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 A
Measures of center and location
1.
2.
3.


Formula
x1 + x2 +.... + xn
x =
n
xi
∑
x =
n
where

- x1 , x2 , ..., xn are the 1st x-value, 2nd xvalue, …. nth 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

n

x f
∑

x = i =1n

i

f
∑
i=
1

i

i


Example
x

f

0
1
2
3
4
5

frequency distribution


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

Productivity
(items/h)
Number of
workers

0-9 10-19 20-29 30-39 40-49 50-59
15

25

30

35

28

17


Weighted mean of a grouped
frequency distribution


Formula:


15
25

20-29
30-39

30
35

40-49
50-59

28
17

Total

xi

xifi


Weighted mean of a grouped
frequency distribution


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




Example
To add
Multiply by
(proportional increases) (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 )

fn


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
bankers

2000 2001 2002 2003 2004 2005 2006
200

220

250

262

312


Example


The average proportional multiplier:

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