Standard Deviation Calculator

Mean — the arithmetic average. Sensitive to outliers; one extreme value pulls it noticeably.

Median — the middle value when sorted. Robust to outliers, which is why income and home-price reports use the median, not the mean.

Mode — the most frequent value(s). A dataset can have no mode (no repeats), one mode (unimodal), or multiple modes (bimodal, multimodal). Especially useful for categorical or discrete data.

Range — max minus min. Quick spread but completely outlier-driven.

Variance & standard deviation — average squared deviation from the mean (variance), or its square root (SD). SD is in the same units as the data, which is why it's the workhorse spread statistic.

Interquartile range (IQR) — Q3 − Q1. The spread of the middle 50% of the data. Used in box plots and outlier detection (anything beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR is a candidate outlier).

Standard error of the mean (SEM) — SD ÷ √n. The expected standard deviation of the sample mean if you took many samples. Drives the width of confidence intervals.

Population SD divides by n. Use it ONLY when your data is the entire population you care about (e.g. exam scores for every student in a single class, when the class IS the population).

Sample SD divides by n−1 (Bessel's correction). Use it when your data is a SAMPLE drawn from a larger population (e.g. survey of 100 customers from a base of 100 000). The n−1 denominator corrects for the fact that the sample mean is itself estimated from the same data — without it, sample SD systematically underestimates the true population SD.

Almost every real-world calculation uses sample SD, including every standard statistical test (t-tests, regression, ANOVA). The difference shrinks as n grows: at n = 100, sample and population SD differ by 0.5%.

Dataset: 12, 15, 14, 10, 18, 17, 16, 14, 13, 11 (n = 10). Sum = 140, mean = 14.0.

Squared deviations: (12−14)² + (15−14)² + … = 4 + 1 + 0 + 16 + 16 + 9 + 4 + 0 + 1 + 9 = 60.

Sample variance = 60 / 9 = 6.667. Sample SD = √6.667 ≈ 2.582. Population SD would be √(60/10) = √6 ≈ 2.449.

Sorted: 10, 11, 12, 13, 14, 14, 15, 16, 17, 18. Median = (14+14)/2 = 14. Mode = 14 (appears twice). Range = 18 − 10 = 8. Q1 = 12.25, Q3 = 15.75, IQR = 3.5. SEM = 2.582 / √10 ≈ 0.816.

Mixing sample and population SD between studies makes results incomparable. State explicitly which one you used.

Standard deviation assumes a roughly symmetric distribution. For skewed data (income, response times, traffic), report median + IQR instead — the SD will be misleading.

Outliers explode SD because they're squared. One extreme value can multiply SD by 2× or more. Always plot or quantile-check before reporting.

Don't confuse SD with SEM. SD measures spread of the data; SEM measures uncertainty in the estimate of the mean. SEM is always smaller than SD by a factor of √n. Bar charts that show 'mean ± SEM' look much tighter than 'mean ± SD' for the same data — neither is wrong, but they answer different questions.

Quartile definitions vary across software. This calculator uses the linear-interpolation method (R-7 / Excel.QUARTILE / NumPy default). SAS, Minitab, and TI calculators may give slightly different Q1/Q3.