Presentation Chapter 5 MTH1022 Rev#01

7/28/2019 Presentation Chapter 5 MTH1022 Rev#01

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Module Outcomes:MO1 Identify basic mathematical concepts, skills and mathematical

techniques for algebra, calculus and data handling.MO2 Apply the mathematical calculations, formulas, statistical methods andcalculus techniques for problem solving in industry.MO3 Analyse calculus and statistical problems in industry.


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LEARNING OUTCOMESAt the end of this topic, student should be able to :

calculate and interpret the mean, median and modefor ungrouped and grouped data.

describe the symmetry and skewness for a data

distribution.

compute and interpret range, variance and standard

deviation for ungrouped and grouped data.

explain the characteristics and uses of each

measure of dispersion


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Topic 5

Measuresof centraltendency

Ungroupeddata

Groupeddata

Measuresof

dispersion

Ungroupeddata

Groupeddata


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MEASURESOFCENTRALTENDENCY

The measures of central tendency is usually called

the average.

Central tendency is a single value situated at the

centre of a data and can be taken as a summary

value for the data set.

Three types of measure of central tendency:

Mean

Median

Mode


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Measures of centraltendency

Ungroupeddata

Mean Median Mode

Groupeddata

With class interval /witho ut class interval

Mean Median Mode

Skewness

Skewedto the left

/ -veskewness

Normalskewness

Skewedto theright /+ve

skewness


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MEAN

Definition of mean:

a measure of central tendency that is computed by

taking the sum of all data values and then dividing it by

the number of data.

Mean value can be calculated for:

Ungrouped data:

Grouped data:

=

: sum of all values

: number of data

=

: frequency for the class

: midpoint for the class


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EXAMPLEFORUNGROUPEDDATA

Example :

Find the mean of the set of data below.

2, 4, 7, 10, 13, 16, and 18

Solution :

7

18161310742 X

770X

10X


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EXAMPLEFORGROUPEDDATA

Table shows the length of the fish in a pond.

Find mean.

Length (cm) Number of fish

5 - 9 8

10 14 17

15 19 20

20 24 10

25 29 18

30 34 11

35 - 39 6


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Solution :

Length

(cm)

Number of

fish

Midpoint

5 - 9 8 7 56

10 14 17 12 204

15 19 20 17 34020 24 10 22 220

25 29 18 27 486

30 34 11 32 352

35 - 39 6 37 222

)(x)( f fx

90f 1880fx

90

1880

f

fxX 89.20X


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MEDIAN Definition of median:

When all observations are arranged in ascending (or maybe descending)

order, then median is defined as the observation at the middle position

(for odd number observation), or it is the average of two

observations at the middle (for even number observations).

Median can calculated for:

Ungrouped data

Grouped data

Step 1: Arrange the given data in ascending orderStep 2: Get the position of the median

Step 3: Identify the median, or calculate the average of two middle

observations (even number)

= +

:determination of the median class

: lower boundary of the median class

: Cumulative before the median class

: Frequency of the median classC: Class size


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Find the median of the set of data below.

(a) 2, 16, 7, 18, 13, 4 and 10

(b) 2, 16, 7, 18, 13 and 10

Solution :

(a) 2, 4, 7, 10, 13, 16, 18 (Odd Data)

Median = 10

(b) 2, 7, 10,13,16,18 (Even Data)

Median =

=

2

1310

5.11


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Example :


Find median.


5 - 9 8

10 14 17

15 19 20

20 24 10

25 29 18

30 34 1135 - 39 6


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Solution :

Length

(cm)

Class Boundary

Length (cm)

Number of fish Cumulative

Frequency

5 - 9 4.5 9.5 8 8

10 14 9.5 14.5 17 25

15 19 14.5 19.5 20 45

20 24 19.5 24.5 10 55

25 29 24.5 29.5 18 7330 34 29.5 34.5 11 84

35 - 39 34.5 39.5 6 90

)( f


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Median Class = observation

= observation

= 45 th observation

Median =

=

= 19.5

2

N

2

90

)5(20

25455.14

+

2


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MODE

Definition of mode:

The observation (or the number) which has the largestfrequency or most frequently.

