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

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... Weighted mean Sim

Chapter 3

Statistical measures

Measure center and location

Measure

variation/dispersion

Statistical measures

Center and

location

Variation/ Dispersion

- Coefficient of variation

Part A Measures of center and

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

 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

 Is the arithmetic mean appropriate

to a simple frequency distribution?

n

i i

x f x

(x): Number of newspapers/magazines/journals a student read a week

(f): Number of students

Weighted mean of a simple

Weighted mean of a grouped frequency

distribution

 Example: The following data relates

to the productivity of workers in a

15 25 30 35 28 17

Weighted mean of a grouped frequency

i i

x f x

Weighted mean of a grouped frequency

distribution

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

increase by 15%.

How many of them are likely to attend this Tuesday?

 The number of students likely to attend this Tuesday

 Proportional increase?

 Proportional multiplier?

3 Geometric mean

 A specialized measure, used to average proportional increases.

- Step 2: Calculate the geometric

3 Geometric mean

- Step 2: Calculate the geometric

mean multiplier

(ii) Weighted geometric mean

multiplier: applied when each

proportional increase repeatedly

n i

n i

n

f

f

i i





- Step 3: Subtract 1 from the gm

multiplier to obtain the average

 The number of bankers of a small

bank over the period 2000-2006 is

presented in the table below:

 The average proportional multiplier:

 The average proportional increase in

the number of bankers over the period is:

1

(1 ) i i

n

f

f

i i





4 Harmonic mean

 Read at home

Characteristics of the

mean

5 Median

 Median of a data set is the value of the item in the middle when the data items are arranged in ascending order.

 The median is considered as an

alternative average to the mean

 Example: The productivity of 5 workers (items/h): 20, 22, 24, 100, 22

Calculate the mean? What is the

problem?

Median for a simple frequency distribution

 Step 1: Find the middle item(s)

 Step 2: Find the value(s) that

correspond to the middle item(s)

- For an even-number data set: 2m, the median is the average value of mth item and (m+1)th item

Median for a simple frequency distribution

 Step 2: Find the value(s) that

correspond to the middle item(s)

- For an odd-number data set: 2m+1, the median is the value of (m+1)th item

1

M  x 

Median for a grouped frequency distribution

 Step 1: Find the middle item(s)

 Step 2: Find the class(es) containing the middle item(s)

 Step 3: Estimating the median by

formula

Lower limit of the median class

Cumulative frequency of class immediately prior to the median class Actual frequency of the median class Median class width

Amount

of food per person

Middle items?

Median

class?

Estimating the median

Characteristics of the

median

The mode of a simple frequency distribution

 Mode is the value

which has the

The mode of a grouped frequency distribution

1 0

Modal class width

Frequency of modal class

Frequency of the class immediately prior to the modal class

Frequency of the class immediately following to the modal class

Modal class?

Amount

of food (kg/perso

n)

Number

of people

400-500 10 500-600 30 600-700 45 700-800 80 800-900 30 900-1000 5

Estimate the Mode by the

formula

Graphical estimation of

the mode

Characteristics of the

mode

Graphical comparison of mean,

median and mode

n

Mode

 Moderately Skewed Left

0

Graphical comparison of mean,

median and mode

Mode

Medi an Mea

n

 Moderately Right Skewed

0

Mean

Medi an

Mod e

Graphical comparison of mean,

median and mode

 Highly Skewed Right

0

Graphical comparison of mean,

median and mode

7 Percentile and quartile

 Read at home