Statistics Calculator — Mean, Median, Mode, Standard Deviation Explained

Measures of Central Tendency

Central tendency describes where the "middle" of a dataset is. Three statistics do this in different ways.

Mean (arithmetic average)

Sum all values and divide by the count: mean = Σx / n. For the dataset {4, 7, 13, 2, 7, 9, 4, 3, 7, 12}: sum = 68, n = 10, mean = 6.8. The mean is sensitive to outliers — one very large value pulls it significantly.

Median

The middle value when sorted. For an odd count, it's the middle element. For an even count, it's the average of the two middle elements. Sorted: {2, 3, 4, 4, 7, 7, 7, 9, 12, 13}. Median = (7 + 7) / 2 = 7. The median is resistant to outliers — adding 1000 to the dataset does not change the median.

Mode

The most frequently occurring value. In the dataset above, 7 appears three times — more than any other value — so mode = 7. A dataset can have no mode (all values unique), one mode (unimodal), or multiple modes (bimodal, multimodal).

Measures of Spread

Range

The simplest spread measure: maximum − minimum. For the dataset above: 13 − 2 = 11. Easy to compute but sensitive to outliers — one extreme value inflates the range.

Variance

Variance measures the average squared deviation from the mean. It answers: "On average, how far are values from the mean?"

Population variance: σ² = Σ(x − μ)² / n

Sample variance: s² = Σ(x − x̄)² / (n − 1)

The (n−1) denominator (Bessel's correction) compensates for the fact that a sample tends to underestimate the true population spread. Use sample variance whenever your data is a subset of a larger population.

Standard Deviation

Standard deviation is the square root of variance, returning the result in the same unit as the original data. If exam scores have a mean of 75 and a standard deviation of 10, most students scored between 65 and 85 (within one standard deviation of the mean).

The statistics calculator computes both sample (s) and population (σ) standard deviation — toggle between them with a single click.

Quartiles and IQR

Quartiles divide a sorted dataset into four equal parts:

• Q1 (25th percentile) — 25% of values fall below this point
• Q2 (50th percentile) — the median
• Q3 (75th percentile) — 75% of values fall below this point

The IQR (Interquartile Range) = Q3 − Q1. It represents the spread of the middle 50% of the data and is resistant to outliers. Box plots are built from Q1, median, Q3, and the IQR.

Detecting outliers with IQR

The standard Tukey method: a value is an outlier if it falls below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Values beyond Q1 − 3 × IQR or Q3 + 3 × IQR are extreme outliers.

Sample vs Population: When It Matters

Common Questions

When does the mean mislead?

When the data is skewed or has outliers. Average household income in a city is often reported as mean — but a few billionaires push the mean far above what a typical household earns. Median household income is more informative in this case.

What does a high standard deviation mean?

High standard deviation means values are spread widely around the mean — high variability. Low standard deviation means values are clustered close to the mean. In a test context: high SD on exam scores means students had very different results; low SD means scores were tightly grouped.