One table, three averages

One table can give the mean, the median and the mode.

One table

Here are 50 made-up readings, in minutes played.

The mean. The class marks x are 2.5, 7.5, 12.5, 17.5, 22.5. The f × x values are 17.5, 67.5, 150, 245 and 180. Direct: Σ f x = 660. So the mean is 660 ÷ 50 = 13.2.

Assumed mean: take a = 12.5. The d values are −10, −5, 0, 5 and 10. The f × d column adds up to 35. So 12.5 + 35 ÷ 50 = 13.2.

Step deviation: take h = 5. The u values are −2, −1, 0, 1 and 2. The f × u column adds up to 7. So 12.5 + 5 × (7 ÷ 50) = 13.2.

The median. The totals climb 7, 16, 28, 42, 50. Half of 50 is 25, first reached in 10-15. So l = 10, cf = 16, f = 12 and h = 5. The formula gives 10 + ((25 − 16) ÷ 12) × 5 = 13.75. Step by step: 25 − 16 = 9. Then 9 ÷ 12 = 0.75. Then 0.75 × 5 = 3.75.

The mode. The tallest class is 15-20. So l = 15 and h = 5. Here f₁ = 14 is its frequency. f₀ = 12 is the class before it. f₂ = 8 is the class after it. The formula gives 15 + ((14 − 12) ÷ (28 − 12 − 8)) × 5 = 16.25. Step by step: 14 − 12 = 2, and 28 − 12 − 8 = 8. Then 2 ÷ 8 = 0.25. Then 0.25 × 5 = 1.25.

One table answers all three. Mean: sum of f × mark, over n. Median: count to n ÷ 2. Mode: read the tallest row and its neighbours.

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Try the lab to finish this lesson.