Dispersion Calculator - Variance, Std Dev & IQR
Calculate the full range of statistical dispersion measures — range, variance, standard deviation, IQR, coefficient of variation, and MAD — from any numerical data set.
Enter your numbers as a comma-separated list and click Calculate to see all dispersion and central tendency statistics instantly.
Dispersion Calculator - Variance, Std Dev & IQR
Calculate the full range of statistical dispersion measures — range, variance, standard deviation, IQR, coefficient of variation, and MAD — from any numerical data set.
About the Dispersion Calculator
Statistical dispersion describes how spread out the values in a data set are. While measures of central tendency — mean, median, and mode — tell you where the centre of a distribution is, measures of dispersion tell you how much the individual data points deviate from that centre. Two data sets can share the same mean but have very different distributions; dispersion measures are what reveal this difference.
The range is the simplest measure of dispersion: the difference between the largest and smallest values. It is easy to compute and interpret, but it is sensitive to outliers because it depends entirely on the two extreme values and ignores everything in between.
Variance measures the average squared deviation from the mean. The sample variance divides the sum of squared deviations by (n−1) — using Bessel's correction to give an unbiased estimate of the population variance when working with a sample. The population variance divides by n and is appropriate only when your data set is the entire population of interest. Standard deviation is the square root of variance, expressed in the same units as the original data.
The interquartile range (IQR) measures the spread of the middle 50% of the data. It is calculated as Q3 − Q1 where Q1 is the 25th percentile and Q3 is the 75th percentile. Because it ignores the bottom and top 25% of values, the IQR is resistant to outliers and is the preferred measure of spread for skewed distributions.
The coefficient of variation (CV) expresses the sample standard deviation as a percentage of the mean, making it a dimensionless measure that allows you to compare the relative variability of data sets measured in different units or at different scales. A CV of 10% means the standard deviation is 10% of the mean — a useful benchmark for comparing measurement precision across experiments.
The median absolute deviation (MAD) is the median of the absolute deviations from the median. It is even more robust to outliers than the IQR and is used in robust statistics and anomaly detection. Like the IQR, the MAD is resistant to extreme values and provides a reliable measure of spread for data sets with heavy tails or skewed distributions.
Dispersion Calculator Examples
Three real-world data sets illustrating different dispersion patterns.
| Data Set | Key Metrics | Interpretation |
|---|---|---|
| 85, 92, 78, 88, 76, 95, 89, 72 | Mean=84.375, SD≈8.19, IQR=12.25 | Class test scores. CV≈9.71% indicates moderate relative spread. IQR of 12.25 shows the middle 50% of students scored within a 12-point band. |
| 1.2, -0.5, 2.1, 0.8, -1.9, 1.5, 2.5, -0.2, 0.3, 1.7, -1.1, 2.3 | Mean=0.725, SD≈1.40, IQR=2.075 | Monthly stock returns (%). High CV (>100%) reflects substantial volatility relative to the small positive mean return. |
| 502, 499, 505, 498, 501, 503, 497, 500 | Mean=500.625, SD≈2.67, CV≈0.53% | Product weights (g) in a quality control batch. Very low CV confirms tight manufacturing consistency around the 500 g target. |
How to Use the Dispersion Calculator
- Type or paste your data values into the text area, separated by commas, spaces, or newlines.
- Click Calculate. The calculator parses the numbers, ignoring any non-numeric tokens.
- Review the central tendency measures (count, mean, median) to understand where your data is centred.
- Check the dispersion metrics: range for overall spread, standard deviation for average deviation from the mean, and IQR for the middle-50% spread.
- Use the coefficient of variation to compare relative variability across different data sets, and the MAD for a robust outlier-resistant measure of spread.
Dispersion Calculator FAQ
When should I use the IQR instead of the standard deviation?
Use the IQR when your data is skewed, contains outliers, or comes from a non-normal distribution. The IQR only considers the middle 50% of data and is unaffected by extreme values. The standard deviation takes all values into account, so one outlier can inflate it substantially. For normally distributed data without outliers, both measures are informative.
What is the difference between sample variance and population variance?
Population variance divides the sum of squared deviations by n (the total count), appropriate when you have data for the entire population. Sample variance divides by n−1 (Bessel's correction), which corrects for the bias that arises when estimating population variance from a sample. For large samples the difference is negligible; for small samples (n < 30) it matters more.
What does the coefficient of variation tell me?
The coefficient of variation (CV) expresses the standard deviation as a percentage of the mean, giving a scale-independent measure of relative variability. A CV of 5% means the data is relatively consistent; a CV of 50% means it is highly variable relative to its average. CV is particularly useful for comparing the variability of measurements in different units, such as comparing the consistency of two different manufacturing processes.
How is the median absolute deviation (MAD) calculated?
The MAD is the median of the absolute values of the deviations from the median: MAD = median(|xi − median(x)|). It is more resistant to outliers than the standard deviation because it uses the median of deviations rather than the mean. A commonly used robust estimate of standard deviation is 1.4826 × MAD, which equals the standard deviation for a normal distribution.
Why might the mean and median differ significantly?
When the mean and median differ substantially, the distribution is skewed. A mean much larger than the median indicates right skew (a few very large values pulling the mean up). A mean much smaller than the median indicates left skew. In skewed distributions, the median is usually a better measure of central tendency than the mean, and the IQR is a better measure of spread than the standard deviation.
Can I use this calculator for very large data sets?
The calculator handles any size data set you can enter, but very large inputs may be slow to parse. For best performance, use comma-separated values on a single line or spread across lines. The calculations use numerically stable algorithms that avoid overflow and underflow for typical data ranges. If you are analysing millions of values, a dedicated statistics package such as R or Python pandas would be more efficient.