How to use the descriptive statistics calculator
- Paste or type your numbers, separated by commas, spaces or new lines. Up to 5,000 values are accepted.
- Choose a quartile method. Inclusive matches Excel QUARTILE.INC, exclusive matches QUARTILE.EXC, and "median of halves" splits the sorted list in two.
- Read the headline mean and median, then the full summary table: count, sum, mode, range, quartiles, IQR, standard deviation, variance, standard error, coefficient of variation, geometric mean and outliers.
- Download the CSV if you want the summary and the sorted values in a spreadsheet.
Formula
Mean = sum / n
Sample variance s^2 = sum of (x - mean)^2 / (n - 1), population variance = same sum / n
Inclusive quartile position = (n - 1) x p, exclusive position = (n + 1) x p, with linear interpolation
IQR = Q3 - Q1, outlier fences = Q1 - 1.5 x IQR and Q3 + 1.5 x IQR
Coefficient of variation = sample standard deviation / |mean| x 100
Quartiles have no single universal definition. The method you pick is stated in the steps, so you can match another tool or textbook. Variance uses a two-pass calculation with a rounding correction.
Worked example
Take the data set 2, 4, 4, 4, 5, 5, 7, 9. The sum is 40 and n is 8, so the mean is 5. The median is the average of 4 and 5, which is 4.5, and the mode is 4, which appears 3 times.
The population standard deviation is 2 and the sample standard deviation is 2.1380899.
With the inclusive method Q1 is 4 and Q3 is 5.5. The exclusive method gives 4 and 6.5, and the median-of-halves method gives 4 and 6. For 1, 2, 3, 4, 5, 100 the fences are -1.5 and 8.5, so 100 is flagged as an outlier.
What each measure tells you
Mean, median and mode describe the center of the data, while range, IQR and standard deviation describe its spread. The median is less affected by extreme values than the mean, so when they differ a lot the data is probably skewed. The coefficient of variation expresses spread as a share of the mean, which helps compare data sets measured in different units. The geometric mean applies only when every value is above zero and is useful for growth rates and ratios.
Sample or population
If your numbers are every value you care about, use the population figures. If they are a sample drawn from a larger group, use the sample figures, which divide by n minus 1 to correct for the fact that a sample tends to understate spread. The page shows both. For a step-by-step deviation table, use the standard deviation calculator, and for a plain average with weights see the average calculator.
Quartile methods
Different software uses different quartile rules, and results can differ for small data sets.
- Inclusive (QUARTILE.INC, R type 7) treats the data as spanning 0 to 100 percent and always works for n of 1 or more.
- Exclusive (QUARTILE.EXC, R type 6) needs more values and is undefined for very small samples, which the tool states instead of guessing.
- Median of halves removes the middle value for odd n and takes the median of each half.
Assumptions and limits
- Entries that are not numbers are ignored and listed in a warning. Values above 1e150 in size are rejected.
- Outliers are flagged with the 1.5 x IQR rule, which is a convention for spotting unusual values, not proof that they are errors.
- Mode is shown only when some value repeats. If several values tie, up to six are listed.
- The standard deviation, variance and standard error need at least 2 numbers. With one value they are shown as not defined.
- Everything runs in your browser. Nothing you enter is sent anywhere.
Frequently asked questions
What is the difference between mean and median?
The mean adds all values and divides by the count, so a very large value pulls it up. The median is the middle value after sorting, so it ignores how extreme the ends are. Comparing the two hints at skew.
Why do different tools give different quartiles?
There are several accepted ways to place a quartile in a data set, such as inclusive, exclusive and median of halves. They agree for large data sets and can differ for small ones, which is why this tool states the method.
How do I find outliers in a list of numbers?
Compute Q1 and Q3, subtract to get the IQR, and flag values below Q1 minus 1.5 times IQR or above Q3 plus 1.5 times IQR. The summary table lists any values outside those fences.
When should I use the sample standard deviation?
Use it when your numbers are a sample from a bigger group, such as 30 students from a school. Use the population version only when your list is every member of the group.
What does the coefficient of variation mean?
It is the sample standard deviation divided by the absolute mean, shown as a percent. It lets you compare the relative variability of data sets with different units or very different averages.