When should I report the median instead of the mean?

When the data are skewed or have outliers, which is common for length of stay, wait times and costs. In the example, one 21-day stay pulls the mean to 4.67 days while the median stays at 3. Without that stay the mean would be 3.5.

Sample or population standard deviation?

Use the sample standard deviation (divide by n − 1) when your data are a sample used to describe a larger process, which is almost always the case in quality work. The population version (divide by n) applies only when you have every member of the group and no wider inference is intended.

How are the quartiles calculated?

By linear interpolation between ranked values, the same method as Excel’s QUARTILE.INC. Other software may use a slightly different method, so small differences in Q1 and Q3 are normal.

Free tool · CPHQ® exam topic

Descriptive statistics calculator

Paste a list of numbers to get measures of central tendency and spread: mean, median, mode, standard deviation, quartiles and more.

Result

Count (n): 15
Mean: 4.667
Median: 3
Mode: 3
Standard deviation (sample): 4.67
Variance (sample): 21.81
Minimum: 2
Maximum: 21
Range: 19
Q1 (25th percentile): 3
Q3 (75th percentile): 4.5
Interquartile range: 1.5
Coefficient of variation: 100.1%
Standard deviation (population): 4.512

Formulas

Mean = sum of the values ÷ n.
Median = the middle value of the sorted data (the average of the two middle values when n is even).
Mode = the most frequent value; there can be several, or none if every value occurs once.
Range = maximum − minimum.
Sample variance s² = Σ(x − x̄)² ÷ (n − 1); standard deviation s = √s².
Interquartile range = Q3 − Q1, the spread of the middle half of the data.
Coefficient of variation = s ÷ x̄, spread relative to the mean, handy for comparing measures on different scales.

Worked example

Fifteen discharges have lengths of stay of 2, 3, 3, 4, 2, 5, 3, 6, 4, 3, 2, 4, 3, 5, 21 days. The mean is 4.67 days, but the median is 3 and the mode is 3: the single 21-day stay drags the mean up. The standard deviation is 4.67 days, while the interquartile range is just 1.5.

That is why skewed measures like length of stay are usually summarized with the median and IQR. To see whether a measure is changing over time, plot it on a control chart rather than comparing two averages.

Test yourself: a CPHQ-style question

Which measure of central tendency is MOST resistant to the effect of extreme outliers?

Mean
Mode
Median
Range

Show answer

C. Median

The median (the middle value when data are ordered) is not pulled by extreme values, so it represents the typical case better when a dataset contains outliers, as healthcare data such as length of stay often do. Range is a measure of spread, not central tendency.

Questions like this come up in the Health Data Analytics domain of the CPHQ® exam. The 249 questions in the app each come with a written explanation.

FAQ

When should I report the median instead of the mean?: When the data are skewed or have outliers, which is common for length of stay, wait times and costs. In the example, one 21-day stay pulls the mean to 4.67 days while the median stays at 3. Without that stay the mean would be 3.5.
Sample or population standard deviation?: Use the sample standard deviation (divide by n − 1) when your data are a sample used to describe a larger process, which is almost always the case in quality work. The population version (divide by n) applies only when you have every member of the group and no wider inference is intended.
How are the quartiles calculated?: By linear interpolation between ranked values, the same method as Excel’s QUARTILE.INC. Other software may use a slightly different method, so small differences in Q1 and Q3 are normal.

Sources

Checked against these sources on October 29, 2026. For education and CPHQ® exam preparation only, not a substitute for your organization’s policies, official measure specifications or professional judgment. Calculations run in your browser; nothing you enter is sent anywhere.