Bài giảng nguyên lý thông kê chương 3 numerical measures part a 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 A
Measures of center and location
1.
2.
3.

4.
5.
6.
7.

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’.
The sum of all values

Arithmetic mean =

The number of values

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

Weighted mean of a simple
frequency distribution
xi

fi

10
12
13
14
16

2
8
17
5
1






Is the arithmetic mean
appropriate to a simple
frequency distribution?
Why?
Formula:

n

x f
∑

x = i =1n

i

f
∑
i=
1

i

i


Example
x

f

0
1
2
3
4
5

12
18
30
20
15
5

Total

100

xf
(x): Number of
newspapers/magazines
/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:

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:

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
Productivity Number
(items/h) of workers
0-9
10-19

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:

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
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 )

3. Geometric mean
-

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

gmm =

∑ fi
i =1

(1 + p1 ) (1 + p2 ) ...(1 + pn )
f1

n

fi
fi
∑
gmm =
∏ (1 + pi )
i =1

f2

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

284

300

312


Example
Year

2000 2001 2002 2003 2004 2005 2006

No of 200
bankers
Proport
ional
multipli
ers

220

250

262

284

300

312


Example


The average proportional multiplier:

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