By Karen G Blaettler | Updated Aug 30, 2022
Master the core statistics that let you summarize and compare data sets with confidence. This guide walks you through the formulas, calculations, and interpretation of mean, median, mode, range, and standard deviation.
The mean is the arithmetic average of a data set. It reflects the central tendency of the values.
Mean = Σx / n
Data set: 20, 24, 25, 36, 25, 22, 23
Sum: 20+24+25+36+25+22+23 = 175
Number of values (n): 7
Mean: 175 ÷ 7 = 25
The median is the middle value when the data are ordered from lowest to highest. It is robust to outliers.
Ordered set: 20, 22, 23, 24, 25, 25, 36
With 7 values, the median is the 4th value: 24.
For an even number of values, average the two middle numbers. Example: 22, 23, 25, 26 → (23+25)/2 = 24.
The mode is the value(s) that appear most frequently. A data set can be unimodal, multimodal, or have no mode.
In the example, 25 appears twice while all others appear once. Mode = 25.
Other scenarios:
The range measures spread by subtracting the smallest value from the largest.
Minimum: 20, Maximum: 36
Range = 36 – 20 = 16
A large range often signals an outlier; in this set, 36 stands out.
Standard deviation quantifies how much the values deviate from the mean. Smaller values indicate tighter clustering.
SD = √(Σ(xᵢ – μ)² / (n – 1))
Values within ±1 SD of the mean (20–30) are typical. Values beyond ±2 SDs (≈10–40) are extreme; 36 exceeds two SDs, flagging it as an outlier.
By mastering these measures, you can describe, compare, and interpret data sets with authority and precision.
