All calculators

Every calculator here runs in your browser and shows its working. Pick by what you have: a list of numbers, a frequency table, several groups, or summary figures from a published result.

From a list of numbers

• Standard deviation

  Sample or population σ from a list of numbers, with a dot plot and full working.

• Variance

  The squared spread, before the final square root. Sample and population.

• Relative SD (%RSD)

  Spread as a percentage of the mean. The laboratory precision measure.

• Coefficient of variation

  The same quantity as %RSD, reported as a ratio.

• Standard error

  How precisely the mean is known, and recovery of SD from a published SE.

• Mean, median, mode, range

  The full descriptive summary, with every tied mode reported.

• Average (mean)

  The arithmetic mean on its own, with the sum and count that produced it.

• Median

  The middle value, sorted and shown — the average that ignores outliers.

• Sum of squares

  Σ(x − x̄)² by both formulas, with the variance and SD that follow from it.

• Root mean square

  RMS of any list, the sum and mean of squares, and how RMS relates to the SD.

Spread, shape and position

• Standard deviation graph

  Histogram of your data with the fitted normal curve and the ±1, ±2 and ±3 SD bands shaded.

• Histogram maker

  Equal-width bins by Sturges’ rule, counts on every bar and a normal curve fitted to your data.

• IQR & outliers

  Quartiles, interquartile range and Tukey outlier fences, with a box plot.

• Five-number summary

  Min, Q1, median, Q3 and max, drawn as a box and whisker plot.

• Percentiles

  Any percentile from your data, plus the standard-deviation conversion table.

• Mean absolute deviation

  Spread without squaring, compared against the standard deviation.

From tables, groups or weights

• Grouped data SD

  From class midpoints and frequencies, when the raw values are gone.

• Pooled SD

  One spread estimate across two or more groups, weighted by degrees of freedom.

• Weighted SD

  When values carry different weights — allocations, survey weights, precision.

Distributions

• Normal distribution

  Probability between any two values, with the area shaded on the curve.

• Binomial distribution

  Mean, variance and σ = √(npq), with the full distribution plotted.

• Empirical rule

  The 68–95–99.7 bands, with their exact coverages.

• Bell curve generator

  Draw and label a normal curve from any mean and standard deviation.

• Poisson distribution

  P(X = k) and cumulative probabilities for a rate λ, with σ = √λ and the mass function plotted.

Inference

• Z-score

  How many standard deviations a value sits from the mean, with percentile.

• Confidence interval

  An interval for a mean, using the t distribution.

• T-test

  One-sample, two-sample or paired, with Cohen’s d and an interval.

• ANOVA

  Three or more group means at once, with the full sum-of-squares table.

• Chi-square

  Goodness of fit on counts, with each category’s contribution.

• Critical value

  Critical values for z, t, chi-square and F at any α, one- or two-tailed, with the rejection region.

• P-value

  Convert a z, t, chi-square or F statistic into a p value.

• Correlation coefficient

  Pearson’s r and r², with Spearman’s rho and a scatter plot.

• Linear regression

  The least-squares line, its slope, and how much it explains.

Applied: finance, quality and planning

• Portfolio SD

  Two-asset portfolio risk from weights, SDs and correlation, with the diversification benefit shown.

• Cpk & process capability

  Cp, Cpk, Cpu and Cpl from a mean, SD and spec limits, with expected ppm out of spec and sigma level.

• Sample size

  Minimum sample for a survey proportion or a mean, from the margin of error and confidence level.

Reference and guides

• What is standard deviation?

  What the number means, its units, and how to read high, low and zero.

• How to calculate it

  Six steps on real numbers, both formulas worked in full, and practice problems.

• SD vs variance

  The square root that separates them, and when each one is the right report.

• The formula

  Both formulas taken apart term by term, and why each step is there.

• Symbols and notation

  σ, s, μ, x̄, Σ — what each means and how to type it.

• Sample vs population

  Which denominator to use, and how much it changes the answer.

• Standard deviation in Excel

  STDEV.S, STDEV.P, conditional SD, and the errors that catch people.

• Standard deviation on a TI-84

  Keystrokes, Sx versus σx, and frequency lists.

• Standard deviation in Python

  NumPy, pandas and the statistics module, and the ddof trap.

• Standard deviation in R

  sd(), var(), population SD, NA handling and by-group.

• SD in psychology

  IQ and T-score scales, APA reporting and Cohen’s d.

Written and reviewed by our editorial team. Last updated September 5, 2026. Method and sources: how these numbers are computed.