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Standard deviation of a list and of a simple frequency table
Mathematics Prep 3 Term 1, Unit 4, Lesson 1
GoalCalculate σ for a list and for a simple frequency table.
New words
mean · the sum of the values divided by their number, x̄deviation · a value minus the mean, x − x̄standard deviation · σ = √(Σ(x − x̄)² ÷ n): the typical distance from the meanfrequency · f: how many times a value occurs
Explanation
The standard deviation σ shows how far values typically lie from the mean. Steps: (1) find the mean x̄; (2) find each deviation x − x̄; (3) square the deviations (the plain deviations always add to 0, so we must square them); (4) add the squares and divide by n; (5) take the square root. Example 4, 6, 8, 10, 12: mean 8, squares 16, 4, 0, 4, 16 add to 40, 40 ÷ 5 = 8, σ = √8 ≈ 2.83. For a frequency table we weight each square by its frequency: σ = √(Σ f(x − x̄)² ÷ Σ f). The unit of σ is the unit of the data. If all values are equal, σ = 0; adding the same number to every value does not change σ.
| x | x − 8 | (x − 8)² |
|---|---|---|
| 4 | −4 | 16 |
| 6 | −2 | 4 |
| 8 | 0 | 0 |
| 10 | 2 | 4 |
| 12 | 4 | 16 |
| Sum | 0 | 40 |
Worked examples
Data 3, 5, 6, 9, 12: mean = 35 ÷ 5 = 7; deviations −4, −2, −1, 2, 5; squares 16, 4, 1, 4, 25 add to 50; 50 ÷ 5 = 10; σ = √10 ≈ 3.16.
| x | f | x − mean | (x − mean)² | f × (x − mean)² |
|---|---|---|---|---|
| 1 | 2 | −1.4 | 1.96 | 3.92 |
| 2 | 3 | −0.4 | 0.16 | 0.48 |
| 3 | 4 | 0.6 | 0.36 | 1.44 |
| 4 | 1 | 1.6 | 2.56 | 2.56 |
| Sum | 10 | 8.4 |
Add 10 to every value of 4, 6, 8, 10, 12. The new data 14, 16, 18, 20, 22 have mean 18 and the same deviations, so σ is still √8.
Class activity
Wobbly darts. Two friends throw 5 darts each at a target (invented distances in cm from the centre). Calculate each σ to see who is steadier.
Practice
1
Find σ for 3, 5, 5, 7, 10. (The mean is 6.)
Squares 9, 1, 1, 1, 16 add to 28; 28 ÷ 5 = 5.6; σ = √5.6 ≈ 2.37.
2
Find σ for the scores x = 5, 6, 7, 8 with f = 1, 3, 4, 2.
Mean 6.7; Σ f(x − x̄)² = 8.1; σ = √0.81 = 0.9.
3
What is σ for 9, 9, 9, 9?
All deviations are 0, so σ = 0.
4
The data 2, 4, 6 have σ = √(8/3). What is σ of 12, 14, 16?
The same, √(8/3): adding 10 does not change σ.
5
Heights of seedlings in cm: 8, 10, 12. What unit has σ?
Centimetres, like the data.