Category
Descriptive Statistics: Summarising a Column of Numbers
Before any test, describe the data. These tools summarise a column of numbers and show you its shape, which is what decides which test is appropriate in the first place.
Centre, spread, shape — in that order
Every summary of a data set answers three questions. Where is it? — the mean, median and mode. How spread out is it? — the standard deviation, variance, range and mean absolute deviation, or the interquartile range. What shape is it? — skewness, kurtosis, and whether there are outliers.
The three interact. On symmetric data the mean and median agree and either will do. On skewed data they separate, and the gap is informative: mean income exceeds median income everywhere in the world, because a long right tail pulls the mean and leaves the median where it is.
Why the mean and the median can tell opposite stories
The mean uses every value's magnitude; the median uses only their order. That makes the mean efficient when the data are well behaved and misleading when they are not. One mistyped value — a weight of 700 kg instead of 70 — moves the mean of a sample of fifty by more than 12 kg and moves the median by nothing at all.
The same split runs through the spread measures. The standard deviation squares every deviation, so an outlier dominates it; the median absolute deviation does not. When a robust and a non-robust summary disagree sharply, that disagreement is a finding: something extreme is in the data.
Sample or population?
Nearly every formula here has two versions. The population version divides by n; the sample version divides by n − 1, because deviations measured from the sample mean are systematically too small. Use the sample version unless your data genuinely are the entire population — every employee, every transaction, every item — rather than a sample drawn from one. This site defaults to the sample version and shows both.
Then look at it
Numbers summarise; pictures reveal. Two data sets can share a mean, a standard deviation and a correlation and look nothing alike. The box plot, histogram and stem-and-leaf plot makers all show the distribution these numbers came from, and each calculator here draws one automatically.
The 17 calculators in this category
- Percentage Change CalculatorCalculate percentage change between two values, or across a whole series, with the reverse change, the multiplier, percentage difference and the percentage-points distinction spelled out.
- Standard Deviation CalculatorCalculate sample or population standard deviation from any data set, with the mean, the deviations, the sum of squares, the variance and a histogram of the distribution.
- Z-Score CalculatorConvert a value to a z-score and a percentile, or z-score a whole column.
- Weighted Average CalculatorCompute a weighted mean from any list of values and weights, with each item's contribution shown, the unweighted mean for comparison and a chart of the weights.
- Percentile and Quartile CalculatorFind any percentile or quartile of a data set, or the percentile rank of a value, with all four common quantile methods compared side by side so results can be reproduced.
- Variance CalculatorCompute sample and population variance from raw data, with every deviation and squared deviation shown, the standard deviation, and a histogram of the distribution.
- Mean, Median and Mode CalculatorFind the mean, median, mode, range and standard deviation of any data set, with the frequency table, the sorted values and a histogram showing why the three centres differ.
- Confidence Interval CalculatorConfidence interval for a mean or a proportion, from raw data or summary statistics.
- Geometric Mean CalculatorCompute the geometric mean of a set of values, or the compound growth rate between two figures, with the arithmetic and harmonic means shown for comparison.
- Coefficient of Variation CalculatorCalculate the coefficient of variation (relative standard deviation) from raw data or from a mean and standard deviation, with the interpretation and the limits explained.
- Range and Mean Absolute Deviation CalculatorCompute the range, interquartile range, mean absolute deviation and median absolute deviation together, with a table showing which measures survive outliers and which do not.
- Outlier CalculatorFind outliers with Tukey's IQR fences, the z-score rule or the robust modified z-score, with the fences shown, the flagged values listed and the summary with and without them.
- Five-Number Summary CalculatorCompute the five-number summary of any data set with the interquartile range, the outlier fences and the box plot it describes.
- Descriptive Statistics CalculatorPaste a column and get mean, median, mode, variance, standard deviation, quartiles, IQR, skewness, kurtosis, standard error and more — with a histogram and a box plot.
- Frequency Table CalculatorBuild a frequency table from raw data: counts, relative frequencies, cumulative counts and percentages, ungrouped or grouped into classes with midpoints.
- Margin of Error CalculatorCompute the margin of error for a survey proportion or a mean, with the finite population correction, a table of other confidence levels and the sample-size curve.
- Standard Error CalculatorCompute the standard error of the mean from data or from n and SD, or the standard error of a proportion, with the 1/√n curve showing what more data buys.
Questions people ask
Should I divide by n or n − 1?
By n − 1 when the data are a sample and you want to estimate the population variance — that is Bessel's correction, and it exists because deviations from the sample mean are too small on average. Divide by n only when the data are the whole population. Every variance and standard deviation result here reports both.
Why does my calculator give a different quartile from a spreadsheet?
Because there are at least four accepted definitions and they genuinely disagree on small samples. Excel's PERCENTILE.INC and R's default use linear interpolation; PERCENTILE.EXC and R's type 6 use a different position; box plots traditionally use Tukey's hinges. The percentile calculator shows all four side by side and names the one it used.
What counts as an outlier?
There is no universal answer — only rules. The box-plot rule flags anything beyond 1.5 × IQR from the quartiles; the z-score rule flags |z| > 3; the modified z-score uses the median and is the most reliable on small samples. The outlier calculator applies all three, and none of them tells you whether to delete the point. That needs a reason from outside the data.