Bài giảng nguyên lý thông kê chương 3 numerical measures part b student

Chapter 3

Statistical measures
Measure center and location
Measure variation/dispersion


Summary
Statistical measures
Center and location

Variation/Dispersion

- Mean (arithmetic,
weighted, geometric)

- Range

- Mode, Median
- Percentile, Quartile

- Variance
- Standard deviation
- Coefficient of variation


Part B
Measures of variation/dispersion
1.
2.
3.

4.
5.

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

20

30

40

50

60

Group
B

38

39

40

41

42

n

∑x

d A = i =1

i

n

−x


Example
Group
A

20

30

40

50

60

Group
B

38

39

40

41

42

n

∑x

d B = i =1

i

n

−x


Example


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

Productivity
(kg/person)
<10

Number of
workers

7

10 – 20

18

20 – 30

25

30 – 35

20

35 – 40

18

≥ 40

12

Total

100


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

σ2 =

2
(
x
−
x
)
fi
∑ i
i =1

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

20

30

40

50

60

Group
B

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?

Productivity
(kg/person)
>10

Number of
workers
7

10 – 20

18

20 – 30

25

30 – 35

20

35 – 40

18

≥ 40

12

Total

100


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 = x 2 − ( x )2


Example
Group
A

20

30

40

50

60

Group
B

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?

Productivity
(kg/person)
>10

Number of
workers
7

10 – 20

18

20 – 30

25

30 – 35

20

35 – 40

18

≥ 40

12

Total

100


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