Statistical Skills: Mean, Median, Mode, Range and IQR

🧮 Statistical Skills: Mean, Median, Mode, Range and IQR

Geography at GCSE level requires you to calculate and interpret basic statistical measures. More importantly, you need to know WHICH measure to use and WHY — because choosing the wrong average can give a misleading picture of a dataset.

The Three Averages

Measure	How to Calculate	When to Use It	Weakness
Mean	Add all values, divide by number of values	When data is roughly evenly distributed with no extreme outliers	Distorted by outliers — a single very large or very small value pulls the mean towards it
Median	Order values from smallest to largest; find the middle value (or average of two middle values if even count)	When data has outliers; income and wealth data; house prices	Ignores the actual values at the extremes — misses the magnitude of inequality
Mode	Most frequently occurring value	Non-numerical categories (e.g., most common land use type); when the "typical" response matters more than the average	A dataset can have no mode (all values different) or multiple modes; tells you nothing about spread

Why Median Beats Mean for Income Data

This distinction carries marks in exams and illustrates a genuinely important principle. Consider a street with 9 households earning £25,000 per year and one household earning £1,000,000 per year.

• Mean income: (9 × £25,000 + £1,000,000) ÷ 10 = £1,225,000 ÷ 10 = £122,500
• Median income: The 5th value in the ordered list is £25,000

The mean says the "average" person on this street earns £122,500. The median says they earn £25,000. The median is far more representative of the actual experience of people on this street — the mean has been pulled upward by a single outlier. This is why economists and geographers use median household income, not mean household income, when analysing inequality and living standards.

Measures of Spread: Range and Interquartile Range

Averages tell you about the centre of a dataset. Measures of spread tell you how much variation there is around that centre — how consistent or variable the data is.

Range = Highest value − Lowest value. Simple to calculate, but vulnerable to the same problem as the mean: a single extreme outlier massively increases the range even if most values cluster tightly together.

Interquartile Range (IQR) = Q3 − Q1, where Q1 is the lower quartile (25th percentile) and Q3 is the upper quartile (75th percentile). The IQR measures the spread of the middle 50% of data, ignoring the extreme top and bottom 25%. It is much more resistant to outliers than the range and gives a better picture of how spread out the "typical" data values are.

Worked Example: River Discharge Data

Monthly mean discharge (m³/s) for a river: 12, 15, 22, 31, 45, 58, 62, 55, 38, 28, 19, 14

• Mean: (12+15+22+31+45+58+62+55+38+28+19+14) ÷ 12 = 399 ÷ 12 = 33.25 m³/s
• Median: Order the values: 12, 14, 15, 19, 22, 28, 31, 38, 45, 55, 58, 62. The two middle values (6th and 7th) are 28 and 31. Median = (28+31) ÷ 2 = 29.5 m³/s
• Range: 62 − 12 = 50 m³/s
• Q1 (lower quartile — median of lower half): median of 12, 14, 15, 19, 22, 28 = (15+19)/2 = 17 m³/s
• Q3 (upper quartile — median of upper half): median of 31, 38, 45, 55, 58, 62 = (45+55)/2 = 50 m³/s
• IQR: 50 − 17 = 33 m³/s

Interpretation: The river's discharge has a mean of 33.25 m³/s but peaks at 62 m³/s in summer (July) and falls to 12 m³/s in winter (January). The high range of 50 m³/s indicates significant seasonal variability — this has implications for flood risk management and water supply planning. The IQR of 33 m³/s shows that even the "middle 50%" of monthly discharges vary by 33 m³/s, confirming that this is not a stable, predictable river system.

Percentage Change

Percentage change allows fair comparison between values of different original sizes — it standardises the change as a proportion of the starting point.

Percentage change formula: ((New value − Original value) ÷ Original value) × 100

Example: A city's population increases from 2.4 million to 3.1 million over 20 years.

Percentage change = ((3.1 − 2.4) ÷ 2.4) × 100 = (0.7 ÷ 2.4) × 100 = +29.2%

Why percentage change matters: If City A grows from 100,000 to 150,000 people (+50,000) and City B grows from 2,000,000 to 2,050,000 (+50,000), City A has grown by 50% while City B has grown by only 2.5%. The absolute change is the same, but the relative change is dramatically different. Percentage change captures this difference; raw change does not.