1. Variance and Standard Deviation
GCSE Statistics
Variance and Standard Deviation

2. We are back to looking at measures of dispersion (spread)

4. Advantages and disadvantages of measures of dispersion
To increase our level of accuracy and sophistication further, it is important that we try to include all the pieces of data in our calculation. This leads us to variance and standard deviation which are based on the mean.
Advantages and disadvantages of measures of dispersion
Book page 148
Measure
Advantages
Disadvantages
Range
A reasonably good indicator
Badly affected by extreme values
Inter-quartile range
Not affected by extreme values
Often used with skewed data
Does not tell you what happens beyond the quartiles
Variance
Good measure
All values used
Used when data are fairly symmetrical
Mathematical properties not useful (SD preferred)
Not so good if data are strongly skewed
Standard deviation
Can be used in mathematical calculations of other statistics

5. Var(x) = Σ(𝑥 − 𝑥) ² 𝑛 = Σ 𝑥 2 𝑛 − 𝑥 2
The deviation or dispersion of an observation, x, from the mean , 𝑥 is given by x - 𝑥
Variance will measure the total dispersion of n items
Var(x) = Σ(𝑥 − 𝑥) ² 𝑛 = Σ 𝑥 2 𝑛 − 𝑥 2
Page 145 in the text and on the formulae sheet in the exam

6. Standard deviation (SD) = Σ(𝑥 − 𝑥) ² 𝑛 = Σ 𝑥 2 𝑛 − 𝑥 2
Standard deviation is the square root of variance
It is the average distance each piece of data is from the mean
Standard deviation (SD) = Σ(𝑥 − 𝑥) ² 𝑛 = Σ 𝑥 2 𝑛 − 𝑥 2
Page 146 in the text and on the formulae sheet in the exam

7. Using the first formulae for Standard deviation (SD) = Σ(𝑥 − 𝑥) ² 𝑛

9. Using the easier formulae standard deviation (SD) = Σ 𝑥 2 𝑛 − 𝑥 2

11. Standard deviation (SD) = Σ𝑓(𝑥 − 𝑥) ² Σ𝑓
And now for data in a frequency table
This time your formula looks like this:
Standard deviation (SD) = Σ𝑓(𝑥 − 𝑥) ² Σ𝑓
Or the easier version
Standard deviation (SD) = Σ 𝑓𝑥 2 Σ𝑓 − Σ𝑓𝑥 Σ𝑓 ²

12. So this time we draw a slightly different table
To use the easy version of the formulae our table would look like this

14. Your turn Exercise 4I page 149 Other sources of material
10 Ticks level 9 pack 3 (the notes come from here)