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STATISTICS

Name

........................................................................................

Statistics

zefry@sas.edu.my

CHAPTER 7 : STATISTICS

7.1 MEASURES OF CENTRAL TENDENCY (i) Ungrouped data data that is not grouped into classes (ii) Grouped data - data that is grouped in certain classes

7.1.1 Calculating the mean of ungrouped data

N

xx

Example 1:

Find the mean of 58, 67, 45, 73 and 77

Example 2:

Number 1 2 3 4 5

Frequency 5 8 4 6 2

Find the mean of the number.

7.1.2 Determining mode of ungrouped data

Mode is the value which appears the most number of times in a set of data (value that has the

highest frequency)

Example 1: Determine the mode for the following sets of number.

(a) 2, 5, 6, 2, 6, 7, 2, 4, 8, 2 Answer: ..

(b) 5, 5, 8, 10, 4, 4 Answer: ..

(c) 2, 2, 3, 4, 4 , 4, 6, 6 Answer: ..

Note: It is possible that a set of data either has more than one mode or has no mode.

Where

x - mean of the set data. x values in the set of data N total number of data

Statistics

zefry@sas.edu.my

Example 2: Determine the mode of the number of pens a student has.

Number of pens 1 2 3 4 5

Number of students 6 7 5 3 3

Answer: ..

7.1.3 Determining the median of ungrouped data

When the values in set of data are arranged in either ascending or descending order, the value that

lies in the middle is the median.

Example 1: 3, 4, 5, 6, 7, 8, 9

Example 2: 21, 20, 19, 18, 17, 16, 15, 14

Activity 1:

1. Calculate the mean, mode and median for the following sets of data.

(i) 1, 4, 5, 8, 9, 8, 8, 7, 4 (ii) 5, 8, 12, 10, 5, 3, 7, 5, 20, 10

2. (i) Find the mean of 6, 8, 4, 9 and 11

(ii) Find the value of x if the mean of 4, 5, 6, 7, 11 and x is 7.

3. Find the mode of each following sets of data.

(i) 8, 6, 10, 8, 5

(ii) 2, 2, 5, 5, 11, 11

Median

4 numbers 4 numbers

Median =

Statistics

zefry@sas.edu.my

4. Find the mean, mode and the median of the following data

Time ( hours ) 12 13 14 15 16

Number of cars 3 5 10 6 6

5. The following frequency distribution table shows the score of a group of students in a

quiz. Find the mean , mode and median

Score 5 6 7 8 9 10

Number of students 5 6 4 3 8 4

7.1.4 Determine modal class of grouped data from the frequency distribution table.

Modal class of a set of data is the value of class which occurs most frequently. The value of mode

can be obtained by drawing the histogram

Example 1:

The following table below shows the mark for 50 students in their Additional Mathematics test.

Find the modal class for the students.

Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79

Number of students 3 6 13 10 7 7 4

7.1.5 Find mode from a histogram

Note: In drawing a histogram, class boundaries are used.

HISTOGRAM

Plot: frequency against boundary of the class

Frequency

Boundary of the class

Statistics

zefry@sas.edu.my

Example 1:

The following table below shows the mark for 50 students in their Additional Mathematics test. .

Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79

Number of students 3 6 13 10 7 7 4

Table 3

Complete the following table.

Mark Frequency , f Lower class boundary Upper class boundary

10 19 3

From the table, draw a histogram, hence determine the mode from the histogram.

Calculate mean of grouped data

f

fxx where f is the frequency for each class

x is the corresponding class midpoint

Note :

Class mid point = 2

limitupper limit lower

Example 1:

1. The table below shows the marks obtained by 30 students in a Mathematics test.

Marks 30 - 39 40 - 49 50 - 59 60 -69

Number of students 4 8 12 6

Find the mean for the marks.

Statistics

zefry@sas.edu.my

Solution :

7.1.7 Calculating median of grouped data from the cumulative frequency distribution table

Median, m = L + Cf

FN

m

2

1

, where

Note: Size of class interval = upper class boundary lower class boundary

Activity :

The following table shows the mark for 40 students in their Additional Mathematics test. Find

the median by using the formula

Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79

Number of students 3 5 7 9 10 4 2

TABLE 1

Marks Number of students, f Class mark ,x fx

f fx

L = Lower boundary of the class in which the median

lies.

