Basic statistics calculator

Center: mean, median, mode

The mean is the arithmetic average. It uses every value, which makes it sensitive to extreme ones. The median is the middle value when the data are sorted, or the average of the two middle values when there is an even number. It ignores how far out the extremes are. The mode is the most frequent value, and is most useful for counts and categories; continuous measurements often have no repeated value at all.

When the mean and median are close, the data are roughly symmetric. When the mean is pulled well above the median, the data have a long tail to the right, and the median is the better description of a typical value. The calculator says which.

Spread: range, standard deviation, IQR

The range is the largest value minus the smallest. It is easy to compute and depends entirely on the two most extreme values. The standard deviation uses every value: roughly, the typical distance of a value from the mean. The interquartile range is the spread of the middle half of the data, from the first quartile (Q1) to the third (Q3), and like the median it is not moved by extremes.

Sample or population: n − 1 or n

The sample standard deviation, s, divides the sum of squared deviations by n − 1. The population standard deviation, σ, divides by n. Use s when your data are a sample from a larger process, which in quality work is almost always: thirty parts measured from a run of thousands. Dividing by n − 1 corrects for the fact that a sample's values sit closer to their own mean than to the true process mean. Spreadsheets give both: STDEV.S is s, STDEV.P is σ.

How the quartiles are calculated

There are several conventions for quartiles, and they give slightly different answers on small data sets. This calculator uses linear interpolation between sorted values, the same method as Excel's QUARTILE.INC. Values more than 1.5 × IQR beyond the quartiles are flagged as possible outliers. Find out what happened to them before deleting anything; an outlier is often the most useful value in the set.

On the Yellow Belt exam

Basic statistics is topic III.A of the 2022 CSSYB Body of Knowledge. Expect to calculate a mean, median, mode, range and standard deviation by hand from a short list, and to say which measure of center or spread suits a given situation.

Common questions

Should I use the sample or the population standard deviation?

Use the sample standard deviation (dividing by n − 1) when your data are a sample used to estimate a process, which is nearly always the case in quality work. Use the population version (dividing by n) only when you have every value there is and are not estimating anything beyond it.

When is the median better than the mean?

When the data are skewed or have outliers. Cycle times, flatness, runout and repair costs usually have a long tail that drags the mean; the median stays with the typical value. For roughly symmetric data the mean is the better estimate.

What is the interquartile range?

The IQR is Q3 − Q1, the spread of the middle half of the data. Unlike the range, a single extreme value barely moves it. It sets the box in a box plot, and values more than 1.5 × IQR beyond the quartiles are commonly flagged as possible outliers.

Why do my quartiles differ from Excel's?

There are several accepted ways to calculate quartiles, and with small samples they give slightly different answers. This page states the method it uses; the difference shrinks as the sample grows.

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