Three ways to describe the middle
"Average" usually means the mean, but it is only one way to summarize a set of numbers. The mean uses every value and is what you need for totals — a mean of $40 a day over 30 days is $1,200. The median ignores how extreme the ends are, which makes it the honest "typical" value for skewed data like house prices. The mode is the only average that works for categories (the most common shoe size), and it shows where values bunch up.
The formulas
Mean = (x₁ + x₂ + … + xₙ) / n Median = middle value of the sorted list (average of the two middle values if n is even) Mode = most frequent value(s) Range = maximum − minimum Geometric mean = (x₁ × x₂ × … × xₙ)^(1/n) Harmonic mean = n / Σ(1/x)
Worked example
Data: 3, 7, 7, 2, 9, 4, 7, 5. Sorted: 2, 3, 4, 5, 7, 7, 7, 9 (n = 8, sum = 44).
Mean = 44 / 8 = 5.5. The two middle values are 5 and 7, so the median = 6. 7 appears three times, so the mode = 7. Range = 9 − 2 = 7.
Lower half 2, 3, 4, 5 → Q1 = 3.5; upper half 7, 7, 7, 9 → Q3 = 7; IQR = 3.5.
Notes
- Geometric and harmonic means are only defined here for all-positive data; they are used for growth rates and for averaging rates such as speeds.
- Spreadsheet quartile functions use interpolation methods that can give slightly different Q1 and Q3 for small data sets.
- Results are rounded to six decimal places.
Sources: NIST/SEMATECH, e-Handbook of Statistical Methods, §1.3.5.1 Measures of Location; OpenStax, Introductory Statistics 2e, §2.5 Measures of the Center of the Data.
Frequently asked questions
What is the difference between mean, median and mode?
The mean is the arithmetic average: add everything and divide by how many there are. The median is the middle value once the numbers are sorted. The mode is the value that occurs most often. In 2, 3, 4, 5, 7, 7, 7, 9 the mean is 5.5, the median 6 and the mode 7.
When is the median better than the mean?
When the data is skewed or has outliers. Household income is the classic case: a few very high incomes pull the mean up, so the median is the better "typical" figure, which is why the Census Bureau reports median household income.
What if there are two middle numbers?
With an even count, the median is the average of the two middle values. For 2, 3, 4, 5, 7, 7, 7, 9 the middle pair is 5 and 7, so the median is 6.
Can there be more than one mode, or none?
Yes. If two values tie for most frequent, the data is bimodal and both are modes. If every value appears once, there is no mode, and the calculator says so.
What are the quartiles and IQR?
Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half (the method taught in most U.S. schools; for an odd count the overall median is left out of both halves). The interquartile range, Q3 − Q1, is the spread of the middle 50% and is not affected by outliers.