whiskers stop at the most extreme entered values still inside the fences. Separate points show potential outliers. It does not remove or replace any observation.
Robust Spread and Distribution Shape
The mean absolute deviation reported here is the average absolute distance from the arithmetic mean. The median absolute deviation is the median of the absolute distances from the dataset median. It is shown as the raw, unscaled MAD. Median-based measures often resist a small number of extreme values better than the range or standard deviation.
Adjusted Fisher-Pearson sample skewness describes asymmetry. A positive result usually indicates a longer right tail, while a negative result usually indicates a longer left tail. A value near zero does not guarantee a normal or perfectly symmetric distribution.
Unbiased sample excess kurtosis compares tail weight and concentration with a normal distribution convention centered near zero. Positive excess kurtosis can indicate heavier tails, while a negative value can indicate lighter tails. Software packages use different bias corrections and kurtosis definitions, so always compare results only after confirming the convention.
Skewness needs at least three observations, kurtosis needs at least four, and both require nonzero variation. The calculator reports them as unavailable when these conditions are not met.
How to Read the Results
| Measure | Best used for | Important caution |
|---|---|---|
| Mean | Overall arithmetic center using every value | Extreme values can move it sharply. |
| Median | Middle position and skewed data | It does not reflect the size of every deviation. |
| Mode | Most frequent observed value or values | A numerical dataset may have no repeated mode. |
| Range | Fast total-span check | It depends only on the minimum and maximum. |
| IQR | Spread of the middle half | Its value depends on the quartile convention. |
| Standard deviation | Typical spread around the mean | Choose the correct sample or population formula. |
| Median absolute deviation | Robust spread around the median | This page reports raw MAD, not a scaled estimate. |
| Skewness and kurtosis | Numerical clues about distribution shape | Small samples make these measures unstable. |
Interpret measures together. A mean close to the median, modest skewness and balanced box plot provide mutually supporting evidence of symmetry, but they do not establish a particular probability distribution. Large gaps between center measures or a compressed box beside a distant outlier deserve investigation.
Common Data and Calculation Mistakes
- Using commas as thousands separators: enter 12500, not 12,500, because the comma separates observations.
- Choosing sample by habit: first decide whether your values are a subset or the complete population of interest.
- Comparing different quartile methods: small datasets often show visible differences between conventions.
- Deleting flagged outliers automatically: inspect the source and reason for an unusual value before changing the dataset.
- Rounding inputs early: retain available precision and round only the final displayed answer.
- Mixing units: convert all measurements to the same unit before calculating a summary.
- Treating descriptive results as inference: a sample summary alone does not provide a confidence interval, p-value or causal conclusion.
For high-stakes work, save the original data, document cleaning decisions and reproduce the calculation in the software required by your course, laboratory, employer or regulator.
Calculation Method and Review Sources
The formulas and terminology on this page were reviewed against the NIST measures of location, NIST measures of scale, and NIST notes on skewness and kurtosis. Quartile and spread explanations were also checked against OpenStax guidance on measures of data location and measures of data spread.
The tool states its quartile, skewness, kurtosis, sample and population conventions because these choices can cause correct software packages to return different numbers. The page does not infer missing observations, remove outliers or send entered values to a server.
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Frequently Asked Questions
What does a statistics calculator calculate?
It summarizes a numerical dataset with measures of center, spread, position, frequency and shape. This tool includes mean, median, mode, range, quartiles, variance, standard deviation, absolute deviations, outlier fences, skewness and excess kurtosis.
How should I enter numbers into the calculator?
Separate numbers with commas, spaces, semicolons, tabs or new lines. Fractions and scientific notation are accepted. Do not add units, labels or thousands separators.
What is the difference between sample and population statistics?
Population statistics describe the complete group of interest and divide variance by n. Sample statistics estimate a larger population from a subset and divide variance by n minus 1.
Why does sample variance divide by n minus 1?
Using n minus 1 applies Bessel's correction. It compensates for the downward bias that appears when a sample mean is used to estimate variability in a larger population.
What is the difference between mean, median and mode?
The mean is the arithmetic average, the median is the middle ordered value, and the mode is the most frequent value. Each describes center from a different perspective.
Can a dataset have more than one mode or no mode?
Yes. Several values are modes when they share the highest repeated frequency. When every value occurs once, this calculator reports no repeated mode.
How are quartiles and the interquartile range calculated?
Q1 and Q3 mark the lower and upper quarter positions in ordered data. The interquartile range equals Q3 minus Q1 and describes the spread of the middle 50 percent.
Why do quartile results differ between calculators?
Several accepted quartile conventions exist. This tool states the selected method and offers inclusive Type 7 interpolation or the median-of-halves method.
Why is sample standard deviation undefined for one value?
Sample variance divides by n minus 1. With one observation, that denominator is zero and there is no information about variation among sampled observations.
How do outliers affect the mean, range and standard deviation?
Extreme values can move the mean, widen the range and increase standard deviation. Review the median, IQR and median absolute deviation alongside them, and investigate flagged values before excluding anything.
Calculation and Financial Disclaimer
This free tool provides mathematical summaries for informational and educational use. It does not provide statistical, academic, financial, medical, legal or professional advice. Verify important results, assumptions and data quality independently.