Statistics Formulas
Mean = Σx / n
Median = middle value of sorted data
Mode = most frequent value(s)
Range = max − min
Variance = Σ(x − mean)² / n
Std Dev = √Variance
Statistics Calculator
Calculate descriptive statistics for any data set. Enter numbers separated by commas or spaces to get mean, median, mode, range, standard deviation, variance, and more instantly.
Descriptive Statistics Calculator
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population std dev (σ) divides by n; sample std dev (s) divides by n−1 (Bessel's correction). Use population when you have the entire data set; use sample when your data is a subset of a larger population. This calculator shows population std dev.
What does standard deviation tell you?
Standard deviation measures how spread out values are from the mean. A low std dev means data points are clustered near the mean; a high std dev means they're spread widely. About 68% of values fall within 1 std dev of the mean in a normal distribution.
Mean, median and mode
- Mean — the total divided by the count. Uses every value, but is pulled by extremes.
- Median — the middle value when sorted. Unaffected by outliers.
- Mode — the most frequent value. The only one that works for categories.
When the mean misleads
Take salaries of $30k, $32k, $35k, $38k and $400k. The mean is $107k — a figure nobody in the group earns, and which describes the data badly. The median of $35k is far more representative.
This is why income and house price statistics are usually reported as medians. Skewed distributions with a long tail make the mean a poor summary.
Standard deviation
Standard deviation measures spread: how far values typically sit from the mean. Two datasets can share a mean while being completely different — one tightly clustered, the other widely scattered.
A low standard deviation means values bunch near the average; a high one means they are dispersed.
Population vs sample
This distinction matters and is easy to miss. Population standard deviation divides by n; sample standard deviation divides by n − 1.
The reduction, known as Bessel's correction, compensates for the fact that a sample's own mean sits closer to its own data than the true population mean does, which otherwise biases the estimate downward.
Use the population formula only when your data genuinely is the entire group. If you are inferring from a subset, use the sample formula.
Always look at the distribution
Summary statistics can hide important structure. Datasets with very different shapes can share a mean, median and standard deviation. Plot the data before concluding anything from the numbers alone.