How this instrument works
Dispersion (also called spread or variability) describes how far a data set's individual values scatter away from its center, and there's no single 'correct' way to measure it — different measures answer subtly different versions of the question, and each has its own strengths and blind spots. This instrument bundles five of the most commonly used dispersion measures together, each of which this site's own dedicated calculators already compute individually, gathered here under a single 'how spread out is this' question for anyone looking for the full picture at once.
Range is the simplest and crudest — just the gap between the highest and lowest value — fast to compute but entirely determined by two extreme points, ignoring everything in between. Sample variance and its square root, sample standard deviation, use every value in the data set, measuring the typical squared (and then un-squared) distance from the mean; standard deviation is the more interpretable of the two, since it's expressed in the same units as the original data rather than squared units. Coefficient of variation expresses that standard deviation as a percentage of the mean, which makes it useful for comparing spread across data sets with very different scales or units — a coefficient of variation of 10% means roughly the same relative consistency whether the underlying numbers are in the single digits or the millions.
Interquartile range (IQR), the fifth measure here, ignores the extreme top and bottom 25% of the data entirely and reports only the width of the middle 50%, making it far less sensitive to a handful of outliers than range, variance, or standard deviation, all of which respond strongly to extreme values. Reading several of these together, rather than picking just one, usually tells a fuller story: a large range with a small IQR points to a few extreme outliers sitting far from an otherwise tightly clustered core, for instance, a pattern no single measure alone reveals as clearly.
- Enter your data set into Data set, separated by commas, spaces, or new lines — at least 2 numbers are required.
- Read Range for the simple gap between the highest and lowest value.
- Read Sample variance and Sample standard deviation for the two most commonly cited spread measures, using every value in the data set.
- Read Coefficient of variation (%) to compare spread relative to the mean, useful across data sets with different scales.
- Read Interquartile range (IQR) for a spread measure that ignores the most extreme 25% at each end, less sensitive to outliers than the others.
Worked example — eight test scores, five measures at once
Entering the data set 2, 4, 4, 4, 5, 5, 7, 9 (n=8, mean=5), Range reads 7, computed as 9 − 2. Sample variance reads 4.571429 and Sample standard deviation reads 2.138090, computed as 32/7 and its square root respectively — both already verified on this site's dedicated variance and standard-deviation calculators for the same data set.
Coefficient of variation (%) reads 42.7618%, computed as (2.138090/5)×100 — the standard deviation amounts to a bit under half the mean, indicating fairly substantial relative spread for this small data set. Interquartile range (IQR) reads 1.5, computed as Q3 (5.5) minus Q1 (4.0) — noticeably tighter than the full range of 7, since the IQR ignores the low outlier (2) and high outlier (9) sitting at either extreme, focusing only on the tightly clustered middle 50% of the eight scores.
Questions
Which measure of dispersion should I actually use?
It depends on what you're trying to learn. Range is fine for a quick, rough sense of spread but is entirely driven by two extreme points; standard deviation is the most commonly used general-purpose measure, using every data point, but it's sensitive to outliers since it squares deviations before averaging; coefficient of variation is the right choice specifically when comparing spread across data sets with different units or scales; interquartile range is the best choice when a few extreme outliers shouldn't be allowed to dominate the picture. Looking at more than one together, as this instrument does, usually gives a more complete picture than any single measure alone.
Why does this calculator bundle five separate measures instead of just picking one?
Because no single dispersion measure tells the whole story on its own, and each of these five is already available individually elsewhere on this site (range, variance, standard deviation, coefficient of variation, and IQR each have their own dedicated calculator) — this page simply gathers all five together for anyone who wants the full spread picture in one pass rather than running five separate calculators on the same data set.
Why is the coefficient of variation undefined when the mean is zero?
Because CV is defined as standard deviation divided by the mean, and dividing by zero has no meaningful result — a data set whose values average to exactly zero (say, a set of temperature deviations or profit/loss figures centered on zero) can still have a perfectly well-defined standard deviation, just no meaningful CV percentage to express it as. This calculator flags that specific case rather than returning a nonsensical or undefined percentage.
Why is the sample standard deviation used here instead of the population standard deviation?
Sample standard deviation (dividing the sum of squared deviations by n−1 rather than n) is the more common default when the data on hand represents a sample drawn from some larger population you're trying to describe, which is the typical framing for a general-purpose dispersion tool like this one. If your data set already is the entire population of interest, the population version would be very slightly smaller, converging to the same value as sample size grows large.
Does a bigger range always mean more variability than a smaller one?
Not necessarily, since range depends entirely on just the two most extreme values and ignores everything else in the data set. A data set can have a large range driven by a single extreme outlier while its standard deviation and IQR both indicate a tightly clustered core — this is exactly the kind of situation where looking at multiple dispersion measures together, rather than range alone, reveals a more accurate picture of the data's actual spread.