chapter 4_part2 student.pdf

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QMT412 Pn. Sanizah's Notes 4/8/201

CHAPTER 4 PART 2SUMMARY MEASURES

(for GROUPED DATA)

Prepared by Sanizah Ahmad1

TYPES OF DATA

2

UngroupedData

Measures ofCentralTendency

Measures ofPosition

Measures ofDispersion

GroupedData

Measures ofCentralTendency

Measures ofPosition


1. Measures of Central Tendencyfor grouped data)

Mean

Median

Mode

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MEANFinding the meanfor grouped data:

sizesample

classeachofmidpointfrequency

where

n

xf

n

xfx

m

m

4

n= f

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EXAMPLE 1Using the given frequency distribution, find

the mean. The data represent the numberof miles run during one week for a sampleof 20 runners.

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Frequency distribution

Class Frequency

(f)

Midpoint(xm)

f . xm

5.5-10.5

10.5-15.5

15.5-20.5

20.5-25.5

25.5-30.5

30.5-35.5

35.5-40.5

1

2

3

5

4

3

2

8

13

18

23

28

33

38

8

26

54

115

112

99

76

f=20 f.xm= 490

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Solution

miles5.24

20

490

n

xfx

m

7

n= fn

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MEDIANC

f

ff

Lxm

m

m

1

2~

classmediantheofwidth

classmediantheoffrequency

classmedianthebeforeintervals

classalloffrequencycumulativeclassmediantheofboundaryclasslower

1

C

f

fL

m

m-

m

where

8

Median location=2

n


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Example 2Refer to Example 1.Find the median.

50% of the runners run less than 24.5 miles and the

other 50% run more than 24.5 miles.

102

20

2location,Median n

The median class is 20.5-25.5.

5

5

6

5.20

1

C

f

f

L

m

m

m

5.24

55

6105.20~

x

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MODE

classmodaltheofw idthC

classmodaltheafterfrequency-classmodaloffrequency

classmodalthebeforefrequency-classmodaloffrequency

classmodaltheofboundariesclasslowerL

where

2

1

mo

CLx mo

21

1Mode,

10

Class withthe highest

frequency

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Example 3Refer to Example 1. Find the mode.

The modal class is 20.5-25.5.

5

145

235

5.20

2

1

C

Lmo

8.23

512

25.20x

11

class with the

highest

frequency

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MODE Estimating from histogram

12

Use graph

paper toestimate

value

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Refer textbook and go through

Example 8 pg. 75-79

Find the mean, median and mode.

Estimate the mode from histogram

(refer Figure 4.5 pg. 80).

13 [email protected] 14

Years of

Experience

Class

boundary

No of

employees

(f)

Mid-

point

(xm)

f xm

14 16

58 20

9

12 28

1316 24

1720 16

2124 11

2528 5

f=120

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2. Measures of Positionfor grouped data)

First Quartile (Q1)

Third Quartile (Q3)

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Formula to obtain Q1and Q3

Step 1: Obtain the cumulative frequencies.

Step 2: Identify the first and third quartile classes.

Locationof first quartile -

Locationof third quartile -

Then refer to the cumulative frequency column anddetermine the locations and classes.

Note:Refer textbook pg. 81-82

4

n

n4

3



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Step 3 Find Q1and Q3Find first quartile valueusing this formula

sizeclassquartilefirst

classquartilefirsttheoffrequency

classquartilefirstthebeforefrequencycumulative

nsobservatioofnumber

classquartilefirstth eofboundaryclasslower

where

4

1

1

1

1

1

1

1

11

C

f

f

n

L

Cf

fn

LQ

m-

m


Step 3Find third quartile as follows:

sizeclassquartilethird

classquartilethirdtheoffrequency

classquartilethirdthebeforefrequencycumulative

nsobservatioofnumber

classquartilethirdtheofboundaryclasslower

where

4

3

3

3

1

3

3

3

1

33

C

f

f

n

L

Cf

fn

LQ

m-

m


Refer textbook Example 12 pg. 82

How to find first and third quartile

Using formula

Using ogive or cumulative frequency curve.


3. Measures of Dispersionfor grouped data)

Range

Interquartile Range

Variance

Standard Deviation

Coefficient of Variation

Skewness



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RANGE

21

BB LU

classlowestof

boundarylower

classhighestof

boundaryupperRange

Refer textbook Example 4 pg. 100

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1

2

2

2

n

n

xfxf

s

mm

VARIANCE AND STANDARD DEVIATION

sizesample

classeachofmidpoint

frequency

where

n

x

f

m


Example 4

Class Frequency (f) Midpoint

5.5-10.5

10.5-15.5

15.5-20.5

20.5-25.5

25.5-30.5

30.5-35.5

35.5-40.5

1

2

3

5

4

3

2

8

13

18

23

28

33

38

8

26

54

115

112

99

76

64

338

972

2645

3136

3267

2888

20 490 13310

mxf2

mxf

Refer to Example 1. Find the varianceandstandard deviation.

23

38

768768

19

20

49013310

12

2

2

2

.

.s.

n

n

xfxf

s

mm


Variance

Standard

deviation


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Coefficient of Variation

Indicate the ratio of the standard deviation to the

arithmetic mean expressed as a percent

The larger the percentage, the greater the variation.

Large variation implies less consistency,

small variation implies better consistency.

100meansample

deviationstandardsample CV


Pearson Coefficient of Skewness

Measure the skewness

deviationstandard

median)3(mean

or

deviationstandard

modemean

skewness


SUMMARY

27

GroupedData

Measures ofCentral

Tendency

Mean

Median

Mode

Measures ofPosition

First Quartile (Q1)

Third Quartile (Q3)


Range

Variance

Standard deviation

Coefficient ofVariation (CV)

Pearson coefficient ofskewness

chapter 4_part2 student.pdf

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