Statistics Calculator

The three measures of central tendency

Three common ways to describe the “center” of a dataset:

• Mean: arithmetic average. Sum all values, divide by count. Sensitive to outliers — one billionaire skews the “average income” of a small town.
• Median: the middle value when sorted. For an even-count list, the average of the two middle values. Insensitive to extreme values, making it a better “typical” for skewed data.
• Mode: the most-frequent value. There can be no mode (all unique), one mode (unimodal), or multiple modes (bimodal, multimodal). Mode is most useful for categorical data.

For symmetric distributions: mean = median = mode. For right-skewed (long tail to the right, like income): mean > median > mode. For left-skewed: mean < median < mode.

Standard deviation: the spread metric

Standard deviation (SD) measures how spread out the data is around the mean. Low SD = data clusters tightly around the mean. High SD = data is spread widely.

Sample SD divides by (n−1) — used when your data is a sample drawn from a larger population. Population SDdivides by n — used when you have the entire population. The (n−1) correction (Bessel's correction) accounts for the fact that you're estimating the population mean using the sample mean, which underestimates true variability.

For roughly normal distributions, the empirical “68-95-99.7 rule” applies:

• ~68% of values fall within 1 SD of the mean.
• ~95% within 2 SDs.
• ~99.7% within 3 SDs.

Variance vs standard deviation

Variance is the average squared deviation from the mean. Standard deviation is the square root of variance — same information, but in the original units (instead of squared units). Use SD when reporting; variance shows up in formulas because it's mathematically convenient.

Quartiles and the five-number summary

Quartiles divide sorted data into four equal-sized groups:

• Q1 (25th percentile): 25% of data below, 75% above.
• Q2 (50th percentile): the median. 50/50 split.
• Q3 (75th percentile): 75% below, 25% above.

IQR (Interquartile Range)= Q3 − Q1, the spread of the middle 50%. Used to detect outliers — values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are commonly flagged as outliers (this is the “Tukey fences” rule used in box plots).

The five-number summary (min, Q1, median, Q3, max) is the basis for box plots and gives a robust description of distribution shape.

When mean lies, median tells the truth

Classic example: house prices in a neighborhood. Most homes are $300K–500K; one mansion is $5M. The mean home price is much higher than the typical home price. Median better represents the “typical” value.

US news typically reports median household income (~$80K) rather than mean (~$120K) for this reason. Real estate uses median sale prices. Salary surveys use median compensation. Whenever you read “average,” check whether they mean the mean or the median — the difference can be significant.

Common use cases for this calculator

• Test scores — class average, top quartile, spread of student performance.
• Workout / fitness data — average pace, variation in reps, weight progression.
• Financial returns — average return, volatility (SD), worst/best month.
• Survey results — mean rating, distribution shape, outliers.
• Manufacturing / quality control — process mean, control limits (mean ± 3 SD).
• Sports stats — batting average, season-over-season variation.
• Personal data — sleep duration, weight, daily steps.

For other math tools: Percentage Calculatorfor relative changes, Random Number Generator for sampling, and Unit Converter for measurement conversions.