Measures of Dispersion

Week 4

2

Dispersion

• Two groups of three students

Group 1 4 7 10

Group 2 7 7 7

• Mean mark

Group 1 4 + 7 + 10 = 21/3 = 7

Group 2 7 + 7 + 7 = 21/3 = 7

• Same mean mark, but Group 1’s marks are widely spread, Group 2’s are all the same

• The following diagram reinforces this point

3

4

Range

• The absolute difference between the

highest and lowest value of the raw data

• Group of students 4 7 10

• Range = Maximum – Minimum

10 – 4 = 6

5

Interquartile Range

• This is the absolute difference between

the upper and lower quartiles of the

distribution.

• Interquartile Range =

Upper Quartile - Lower Quartile

• See next powerpoints for estimating

quartiles

6

Quartiles (1)

• Upper quartile: that value for which 25%

of the distribution is above it and 75%

below

• Lower quartile: that value for which 75%

of the distribution is above it and 25%

below

7

Quartiles (2)

• If the data is ungrouped, then put the data

in order in an array

• Find the quartile position , then estimate

its value, as previously for the median

• Upper quartile (Q3): position = 3(n + 1)

4

• Lower quartile (Q1): position = (n + 1)

4

8

Quartiles (3)

Example: ungrouped data:

3, 5, 6, 9, 15, 27, 30, 35, 37

• Lower quartile: position = n + 1 = 9 + 1 = 2.5th

4 4

Lower quartile: value = 5.5

(mid-way between 2nd and 3rd number in array)

• Upper quartile: position = 3(n + 1) = 3(9 + 1) 4 4

= 7.5th

Upper quartile: value = 32.5

(mid-way between 7th and 8th number in array)

9

Quartiles (4)

• Grouped data: use the same approach as

for estimating the median for grouped data

in week 4, except this time use the

quartile positions

10

Semi-Interquartile Range

• This is half the interquartile range. It is

sometimes called the Quartile Deviation

• Semi-Interquartile Range

= Upper Quartile - Lower Quartile

2

11

Example

Using previous ungrouped data

Interquartile range = UQ - LQ

= 32.5 – 5.5 = 27

Semi-interquartile range = UQ - LQ

2

= 32.5 – 5.5

2

= 27 = 13.5

2

12

Mean Deviation

• Average of the absolute deviations from

the arithmetic mean (ignoring the sign)

• When two straight lines (rather than

curved brackets) surround a number or

variable it is referred to as the modulus

and we ignore the sign

13

Mean Deviation of ungrouped data

• X1 = 2, X2 = 4, X3 = 3

• MD =

• MD = =

= ⅔

X X X X X X

n

1 2 3

2 3 4 3 3 3

3

1 1 0

3

14

Variance

• If we square all the deviations from the arithmetic mean, then we no longer need to bother with dropping the signs since all the values will be positive.

• We can then replace the straight line brackets (modulus) for the Mean Deviation with the more usual round brackets.

• Variance is the average of the squared deviations from the arithmetic mean

15

Variance: ungrouped data (1)

• Variance =

• To calculate the variance

1. Calculate the mean value

2. Subtract the mean from each value in turn,

that is, find

3. Square each answer to get

n

XXn

i

i

1

2

X

XX i

2XX i

16

Variance: ungrouped data (2)

4. Add up all these squared values to get

5. Divide the result by n to get

6. You now have the average of the squared deviations

from the mean (in square units)

n

i

i XX1

2

n

XXn

i

1

2

1

17

Standard deviation (SD)

• This is simply the square root of the

variance

• An advantage is that we avoid the square

units of the variance

• Larger SD, larger the average dispersion

of data from the mean

• Smaller SD, smaller the average

dispersion of data from the mean

18

Example 1: variance/standard

deviation

xi x1 - x (x1 – x)2

4

7

10

Total

4 – 7 = - 3

7 – 7 = 0

10 – 7 = 3

(-32) = 9

02 = 0

32 = 9

18

19

Solutions

Variance = square units

Standard deviation is square root of 6

= 2.449 units

6

3

181

2

n

XXn

i

i

20

Example 2: variance/standard

deviation

xi xi - x (xi – x)2

7

7

7

Total

7 – 7 = 0

7 – 7 = 0

7 – 7 = 0

02 = 0

02 = 0

02 = 0

0

21

Solution

Variance = square units

Standard deviation is square root of 0 = 0

i.e. there is no spread of values

0

3

01

2

n

XXn

i

i

22

Variance of grouped data

where Fi = Frequency of ith class interval

Xi = mid point of ith class interval

j = number of class intervals

2

1

1

1

1

2

2

j

i

j

i

ii

j

i

i

j

i

ii

iF

XF

F

XF

S

23

Price of item (£)

No of items

sold

LCB UCB Fi Xi FiXi FiXi^2

1.5 2.5 15 2 30 60

2.5 3.5 2 3 6 18

3.5 4.5 19 4 76 304

4.5 5.5 10 5 50 250

5.5 6.5 14 6 84 504

60 246 1136

24

2

2

60

246

60

1136

S

S2 = 18.93 – 4.12

S2 = 18.93 – 16.81

S2 = £2.122

S = √ 2.12 = £1.45

25

Co-efficient of variation (C of V)

• A measure of relative dispersion

• Given by i.e. the standard

deviation divided by the arithmetic

mean of the data.

• Data sets with a higher co-efficient of

variation have higher relative

dispersion

X

S