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Standard Deviation

Standard deviation is not part of the Year 10 core curriculum. Offered as preparation for senior mathematics.

Read carefully. The mean tells you the middle. The standard deviation tells you how spread out the data is around it.

Method

Mean
xˉ=∑xn\bar{x} = \frac{\sum x}{n}
Add them all, divide by how many.
Variance
variance=∑(x−xˉ)2n\text{variance} = \frac{\sum (x - \bar{x})^{2}}{n}
The average of the squared distances from the mean.
Standard deviation
σ=variance\sigma = \sqrt{\text{variance}}
The square root of the variance.
What it means
small σ⇒ data close to the mean\text{small } \sigma \Rightarrow \text{ data close to the mean}
A large $\sigma$ means the data is widely spread.

Worked example

e.g. 2; 4; 4; 4; 5; 5; 7; 92;\ 4;\ 4;\ 4;\ 5;\ 5;\ 7;\ 9
  1. Find the mean. Add them all and divide by nnxˉ=408=5\bar{x} = \tfrac{40}{8} = 5
  2. Subtract the mean from each value−3; −1; −1; −1; 0; 0; 2; 4-3;\ -1;\ -1;\ -1;\ 0;\ 0;\ 2;\ 4
  3. Square each one. This removes the minus signs9; 1; 1; 1; 0; 0; 4; 169;\ 1;\ 1;\ 1;\ 0;\ 0;\ 4;\ 16
  4. Add the squares and divide by nn. That is the variancevariance=328=4\text{variance} = \tfrac{32}{8} = 4
  5. Take the square rootσ=4=2\sigma = \sqrt{4} = 2
  6. Read it back. Most values sit within 2 of the mean of 5xˉ±σ=3 to 7\bar{x} \pm \sigma = 3 \text{ to } 7
    The eight values with the mean marked2244444455557799mean 5σ = 2, so most values fall between 3 and 7

Common mistakes

WrongForgetting to square the differences\text{Forgetting to square the differences}
Right∑(x−xˉ)2\sum (x - \bar{x})^{2}
Without squaring, the positive and negative differences cancel and you always get zero. Squaring is what makes the measure work.
WrongStopping at the variance\text{Stopping at the variance}
Rightσ=variance\sigma = \sqrt{\text{variance}}
The variance is in squared units. Taking the root brings it back to the same units as the data, so it can be compared to the mean.
WrongDividing by n−1 when the question wants n\text{Dividing by } n - 1 \text{ when the question wants } n
RightDivide by n for a whole population\text{Divide by } n \text{ for a whole population}
Use nn when you have every value. Use n−1n-1 only when your data is a sample of something larger. Check which the question means.

Practice

Find the mean, then the standard deviation:

1
4; 4; 4; 44;\ 4;\ 4;\ 4
Answerxˉ=4, σ=0\bar{x} = 4,\ \sigma = 0
2
1; 3; 5; 71;\ 3;\ 5;\ 7
Answerxˉ=4, σ=5≈2.24\bar{x} = 4,\ \sigma = \sqrt{5} \approx 2.24
3
2; 4; 62;\ 4;\ 6
Answerxˉ=4, σ=83≈1.63\bar{x} = 4,\ \sigma = \sqrt{\tfrac{8}{3}} \approx 1.63
4
10; 12; 14; 16; 1810;\ 12;\ 14;\ 16;\ 18
Answerxˉ=14, σ=8≈2.83\bar{x} = 14,\ \sigma = \sqrt{8} \approx 2.83
5
3; 3; 5; 93;\ 3;\ 5;\ 9
Answerxˉ=5, σ=6≈2.45\bar{x} = 5,\ \sigma = \sqrt{6} \approx 2.45
6
6; 8; 10; 12; 146;\ 8;\ 10;\ 12;\ 14
Answerxˉ=10, σ=8≈2.83\bar{x} = 10,\ \sigma = \sqrt{8} \approx 2.83
7
Two sets both have mean 50. One has σ=2, the other σ=20. Which is more consistent?\text{Two sets both have mean 50. One has } \sigma = 2, \text{ the other } \sigma = 20. \text{ Which is more consistent?}
AnswerThe one with σ=2—its values sit much closer to the mean\text{The one with } \sigma = 2 — \text{its values sit much closer to the mean}
8
What does σ=0 tell you?\text{What does } \sigma = 0 \text{ tell you?}
AnswerEvery value is identical, so there is no spread at all\text{Every value is identical, so there is no spread at all}
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