Set of data having only one mode is called unimodal

data.

A set of data may have two modes and the set is calledbimodaldata.

In the case of more than two modes the set will be

called multimodaldata.


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MODE

Mode can calculated for:

Ungrouped data

Grouped data

The data should f i rst arranged in ascending

or descendin g order.

= +1

1 +

: Lower boundary of the mode class

1 = (frequency class mode) (frequency before class mode)

= (frequency class mode) (frequency after class mode)

: class size


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Find the mode of the set of data below.

(a) 2, 4, 7, 16, 13, 4 and 10

(b) 2, 16, 7, 16, 13 and 13

(c) 2, 4, 7, 10, 13 and 16

Solution :

(a) Mode = 4(b) Mode = 13 and 16 (Dual Mode)

(c) No mode


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Exercises :

1. Find mean, median and mode for the following data :(a) 15, 18, 21, 25, 20, 18

(b) 3, 11, 9, 6, 17

(c) 0, 1, 2, 7, 3, 2, 1, 1, 2, 1, 2

2. A number ofx is added to the set of data of 2, 4, 7, 10, 13,

16, and 18 the mean becomes 9.5. Find the value ofx.


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EXAMPLEFORGROUPEDDATA


Find mode.


5 - 9 8

10 14 17

15 19 20

20 24 10

25 29 18

30 34 11

35 - 39 6


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Solution :

Mode =

Mode =

Length

(cm)

Class

Boundary

Length (cm)

Number of

fish

5 - 9 4.5 9.5 8

10 14 9.5 14.5 17

15 19 14.5 19.5 20

20 24 19.5 24.5 10

25 29 24.5 29.5 18

30 34 29.5 34.5 11

35 - 39 34.5 39.5 6

)( f

65.155103

3

5.14

cL

21

1


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Exercises :

1. You are given the following table :

Find mean, median and mode.

Daily Salary ( RM) Number of Workers

10 - 14 25

15 19 40

20 - 24 15

25 29 12

30 - 34 8


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Measuring Central Tendency

Of Grouped Data (Without Class Interval)

Mean

Formula

= mid point of the class

= frequency

f

fxX

x

f


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Median

1. Find Median Class

Median Class =

2. Find Cumulative Frequency

3. Write down median

2

1f


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Mode

1. Find Class Mode

Class Mode = the most frequent

occurred

2. Write down mode


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Example :

The table shows the number of magazines read bystudents in a month.

Find mean, median and mode.

Magazines 1 2 3 4 5

Frequency 8 11 9 7 5


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Solution :

Magazines Frequency Cumulativefrequency

1 8 8 8

2 11 22 19

3 9 27 28

4 7 28 35

5 5 25 40

40f

)( f)(x fx

110fx

40

110

f

fxX 75.2X


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Solution :

Median Class = observation

= observation

= th observation

Median = 3

Class Mode = 11

Mode = 2

2

1f

2

140

5.20


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SKEWNESS

Skewness measures the lack of symmetry in a data

distribution. The skewed portion is the long and thin part of the

curve.

A skewed distribution means the data are sparse at

one end of the distribution but piled up at the otherend.


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SKEWNESSINRELATIONTOMEAN, MEDIAN

ANDMODE

The concept of skewness helps to understand the

relationship between the measures: mean, median

and mode.

Mode is the highest point of the curve and the

median is the middle value.

The mean is usually located somewhere towards

the tail of the distribution because the mean

affected by all values.

A bell-shaped ornormal distribution has no

skewness.


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THERELATIONSHIPBETWEENMEAN, MEDIAN

ANDMODE

Measures of central

tendency

Skewness

Mode < Median < Mean positively-skewed orskewedto the right.

Mean = Median = Mode symmetricalor has normal-

skewness

Mean < Median < Mode negatively-skewedor is

skewed to the left.


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SKEWNESS

Mean = median = modeMode < median < mode Mean < median < mode


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MEASURESOFDISPERSION

Measures of dispersion help to understand the

spread or variability of a set of data.

Give additional information to judge the reliability of

the measures of central tendency and helps in

comparing dispersion that is present in varioussamples.

Common measures of dispersion:

Range

Variance

Standard deviation


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DEFINITIONOFCOMMONMEASURESOF

DISPERSION

Range Difference between the largest and the

smallest observations in a set of data

Variance The average of squared distance of eachscore (or observation) from the mean.

Used to measure the spreading of data

Standard deviation Square root of the variance


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Measures of dispersion

Ungrouped data

Sample &Population

Range,Variance,Standarddeviation

Grouped data

(with class intervals / withoutclass intervals)

Sample


Population



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MEASURING DISPERSION

OF UNGROUPED DATA (SAMPLE & POPULATION)

Example : Find the range for the following data

2, 14, 7, 1, 13, 16, 8

Solution : Range = max value min value

= 16 1 = 15

Range

(Distancemeasures of

dispers ion)

Formula

Range = Maximum value Minimum value


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Variance

Formula

Variance Population

Variance Sample

N / n: number of data

N

xx

N

2

22 1

n

xxns

2

22

1

1


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Standard

Deviation

Formula

Standard Deviation Population

OR

Standard Deviation Sample

OR

iancevar

N

xxN

2

21

iancesvar

n

xx

ns

2

2

1

1


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Example :

Find the variance and standard deviation for the data

below:

15, 17, 21, 24 and 31

Solution :

=

= .

= .

108x

24929615764412892252 x

5n


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MEASURING DISPERSION

OF GROUPED DATA (WITH CLASS INTERVAL)

Range

Formula

Range = Highest Class Boundary Lowest

Class Boundary


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Variance

Formula

Variance Population

Variance Sample

= mid-point

= frequency

f

f xf x

f

2

22 1

f

fxfx

fs

2

22

1

1

x

f


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Standard

Deviation

Formula

Standard Deviation Population

OR

Standard Deviation Sample

OR

= mid-point

= frequency

x

f

iancevar

f

fxfx

f

2

21

iances var

f

fx

fxfs

2

2

1

1


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Example :


Find range, variance and standard deviation.


5 - 9 8

10 14 17

15 19 20

20 24 10

25 29 18

30 34 1135 - 39 6


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Solution :

Range = 39.5 4.5 = 35

Length (cm) Class

Boundary

Length (cm)

Number of

fish

5 - 9 4.5 9.5 8

10 14 9.5 14.5 17

15 19 14.5 19.5 20

20 24 19.5 24.5 10

25 29 24.5 29.5 18

30 34 29.5 34.5 11

35 - 39 34.5 39.5 6


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Example :

You are given the following table :

Find variance and standard deviation.

Daily Salary ( RM) Number of Workers

10 - 14 40

15 19 25

20 - 24 15

25 29 12

30 - 34 8


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Solution :

Variance =

Daily

Salary( RM)

Number of

Workers

Midpoint

10 - 14 40

15 19 25

20 - 24 1525 29 12

30 - 34 8

)(x)( f2x fx 2fx

f fx 2fx


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Measuring Dispersion

Of Grouped Data (Without Class Interval)

Range Range = Maximum value Minimum value

Variance Formula as per grouped data (with class

interval)

Standard Deviation Formula as per grouped data (with class

interval)


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Example :

You are given the following table :

Find range, variance and standard deviation.

No. of days 2 3 4 5 6 7 8

Frequency 6 8 5 3 3 3 2


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EXERCISE:

One of the events in the Winter Olympics is the Mens 500

meter Speed Skating. The times for this event are show tothe right. Find the mean, median, mode, range, variance

and standard deviation of times.

Year Time Year Time1928 43.4 1964 40.1

1932 43.4 1968 40.3

1936 43.4 1972 39.44

1948 43.1 1976 39.17

1952 43.2 1980 38.03

1956 40.2 1984 38.03

1960 40.2 1988 36.45

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Presentation Chapter 5 MTH1022 Rev#01

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