Statistics Calculator —
full descriptive analysis.

Field guide

Descriptive statistics explained.

Descriptive statistics summarise a dataset with a small set of numbers that capture its essential shape without examining every data point individually. Instead of reporting that a class of 30 students scored 72, 88, 64, 91… you report that the mean is 78.3, the standard deviation is 9.1, and the range is 47, a complete picture in three numbers.

Measures of central tendency

Central tendency answers: where is the middle of the data?

• Mean (arithmetic mean, x̄): the sum of all values divided by the count: Σx ÷ n. The mean is the most familiar measure but is sensitive to extreme values (outliers). One unusually large number can pull the mean far from the typical value.
• Median: the middle value when data is sorted ascending. For an even number of values, it is the average of the two middle values. The median is robust to outliers — adding one extremely high value does not change the median. Preferred over the mean for describing income, house prices, or any variable with a heavy tail.
• Mode: the value(s) that appear most often. A dataset can have zero modes (all values unique), one mode (unimodal), or multiple modes (multimodal). Mode is the only central-tendency measure that works for categorical data.

Measures of dispersion (spread)

Dispersion answers: how spread out is the data? Two datasets can have identical means but completely different distributions.

• Range: the simplest measure: max − min. Extremely sensitive to outliers; one bad data point inflates the range dramatically.
• Variance (s² or σ²): the average of the squared deviations from the mean. Squaring the deviations ensures positive contributions (deviations can be negative) and gives more weight to large deviations. The formula differs depending on whether you have a sample (divide by n − 1) or the full population (divide by n).
• Standard deviation (s or σ): the square root of variance, restoring the original units. Approximately 68% of data falls within 1 standard deviation of the mean in a normal distribution.
• Interquartile range (IQR): the difference between the 75th percentile (Q3) and the 25th percentile (Q1). The IQR measures the spread of the middle 50% of the data and is robust to outliers. Box plots use the IQR to define whisker boundaries and to flag potential outliers (values beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR).
• Coefficient of variation (CV): standard deviation divided by the absolute mean, expressed as a percentage. CV lets you compare variability across datasets with different units or scales: a CV of 15% is the same whether the data is in dollars, kilograms, or milliseconds.

Sample vs population statistics

The key distinction is whether your data is the entire population or is a sample drawn from a larger one:

The sample formula divides by N − 1 instead of N. This is Bessel's correction. When estimating the population variance from a sample, using N in the denominator systematically underestimates the true variance because the sample mean is computed from the same data, consuming one degree of freedom. Dividing by N − 1 produces an unbiased estimator.

Rule of thumb: use “sample” when your data is a subset of a larger group you want to draw conclusions about. Use “population” when your data covers every member of the group of interest.

The 5-number summary and box plots

The 5-number summary (minimum, Q1, median, Q3, maximum) provides a compact, complete description of a distribution's shape and spread without requiring a histogram. A box plot (also called a box-and-whisker plot) visualises these five numbers:

• The box spans Q1 to Q3 (the IQR).
• A vertical line inside the box marks the median.
• The whiskers extend to the minimum and maximum (in the basic form) or to the most extreme non-outlier (in the Tukey form, where outliers beyond 1.5×IQR are plotted individually).

Box plots are exceptionally useful for comparing distributions side by side; two distributions that look identical in a table of means and standard deviations can reveal very different shapes when their boxes are placed on the same axis.

Skewness and kurtosis

Mean, median, and standard deviation describe where and how wide a distribution is. Skewness and kurtosis describe its shape.

• Skewness measures asymmetry. A perfectly symmetric distribution has skewness = 0. Positive skewness (right-skewed) means the right tail is longer; the mode and median are lower than the mean. Negative skewness (left-skewed) means the left tail is longer. Income distributions are typically right-skewed; exam score distributions are often left-skewed.
• Excess kurtosis measures the “heaviness” of the tails relative to a normal distribution (which has excess kurtosis = 0). Positive kurtosis (leptokurtic) means heavier tails and a sharper peak; extreme values occur more often than in a normal distribution. Negative kurtosis (platykurtic) means lighter tails and a flatter peak.

This calculator uses Fisher's unbiased sample skewness formula and the corresponding excess kurtosis formula. Both require n ≥ 3 and n ≥ 4 respectively.

Frequency distributions and histograms

A frequency distribution groups values into bins of equal width and counts how many values fall in each bin. The histogram above uses Sturges' rule (k = 1 + log₂n, capped at 10) to select the number of bins automatically, enough bins to reveal the shape without so many that individual bars contain too few observations to be meaningful.

The shape of the histogram, whether it is approximately bell-shaped, skewed, bimodal, or uniform, often reveals more about the data than any single summary statistic. Always look at the distribution before relying on the mean alone.