Median, Mode, and Range: Completing the Statistical Picture
While the mean is the most recognised average, the median and mode often tell a more honest story — particularly for skewed data. Range measures spread rather than centre, rounding out a basic descriptive statistics toolkit.
Median: Step-by-Step
1. Sort data ascending
2. Odd count: middle value
3. Even count: mean of two middle values
Odd: [3, 5, 7, 9, 11] → Median = 7
Even: [3, 5, 7, 9] → Median = (5+7)/2 = 6Mode: Most Frequent Value
Unimodal: [2, 3, 3, 4, 5] → Mode = 3
Bimodal: [1, 2, 2, 3, 4, 4] → Modes = 2 and 4
No mode: [1, 2, 3, 4] → no value repeatsRange
Range = Max − Min
[12, 7, 45, 3, 21] → Range = 45 − 3 = 42Why Median Beats Mean for Income Data
If 9 people earn £30,000 and 1 earns £1,000,000, the mean is £127,000 — a figure that represents no-one's experience. The median (£30,000) accurately reflects the typical worker's reality. This is why economists report median household income, not mean.
Calculate statistics: Free Median Calculator | Free Mode Calculator
Calculating Each Measure
- Median: Sort the data. If odd count n: middle value at position (n+1)/2. If even count: average of values at positions n/2 and n/2 + 1. Example: {3,7,7,9,12} — median = 7 (3rd of 5 values).
- Mode: The most frequent value. {3,7,7,9,12} — mode = 7. Multiple modes: {2,4,4,7,7,9} is bimodal (4 and 7). No mode: {1,2,3,4,5}.
- Range: Maximum − minimum. {3,7,7,9,12} — range = 12 − 3 = 9.
Limitations and Context
Range is a crude dispersion measure — it uses only two values and ignores the rest. A dataset {1, 5, 5, 5, 9} has range 8 and is tightly clustered around 5; {1, 3, 5, 7, 9} also has range 8 but is evenly spread. Standard deviation is a more informative measure of spread. The interquartile range (IQR = Q3 − Q1) is a more robust alternative to range because it excludes the extreme 25% of values at each end. Mode is most useful for categorical data (favourite colour, most popular product) where mean and median are not meaningful.
Frequently Asked Questions
When is the median more useful than the mean?
When data is skewed or contains outliers. A salary dataset where most employees earn £30,000–£50,000 but one executive earns £2,000,000 will have a mean of perhaps £80,000 — not representative of a typical salary. The median might be £38,000, which better reflects the majority. House prices, incomes, and wait times are typically reported as medians for this reason.
What does a large range indicate?
A large range indicates high variability — the data spans a wide interval. In manufacturing quality control, a large range means inconsistent output. In finance, a large range for daily stock prices indicates high volatility. However, range alone cannot distinguish between a dataset where values are uniformly spread and one where they are clustered at the extremes.
Can the median equal the mean?
Yes, when data is symmetrically distributed. In a perfectly normal (bell curve) distribution, mean = median = mode. As data becomes skewed, the three measures diverge: positively skewed data (long right tail) has mean > median > mode; negatively skewed data has the reverse order.