Dispersion Calculator
Paste or type your numbers, choose population or sample mode, and get every standard measure of spread at once: range, interquartile range, quartile deviation, variance, standard deviation, coefficient of variation, standard error of the mean, and mean absolute deviation. The step-by-step panel shows every calculation with your actual values.
What are measures of dispersion?
Measures of dispersion describe how spread out the values in a dataset are around a central value. Two datasets can have identical means yet completely different shapes: one tightly clustered, one wildly scattered. Without a measure of spread you only know the center, not the shape. The eight measures this calculator provides cover the most widely used options in statistics, from the simple range to the robust mean absolute deviation, so you can choose the one best suited to your data and audience.
Range, IQR, and quartile deviation
The range is the easiest measure: maximum value minus minimum value. It is intuitive but extremely sensitive to a single outlier. The interquartile range (IQR) improves on this by focusing on the middle 50% of values, from the 25th percentile (Q1) to the 75th percentile (Q3). Because it ignores the top and bottom 25%, a single extreme value cannot distort it. The quartile deviation (also called the semi-interquartile range) is simply IQR divided by 2, giving the average spread from Q1 or Q3 to the median. These three measures are especially useful for skewed data and are the backbone of box-and-whisker plots.
Variance, standard deviation, and population vs. sample
Variance is the average squared deviation from the mean. Squaring removes negative signs and magnifies larger deviations, making it sensitive to outliers. The standard deviation is the square root of variance, which brings the measure back to the original units and is far easier to interpret alongside the mean. When your dataset is a sample drawn from a larger population, you divide the sum of squared deviations by n - 1 (Bessel's correction) instead of n to get an unbiased estimate of the true population variance. When your data is the entire population, use n. Most real-world analyses use sample mode.
Coefficient of variation, SEM, and MAD
The coefficient of variation (CV) expresses standard deviation as a percentage of the mean. This makes it possible to compare the relative variability of datasets measured on different scales, for example the consistency of two manufacturing processes producing parts of very different sizes. The standard error of the mean (SEM) is SD divided by the square root of n; it tells you how precisely the sample mean estimates the true population mean and shrinks as sample size grows. The mean absolute deviation (MAD) is the average of the absolute differences between each value and the mean. Unlike variance, it does not square the deviations, so it is less influenced by outliers and easier to interpret intuitively.
Measures of dispersion: quick comparison
| Measure | Formula | Unit | Robust to outliers? | Best used when |
|---|---|---|---|---|
| Range | Max - Min | Same as data | No | Quick first look; small samples |
| IQR | Q3 - Q1 | Same as data | Yes | Skewed data or data with outliers |
| Quartile Deviation | IQR / 2 | Same as data | Yes | Symmetric skewed distributions |
| Variance | SS / (n or n-1) | Squared units | No | Mathematical derivations; comparing groups |
| Std Deviation | sqrt(Variance) | Same as data | No | Normal distributions; everyday reporting |
| CV | (SD / mean) x 100 | Percent | No | Comparing spread across different scales |
| SEM | SD / sqrt(n) | Same as data | No | Describing precision of a sample mean |
| MAD | mean(|x - mean|) | Same as data | Yes | Data with outliers; robust alternative to SD |
All measures describe how spread out values are, but each has different uses and robustness properties.
Frequently asked questions
When should I use sample mode vs. population mode?
Use sample mode (which divides by n - 1) whenever your data is a subset drawn from a larger group and you want to estimate that group's true variance. This applies to almost all practical research, quality control, and survey analysis. Use population mode (divides by n) only when your dataset genuinely includes every member of the group you care about, for example a company's full employee records when studying that company specifically.
What is the difference between standard deviation and standard error?
Standard deviation measures how spread out individual data points are around the mean of your dataset. Standard error of the mean (SEM) measures how accurately your sample mean estimates the true population mean. SEM = SD / sqrt(n), so it always decreases as you collect more data. A large SD simply means the data varies a lot; a large SEM means your estimate of the mean is imprecise, usually because n is small.
Why is the IQR considered more robust than standard deviation?
Standard deviation uses all values in the calculation, so a single extreme outlier can inflate it dramatically. The IQR only considers the values between Q1 and Q3, completely ignoring the top 25% and bottom 25% of the data. A wildly extreme value can only shift Q1 or Q3 if it displaces enough data points to move the quartile boundary, making IQR far more resistant to distortion.
What does a low or high coefficient of variation mean?
CV below roughly 10% is generally considered low variability: values cluster tightly around the mean, suggesting a consistent or stable process. CV between 10% and 30% is moderate. Above 30% is high variability. These are rules of thumb, not universal thresholds. CV is especially useful when comparing two groups with very different means, since it normalizes spread by the scale of the data.
How is the mean absolute deviation different from standard deviation?
Both measure average deviation from the mean, but MAD takes the absolute value of each deviation instead of squaring it. Squaring in variance gives extra weight to large deviations and makes the formula tractable for calculus, but it can distort the picture when outliers are present. MAD is easier to explain to non-statisticians (it is literally the average distance from the mean) and is more robust to extreme values.
Can this calculator handle large datasets?
Yes. Paste any number of values separated by commas, spaces, or line breaks. All calculations are performed in your browser in real time. For datasets with thousands of values, there is no practical limit imposed by the calculator, though very large inputs may take a moment to process depending on your device.