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Statistics Calculator

Find mean, median, mode, and standard deviation, explore the normal distribution and z-scores, and calculate permutations, combinations, and binomial probabilities - with every step shown

What this tool covers

Statistics describes and summarizes data, while probability quantifies uncertainty about outcomes. This tool covers descriptive statistics (mean, median, mode, variance, standard deviation), the normal distribution (z-scores and area under the curve), and basic combinatorics and probability (permutations, combinations, and the binomial distribution).

Measures of center and spread

Mean

\(\bar{x} = \frac{\sum x_i}{n}\)

Median

middle value of sorted data

Mode

most frequently occurring value(s)

Range

\(\max(x) - \min(x)\)

The mean is sensitive to outliers — a single extreme value can pull it far from where most of the data sits. The median is more robust, since it only depends on the data's order, not its exact values.

\(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\)    vs    \(\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}\) sample variance (n−1, Bessel's correction) vs. population variance (n)
Why divide by n−1 for a sample? A sample's mean is calculated from the same data used to measure spread, which slightly understates the true population variance. Dividing by n−1 instead of n corrects for this bias — it's called Bessel's correction.

Standard deviation is simply the square root of variance, bringing the units back in line with the original data (variance is in squared units, standard deviation is not).

The normal distribution and z-scores

The normal distribution is the familiar bell curve that many natural and measured quantities approximately follow. A z-score converts any value into "how many standard deviations away from the mean" it is:

\(z = \dfrac{x - \mu}{\sigma}\) z-score: x is the value, μ is the mean, σ is the standard deviation

Once converted to a z-score, the standard normal distribution (mean 0, standard deviation 1) tells you what fraction of the data falls above, below, or between any two points — this tool computes that area numerically.

RangeApproximate area
Within 1 standard deviation of the mean≈ 68.27%
Within 2 standard deviations≈ 95.45%
Within 3 standard deviations≈ 99.73%

This is known as the 68-95-99.7 rule (or empirical rule) — a quick way to sanity-check how unusual a value is without doing a full calculation.

Combinatorics: permutations and combinations

Permutation

\(P(n,r) = \dfrac{n!}{(n-r)!}\)

Combination

\(C(n,r) = \dfrac{n!}{r!(n-r)!}\)

Use a permutation when the order of selection matters (arranging 3 books on a shelf out of 10). Use a combination when it doesn't (choosing 3 books to take on a trip out of 10 — the order you pick them in doesn't matter).

Binomial probability

The binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success — like counting heads in 10 coin flips.

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) n = number of trials, k = number of successes, p = probability of success on each trial

This tool also computes the cumulative probability \(P(X \le k)\) — the chance of getting at most k successes — by summing the individual probabilities from 0 up to k.

Mean and variance of a binomial distribution Without needing the full formula, a binomial distribution's mean is simply \(np\) and its variance is \(np(1-p)\) — useful for a quick sanity check on whether a computed probability looks reasonable.

Using the calculator

  1. Choose an operationPick from the sidebar: Descriptive Stats, Normal Distribution, Permutations & Combinations, or Binomial Probability. Each mode shows only the fields it needs.
  2. Enter your data or parametersFor descriptive stats, type a list of numbers with one value per line, or upload a text/CSV file. For the normal distribution, enter the mean, standard deviation, and the value or range you're interested in. For combinatorics or binomial probability, enter n, r (or k), and p as needed.
  3. Press CalculateThe result appears immediately, with the relevant formulas and intermediate steps shown underneath.
  4. Try an example chipIf you're not sure what to enter, click any example below the input fields to load a ready-made problem for the current mode.

Entering a data set

For Descriptive Stats and the histogram view, type your numbers with one value per line, e.g.:

2
4
4
4
5
5
7
9

Decimals and negative numbers are both supported. You can also click Upload file to load values from a .txt or .csv file — any non-numeric cells (like a header row or text labels) are automatically skipped. Use the toggle to choose whether your data is a full population or a sample — this changes how variance and standard deviation are calculated.

Suggested classroom uses

  • Compare mean vs. median sensitivity. Enter a data set, note the mean and median, then add one extreme outlier and recalculate — discuss why the mean shifts dramatically while the median barely moves.
  • Verify the empirical rule. Set up a normal distribution with a chosen mean and standard deviation, then check the area within 1, 2, and 3 standard deviations against the 68-95-99.7 figures.
  • Permutations vs. combinations. Pose the same scenario two ways — "how many ways to arrange 3 of these 5 books" vs. "how many ways to choose 3 of these 5 books" — and use the calculator to show why the answers differ.
  • Binomial vs. normal approximation. For a binomial distribution with a large n, compare its mean and standard deviation against a normal distribution with the same parameters, and discuss when the normal approximation becomes reasonable.

Quick examples by topic

Operation

Descriptive statistics

\(\bar{x}, \text{med}, \sigma\)

Enter one number per line. Decimals and negative numbers are supported.

Result

Enter values and press Calculate to see the result here.