Prepared by: Nurazrin Jupri

differences will be

large

differences will be

small

MATH0102|Nurazrin Jupri

Two groups of three studentsGroup 1 4 7 10Group 2 7 7 7

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

MATH0102|Nurazrin Jupri

MATH0102|Nurazrin Jupri

A measure of the average amount by which the values in a distribution (x) differ from the arithmetic mean

Average of the absolute deviations from the arithmetic mean (ignoring the sign)

MATH0102|Nurazrin Jupri

groupedungroup

ed

Vertical bars = all difference

s are taken as positive

MATH0102|Nurazrin Jupri

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

MATH0102|Nurazrin Jupri

Mean Deviation of grouped data

XMATH0102|Nurazrin Jupri

MATH0102|Nurazrin Jupri

MATH0102|Nurazrin Jupri

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.

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

MATH0102|Nurazrin Jupri

Variance =

To calculate the variance1. Calculate the mean value 2. Subtract the mean from each value in

turn, that is, find 3. Square each answer to get

n

XXn

ii

1

2

X

XX i

2XX i

MATH0102|Nurazrin Jupri

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

ii XX

1

2

n

XXn

i

1

2

1

MATH0102|Nurazrin Jupri

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

MATH0102|Nurazrin Jupri

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

MATH0102|Nurazrin Jupri

Variance = square units

Standard deviation is square root of 6 = 2.449 units

6

3

181

2

n

XXn

ii

MATH0102|Nurazrin Jupri

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

MATH0102|Nurazrin Jupri

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

ii

MATH0102|Nurazrin Jupri

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

iii

j

ii

j

iii

iF

XF

F

XFS

MATH0102|Nurazrin Jupri

LCB UCB F X FX FX^2

5.5 10.5 8 8 64 512

10.5 15.5 4 13 52 676

15.5 20.5 6 18 108 1944

    18   224 3132

MATH0102|Nurazrin Jupri

S2 = 174 – 12.442 S2 = 174 – 154.86 S2 = 19.14 S = √ 19.14 = 4.375

22

18

224

18

3132

S

MATH0102|Nurazrin Jupri