Statistics Calculator

What this calculator does

This statistics calculator produces a complete descriptive summary of a data set in one pass. It reports the count, sum, mean, median, mode, range, smallest and largest values, geometric mean, population and sample variance, population and sample standard deviation, and the sorted data.

The summary is useful at the start of an analysis, when you want to understand the shape of the data before choosing a model or running a test. It also provides both the population and sample versions of variance and standard deviation, so you can select the measure that matches how the data was collected.

How to use this calculator

1. Enter the numbers separated by commas, spaces, semicolons, or new lines.
2. Click Calculate.
3. Read the headline mean and review the summary rows.

The order of the input does not matter, because the calculator sorts the values internally. The sorted row lets you check the minimum, maximum, and median positions directly. The mode row reports every value that ties for the highest frequency.

Formulas

Count: N
Sum: Σ x_i
Mean: μ = Σ x_i / N
Median: middle value of the sorted data
Mode: value(s) with the highest frequency
Range: maximum − minimum

Population variance: σ² = Σ(x_i − μ)² / N
Population standard deviation: σ = √σ²

Sample variance: s² = Σ(x_i − x̄)² / (N − 1)
Sample standard deviation: s = √s²

Geometric mean: (x_1 × x_2 × ... × x_N)^(1/N)

The geometric mean is reported only when every value is positive.

Default example

The default data set is 10, 2, 38, 23, 38, 23, 21, 23. It contains eight values. The sum is 178 and the mean is 22.25. After sorting, the data is 2, 10, 21, 23, 23, 23, 38, 38, so the median is 23.

The value 23 appears three times and is the only mode. The range is 38 − 2 = 36. The population variance is 132.4375 and the population standard deviation is about 11.50815. The sample variance is about 151.35714 and the sample standard deviation is about 12.30273. The geometric mean is about 17.11985.

Assumptions and limitations

• At least two values are required, because the sample variance divides by N − 1 and is undefined for a single observation.
• The data is treated as a list of individual observations. Grouped frequency data must be expanded before entry.
• Population measures assume the data contains every member of the group. Sample measures assume the data is a subset of a larger population.
• The mean and standard deviation are sensitive to outliers. The median and mode are more resistant but convey less information about spread.
• The geometric mean is not defined when any value is zero or negative, and the calculator reports that condition instead of a number.
• A data set may have no mode when all values are unique, or several modes when values tie for the highest frequency.
• Results are computed in double precision, so very large data sets or extreme values may show rounding in the final digits.

Frequently asked questions

Which standard deviation should I use?

Use the population version when your data includes every member of the group you are studying. Use the sample version when the data is a subset and you want to estimate the spread of the larger population. The sample version is larger because it divides by N − 1 rather than N.

Why are the mean and median different?

The mean uses every value and moves toward extreme observations. The median depends only on the middle position after sorting. A large gap between them usually signals skew or outliers. When the mean is much larger than the median, the data likely has a long right tail.

What is the geometric mean used for?

The geometric mean summarizes multiplicative growth or ratios. It is common in finance for average returns and in science for quantities that vary over orders of magnitude. Because it multiplies values, it cannot be calculated when any observation is zero or negative.

Source

Reference: Calculator.net — Statistics Calculator.