Chapter 3 (numerical measures part a) 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.
6.
7.

Part A
Measures of center and
location

Arithmetic mean
Weighted mean
Geometric mean
Harmonic mean
Median
Mode
Percentile, Quartile


1. Arithmetic mean




The mean of a data set is the average of all
the data values

Arithmetic mean of a data set is defined as
‘ the sum of the values’ divided by the
‘number of values’.

Arithmetic
mean =

The sum of all values
The number of
values


Formula
x1  x2  ....  xn
x 
n
xi
�
x 
n
wher
e
-

are the 1st x-value, 2nd x-value,

x2 , ..., xn
….x
nth
1 ,x-value

- n is the number of data values in the set


Example


If a firm received orders worth:
£151, £155, £160, £90, £270 for five
consecutive months, their average
value of orders per month would be
calculated as:


Limits of arithmetic
mean


2. Weighted mean
 Simple

frequency
distribution
 Grouped frequency
distribution


Weighted mean of a simple
frequency distribution
xi

f

10
12
13
14
16

2
8
17
5
1






Is the arithmetic
mean appropriate
to a simple
frequency
distribution?
Why?
n
Formula:�xi f i
x 

i 1

n

�f
i 1

i


Example
x

f

0
1
2
3
4
5

12
18
30
20
15
5

Tota 100
l

xf
(x): Number of
newspapers/maga
zines/journals a
student read a
week
(f): Number of
students


Weighted mean of a simple
frequency distribution


The mean number of
newspapers/magazines/journals a
student read a week is:




Weighted mean of a
grouped frequency
distribution

Example: The following data relates to
the productivity of workers in a
factory:

Productivi

ty
(items/h)
Number
of
workers

0-9

1019

2029

3039

4049

5059

15

25

30

35

28

17



Weighted mean of a
grouped frequency
distribution

Formula:

n

x 

�x
i 1
n

i

�f
i 1

fi
i

Where:
- x: mid-point as representative value
of each class
- f: frequency of each class


Weighted mean of a
grouped frequency
distribution
Productiv Numbe
ity
r of
(items/h) worker
s
0-9
10-19
20-29
30-39
40-49
50-59

15
25
30
35
28
17

xi

xif




Weighted mean of a
grouped frequency
distribution

The average productivity (mean) of
workers in the factory is:


3. Geometric mean




Applicable when the products of data
values are meaningful
Proportional increases and multipliers:
Example:
The number of students attending the
music class last Tuesday was 160. This
Tuesday, the number is expected to
increase by 15%.
How many of them are likely to attend
this Tuesday?


3. Geometric mean


The number of students likely to

attend this Tuesday



Proportional increase?



Proportional multiplier?


Example
To add
(proportional
increases)

Multiply by
(proportional
multipliers)


3. Geometric mean



-

A specialized measure, used to
average proportional increases.
Formula:

Step 1: Express the proportional
increases (p) as proportional
multipliers (1+p)


3. Geometric mean
-

Step 2: Calculate the geometric
mean multiplier
(i) Simple geometric mean multiplier:
applied when each proportional
increase appears once only

gmm  n (1  p1 )(1  p2 )...(1  pn )


3. Geometric mean
-

Step 2: Calculate the geometric mean
multiplier
(ii) Weighted geometric mean
multiplier: applied when each
n
proportional
increase repeatedly
fi
appears�
fn

f1
f2
i 1

gmm 

(1  p1 ) (1  p2 ) ...(1  pn )
n

fi
fi
�
gmm 
�(1  pi )
i 1


3. Geometric mean
-

Step 3: Subtract 1 from the gm
multiplier to obtain the average
proportional increase
average proportional increase = gm
multiplier - 1


Example


The number of bankers of a small
bank over the period 2000-2006 is
presented in the table below:

Year
No of
banke
rs

200 200 200 200 200 200 200
0
1
2
3
4
5
6
200 220 250 262 284 300 312


Example
Year

200 200 200 200 200 200 200
0
1
2
3
4
5

6
No of 200 220 250 262 284 300 312
banke
rs
Propo
rtiona
l
multi
pliers


Example


The average proportional multiplier:

The average proportional increase in
the number of bankers over the period is:
