Sqqs1013 ch2-a122

SQQS1013 Elementary Statistics

DESCRIPTIVE STATISTICSDESCRIPTIVE STATISTICS2.1 INTRODUCTION

Raw data - Data recorded in the sequence in which there are

collected and before they are processed or ranked

Array data - Raw data that is arranged in ascending or descending order.

Here is a list of question asked in a large statistics class and the “raw data” given by one of the students:

1. What is your sex (m=male, f=female)?Answer : m

2. How many hours did you sleep last night?Answer: 5 hours

3. Randomly pick a letter – S or Q.Answer: S

4. What is your height in inches?Answer: 67 inches

5. What’s the fastest you’ve ever driven a car (mph)?Answer: 110 mph

• Quantitative raw data • Qualitative raw data

These data also called ungrouped data.

Chapter 2: Descriptive Statistics 1

Example 2

Example 1


2.2 ORGANIZING AND GRAPHING QUALITATIVE DATA

2.2.1 Frequency Distributions Table

• A frequency distribution for qualitative data lists all categories and the

number of elements that belong to each of the categories.

• It exhibits the frequencies are distributed over various categories

• Also called as a frequency distribution table or simply a frequency

table.

e.g. : The number of students who belong to a certain category is called

the frequency of that category.

2.2.2 Relative Frequency and Percentage Distribution

• A relative frequency distribution is a listing of all categories along with their

relative frequencies (given as proportions or percentages).

• It is commonplace to give the frequency and relative frequency distribution

together.

• Calculating relative frequency and percentage of a category

Relative Frequency of a category

= Frequency of that category Sum of all frequencies


FORMULA

ξ∆Σ λϖβ


Percentage (%) = (Relative Frequency)* 100

A sample of UUM staff-owned vehicles produced by Proton was identified and the make of each noted. The resulting sample follows (W = Wira, Is = Iswara, Wj = Waja, St = Satria, P = Perdana, Sv = Savvy):

Construct a frequency distribution table for these data with their relative frequency and percentage.

W W P Is Is P Is W St WjIs W W Wj Is W W Is W WjWj Is Wj Sv W W W Wj St WWj Sv W Is P Sv Wj Wj W WSt W W W W St St P Wj Sv

Solution:

Category FrequencyRelative

FrequencyPercentage (%)

Wira 19Iswara 8Perdana 4Waja 10Satria 5Savvy 4

Total

2.2.3 Graphical Presentation of Qualitative Data

a) Bar Graphs


Example 3


• A graph made of bars whose heights represent the frequencies of

respective categories.

• Such a graph is most helpful when you have many categories to

represent.

• Notice that a gap is inserted between each of the bars.

• It has

o simple/ vertical bar chart

o horizontal bar chart

o component bar chart

o multiple bar chart

• Simple/ Vertical Bar Chart

To construct a vertical bar chart, mark the various categories on the horizontal

axis and mark the frequencies on the vertical axis

• Horizontal Bar Chart

To construct a horizontal bar chart, mark the various categories on the vertical

axis and mark the frequencies on the horizontal axis.

• • Component Bar Chart


UUM Staff-owned Vehicles Produced By Proton

0 5 10 15 20

Wira

Perdana

Satria

Typ

es o

f V

ehic

le

Frequency


To construct a component bar chart, all categories is in one bar and every

bar is divided into components.

The height of components should be tally with representative frequencies.

Suppose we want to illustrate the information below, representing the number of people participating in the activities offered by an outdoor pursuits centre during Jun of three consecutive years.

2004 2005 2006Climbing 21 34 36Caving 10 12 21Walking 75 85 100Sailing 36 36 40

Total 142 167 191

Solution:

• Multiple Bar Chart

To construct a multiple bar chart, each bars that representative any

categories are gathered in groups.

The height of the bar represented the frequencies of categories.

Useful for making comparisons (two or more values).

The bar graphs for relative frequency and percentage distributions can be

drawn simply by marking the relative frequencies or percentages, instead of the

class frequencies.


Example 4

Activities Breakdown (Jun)

0

50

100

150

200

2004 2005 2006

Year

Nu

mb

er o

f p

arti

cip

ants

Sailing

Walking

Caving

Climbing


b) Pie Chart

• A circle divided into portions that represent the relative frequencies or

percentages of a population or a sample belonging to different

categories.

• An alternative to the bar chart and useful for summarizing a single

categorical variable if there are not too many categories.

• The chart makes it easy to compare relative sizes of each

class/category.

• The whole pie represents the total sample or population. The pie is

divided into different portions that represent the different categories.

• To construct a pie chart, we multiply 360o by the relative frequency for

each category to obtain the degree measure or size of the angle for the

corresponding categories.

Movie Genres

Frequency Relative Frequency Angle Size

ComedyActionRomanceDramaHorrorForeignScience Fiction

54362828221616

0.270.180.140.140.110.080.08

360*0.27=97.2o

360*0.18=64.8o

360*0.14=50.4o

360*0.14=50.4o

360*0.11=39.6o

360*0.08=28.8o

360*0.08=28.8o

Total 200 1.00 360o


Activities Breakdown (Jun)

0

20

40

60

80

100

120

2004 2005 2006

Year

Nu

mb

er o

f p

arti

cip

ants

Climbing

Caving

Walking

Sailing

Example 5


c) Line Graph/Time Series Graph

• A graph represents data that occur over a specific period time of time.