N = total frequency

C = Size of class interval

F = cumulative frequency before the class in which the

median lies.

fm = frequency of the class in which the median lies

Statistics

zefry@sas.edu.my

Solution:

Mark Frequency , f Cumulative

frequency, F

10 19 3

7.1.8 Estimate median of grouped data from an ogive

OGIVE

Plot: cumulative frequency against upper class boundary

Cumulative

frequency

Upper class boundary

Statistics

zefry@sas.edu.my

Refer to table 1

Construct a cumulative frequency table and then draw cumulative frequency curve (ogive)

From the graph, find the median weight.

Mark Frequency , f Upper class

Boundary

Cumulative

frequency, F

Activity 2:

1. The following table shows the marks obtained by 30 students in Mathematics test. Find the mean for the data.

Mark 10-19 20-29 30-39 40-49 50-59 60-69 70-79

Number of students 3 6 13 10 7 7 4

Solution :

Mark Number of students, f Class mark ,x fx

Statistics

zefry@sas.edu.my

2. The data below shows the scores obtained by 30 students in a game.

0 3 3 6 8 9 9 10 10 11

5 0 7 6 9 10 12 17 5 4

2 5 8 8 10 11 7 8 12 11

Using 3 scores as a size of class interval, construct a frequency distribution table for its data.

Then find

(a) the mean (b) median (c) modal class

Note : Students have to draw the frequency distribution table as below.

Number of scores x f fx Cumulative

frequency

3. The table below shows the age of factory workers in 1995.

Age(year) 20-25 26-31 32-37 38-43 44-49 50-55 56-61

Number of workers 5 24 16 20 13 12 10

(i) Draw a histogram and hence estimate the mode of the data. (ii) Find the median without using an ogive.

Statistics

zefry@sas.edu.my

7.1.9 Determine the effects on mode, median and mean for a set of data when

(a) Every value of the data is change uniformly

Measures of

central

tendency

Added by k Subtracted by k Multiplied by k Divided by k

New mean Original mean + k Original mean - k k(Original mean)

k

mean original

New mode Original mode + k Original mode - k K(Original mode)

k

mode original

New median Original median+ k Original median - k K(Original median)

k

median original

1. The mean, mode and median of a set data are 7.4, 9 and 8 respectively. Find the new

mean, mode and median if every value of the set data is

a) divided by 2 b) subtracted by 4 c) multiplied by 3 d) added by 3

2. Find the mean of 12, 14, 16, 18, 20. By using the result, find the mean of 8 ,10 ,12 ,14,16

7.1 MEASURES OF DISPERSION

7.2.1 Finding the measures of dispersion of ungrouped data

Formulae:

(i) Range of ungrouped data = largest value smallest value (ii) Interquartile range = Upper quartile lower quartile

= 3Q 1Q

(iii) Variance, 2 :

2

2)(

N

xxi 2

2

2 xN

xor

, where

N

xx

(iv) Standard deviation, : = iancevar

Statistics

zefry@sas.edu.my

Example :

For the given ungrouped set of data, find

(i) range (ii) interquartile range (iii) variance (iv) standard deviation

(a) 3 , 6 , 8 , 12 , 15 , 16

(b) 3 , 5 , 6 , 7 , 8 , 4 , 8 , 9 , 10 , 12 , 14

Statistics

zefry@sas.edu.my

Activity 3:

1. Find range, interquartile range, variance and standard deviation for each set of the

following data.

(a) 10 , 7 , 19 , 13 , 14 , 10

(b) 4 , 3 , 2 , 7 , 9 , 10 , 12 , 6 , 15

(d) 4 , 12 , 15 , 10 , 7 , 6 , 1

2. Given that the mean of set data 5 , 7 , x , 11 , 12 is 9.

a. Find the value of x. b. Find the variance and the standard deviation of the data.

7.2.2 Finding measu