Chapter 3 (numerical measures part b) student

Chapter 3

Statistical
measures
Measure center and
location
Measure
variation/dispersion


Summary
Statistical
measures
Center and
location

- Mean
(arithmetic,
weighted,
geometric)

- Mode, Median

Variation/Disper
sion

- Range

- Variance
- Standard
deviation

1.
2.
3.
4.
5.

Part B
Measures of
variation/dispersion

Range
Mean deviation
Variance
Standard deviation
Coefficient of variation


1. The range




The range is defined as the numerical
difference between the smallest and
largest values of the items in a set or
distribution
Formula:

R = largest value – smallest
value


Example


Ages of two groups of people on survey:

Group
A

20

30

40

50

60

Group
B

38

39

40

41

42


Advantages and
disadvantages
of the range



Advantages:



Disadvantages:


Implication


2. The mean deviation




The mean deviation is a measure of
dispersion that gives the average
difference (i.e. ignoring ‘-’ signs) between

each item and mean.
Formula:
- For a data set
n

d =

∑x
i =1

i

n

−x


Formulae
- For a frequency distribution
k

d =

∑f
i =1

i

xi − x

k

∑f
i =1

i


Example
Group
A
Group
B

20

30

40

50

60

38

39

40

41

42

n

dA =

∑
i =1

xi − x
n


Example
Group
A
Group
B

20

30

40

50

60

38

39

40

41

42

n

dB =

∑
i =1

xi − x
n


Example


The data in
table below
relates to the
productivity
(kg/person) of

100 workers in
a small factory
Mean
deviation?

Productivi
ty
(kg/perso
n)
<10
10 – 20
20 – 30
30 – 35
35 – 40
≥ 40

Number
of
workers
7
18
25
20
18
12


Characteristics of the mean
deviation






A better measure of dispersion than the
range
Useful for comparing the variability
between distributions
Can be complicated to calculate in
practice if the mean is anything other than
a whole number.


3. Variance






Variance is another statistical measure of
dispersion
It is defined as the average of squared
discrepancies between each data value
and their mean
Formula:


For a set of values

n

σ =
2

∑ (x
i =1

i

− x)

2

n

n

or

σ2 =

2
x
∑i
i =1

n

− ( x )2 = x 2 − ( x )2

The mean of the squares less the square
of the mean


For a frequency
distribution
k
2
(
x
−
x
)
fi
∑
i
2
i =1
σ =

k

∑f
i =1

i

k

or

σ2 =

2
x
∑ i fi
i =1
k

∑f
i =1

− ( x )2 = x 2 − ( x )2

i

The mean of the squares less the square
of the mean


Example
Group
A
Group
B

20

30

40

50

60

38

39

40

41

42

n

σ2 =

2
(
x
−
x
)
∑ i
i =1

n


Example


The data in
table below
relates to the
productivity
(kg/person) of
100 workers in
a small factory
Variance?

Productivi
ty
(kg/perso
n)
>10
10 – 20
20 – 30
30 – 35
35 – 40
≥ 40

Number
of
workers
7

18
25
20
18
12


Characteristics of the
variance






A better measure of dispersion than the
range
Complicated since it multiply the
discrepancies
The unit of the variance is not meaningful


4. Standard deviation




Standard deviation is defined as the
square root of the variance.
Formula

For a set of values
n

∑( x

σ =
n

or

σ=

i =1

n

− x)

n

∑x
i =1

i

2

2

i

− (x ) = x − (x )
2

2

2


For a frequency
distribution
k

∑( x

σ =

i =1

i

− x)

2

fi

k

∑f
i =1

i

k

or

σ =

2
x
∑ i fi
i =1
k

∑f
i =1

i

− ( x )2 =

x2 − ( x )2


Example
Group
A

Group
B

20

30

40

50

60

38

39

40

41

42

σ= σ

2


Example


The data in
table below
relates to the
productivity
(kg/person) of
100 workers in
a small factory
Standard
deviation?

Productivi
ty
(kg/perso
n)
>10
10 – 20
20 – 30
30 – 35
35 – 40
≥ 40

Number
of
workers
7
18
25
20
18

12


Characteristics of Standard
Deviation




Can be regarded as one of the most useful
and appropriate measure of dispersion.
For distribution that are not too skewed:
- 99.7% of the data items should lie within
three standard deviation of the mean
- 95% of the data items should lie within
two standard deviation
- 68% of the data items should lie within
one standard deviation of the mean