Standard deviation with class intervals, and which group is more consistent
Mathematics Prep 3 Term 1, Unit 4, Lesson 1
GoalCalculate σ from a table with class intervals and compare the consistency of two groups.
New words
class interval · a range of values such as 10– (10 up to, but not including, 20)class centre · the middle of the class; it stands for every value in itconsistent · values close together, with a small σestimate · an approximate answer; with classes we do not know the exact values
Explanation
When the data are in class intervals 10–, 20–, 30–, 40– we do not know each value, so we use the class centre: 15, 25, 35, 45. Then work as before with frequencies. Example (waiting minutes of 20 customers): f = 4, 6, 8, 2. Mean = (4×15 + 6×25 + 8×35 + 2×45) ÷ 20 = 580 ÷ 20 = 29. The table gives Σ f(x − 29)² = 1680. So σ = √(1680 ÷ 20) = √84 ≈ 9.17 minutes. This is an estimate, because the centre stands for the whole class. To compare two groups with the same mean, the one with the smaller σ is more consistent. A bus company with mean waiting 29 minutes and σ = 9.17 is less predictable than one with mean 29 and σ = 3.
Class
Centre x
f
f × (x − 29)²
10–
15
4
784
20–
25
6
96
30–
35
8
288
40–
45
2
512
Sum
20
1680
Worked examples
Same table: mean 29, Σ f(x − x̄)² = 1680, n = 20, so σ² = 84 and σ ≈ 9.17 minutes.
Two teams scored the same mean of 70 points per game. Team A has σ = 4.2 and team B has σ = 9.1. Team A is more consistent, because 4.2 < 9.1.
A shop’s daily sales in hundreds of LE are grouped 0–, 10–, 20–, 30– with f = 2, 5, 8, 5. Centres 5, 15, 25, 35; mean = 460 ÷ 20 = 23; Σ f(x − 23)² = 1720; σ = √86 ≈ 9.27 (hundred LE).
Class activity
Compare two classes. Invent the marks of two groups of 5 pupils with the same mean but different spread. Calculate both σ and say which group is more consistent.
Practice
1
Centres of the classes 0–, 10–, 20–, 30– are what numbers?
5, 15, 25, 35.
2
Classes 0–, 10–, 20–, 30– with f = 2, 5, 8, 5. Find the mean.
(10 + 75 + 200 + 175) ÷ 20 = 23.
3
Same data: Σ f(x − 23)² = 1720. Find σ.
√(1720 ÷ 20) = √86 ≈ 9.27.
4
Class A: mean 15 kg, σ = 2.5 kg. Class B: mean 15 kg, σ = 4 kg. Which is more consistent?
Class A, the smaller σ.
5
Why is a σ calculated from classes only an estimate?
Because the centre stands for every value in the class.
