Statistics Calculator

Population standard deviation (σ) divides by N and is used when you have the entire population. Sample standard deviation (s) divides by N−1 (Bessel's correction) and is used when you have a sample — it corrects for bias in estimating the population std dev. For large N the difference is negligible. Most statistics courses and tools default to sample std dev unless otherwise specified. Variance follows the same distinction: population variance = σ², sample variance = s².

The median is the middle value when data is sorted in ascending order. For odd N: the median is the value at position (N+1)/2. For even N: the median is the average of the values at positions N/2 and N/2+1. The median is robust to outliers — a billionaire joining a group of 99 average earners barely changes the median but dramatically changes the mean. Use median for skewed distributions (income, house prices) and mean for symmetric, bell-shaped distributions.

Mode is the most frequently occurring value(s). A dataset can be unimodal (one mode), bimodal (two modes), multimodal (many modes), or have no mode (all values appear once). Example: [1, 2, 2, 3, 3] has modes 2 and 3 (bimodal). Mode is the only measure of central tendency that works for categorical data. In a normal distribution, mean = median = mode. For continuous data, mode is often computed using a histogram or kernel density estimate.

Quartiles divide sorted data into four equal parts. Q1 (25th percentile): 25% of data falls below this. Q2 (50th percentile): the median. Q3 (75th percentile): 75% of data falls below this. IQR (Interquartile Range) = Q3 − Q1: the spread of the middle 50% of data. Outlier detection: values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are considered potential outliers (Tukey's fence). Box plots visualize Q1, Q2, Q3, and outlier boundaries.

Mean: best for symmetric distributions without extreme outliers — test scores, heights, temperatures. Median: best for skewed distributions or when outliers are present — income, house prices, response times. Mode: best for categorical data or finding the most common value — survey responses, product sizes. In practice, always report all three for a complete picture. If mean and median differ significantly, your data is likely skewed or contains outliers worth investigating.