• Line graphs are more popular than all other graphs combined because

their visual characteristics reveal data trends clearly and these graphs

are easy to create.

• When analyzing the graph, look for a trend or pattern that occurs over

the time period.

• Example is the line ascending (indicating an increase over time) or

descending (indicating a decrease over time).

• Another thing to look for is the slope, or steepness, of the line. A line

that is steep over a specific time period indicates a rapid increase or

decrease over that period.

• Two data sets can be compared on the same graph (called a

compound time series graph) if two lines are used.

• Data collected on the same element for the same variable at different

points in time or for different periods of time are called time series data.

• A line graph is a visual comparison of how two variables—shown on the

x- and y-axes—are related or vary with each other. It shows related

information by drawing a continuous line between all the points on a

grid.

• Line graphs compare two variables: one is plotted along the x-axis

(horizontal) and the other along the y-axis (vertical).

• The y-axis in a line graph usually indicates quantity (e.g., RM, numbers

of sales litres) or percentage, while the horizontal x-axis often measures

units of time. As a result, the line graph is often viewed as a time series

graph



A transit manager wishes to use the following data for a presentation showing how Port Authority Transit ridership has changed over the years. Draw a time series graph for the data and summarize the findings.

YearRidership

(in millions)19901991199219931994

88.085.075.776.675.4

Solution:

The graph shows a decline in ridership through 1992 and then leveling off for the years 1993 and 1994.

EXERCISE 1


Example 6

75

77

79

81

83

85

87

89

1990 1991 1992 1993 1994

Year

Rid

ers

hip

(in

mil

lio

ns

)


1. The following data show the method of payment by 16 customers in a supermarket checkout line. ( C = cash, CK = check, CC = credit card, D = debit and O = other ).

C CK CK C CC D O CCK CC D CC C CK CK CC

a. Construct a frequency distribution table.b. Calculate the relative frequencies and percentages for all categories.c. Draw a pie chart for the percentage distribution.

2. The frequency distribution table represents the sale of certain product in ZeeZee Company. Each of the products was given the frequency of the sales in certain period. Find the relative frequency and the percentage of each product. Then, construct a pie chart using the obtained information.

Type of Product

Frequency Relative Frequency Percentage Angle Size

ABCDE

131259

113. Draw a time series graph to represent the data for the number of worldwide airline

fatalities for the given years.

Year 1990 1991 1992 1993 1994 1995 1996No. of fatalities

440 510 990 801 732 557 1132

4. A questionnaire about how people get news resulted in the following information from 25 respondents (N = newspaper, T = television, R = radio, M = magazine).

N N R T TR N T M RM M N R NT R M N MT R R N N

a. Construct a frequency distribution for the data.b. Construct a bar graph for the data.

5. The given information shows the export and import trade in million RM for four months of sales in certain year. Using the provided information, present this data in component bar graph.

Month Export ImportSeptember

OctoberNovemberDecember

28303224

20281714

6. The following information represents the maximum rain fall in millimeter (mm) in each state in Malaysia. You are supposed to help a meteorologist in your place to make an analysis. Based on your knowledge,



present this information using the most appropriate chart and give your comment.

State Quantity (mm)

PerlisKedahPulau PinangPerakSelangorWilayah Persekutuan Kuala LumpurNegeri SembilanMelakaJohorPahangTerengganuKelantanSarawakSabah

435512163721664

1003390223876

10501255986878456

2.3 ORGANIZING AND GRAPHING QUANTITATIVE DATA

2.3.1 Stem-and-Leaf Display

• In stem and leaf display of quantitative data, each value is divided into two

portions – a stem and a leaf. Then the leaves for each stem are shown

separately in a display.

• Gives the information of data pattern.

• Can detect which value frequently repeated.

25 12 9 10 5 12 23 736 13 11 12 31 28 37 614 41 38 44 13 22 18 19

Solution:

2.3.2 Frequency Distributions


Example 7


• A frequency distribution for quantitative data lists all the classes and the

number of values that belong to each class.

• Data presented in form of frequency distribution are called grouped data.

• The class boundary is given by the midpoint of the upper limit of one

class and the lower limit of the next class. Also called real class limit.

• To find the midpoint of the upper limit of the first class and the lower limit

of the second class, we divide the sum of these two limits by 2.

e.g.:+ =400 401

400.52

• Class Width (class size)

Class width = Upper boundary – Lower boundary

e.g. : Width of the first class = 600.5 – 400.5 = 200

• Class Midpoint or Mark


class boundary

FORMULA

ξ∆Σ λϖβ

FORMULA

ξ∆Σ λϖβ


Lower limit + Upper limitclass midpoint or mark =

2

e.g:+ =401 600

Midpoint of the 1st class = 500.52

• Constructing Frequency Distribution Tables

1. To decide the number of classes, we used Sturge’s formula, which is

c = 1 + 3.3 log n

where c is the no. of classes

n is the no. of observations in the data set.

2. Class width,

>

>

Largest value - Smallest value

Number of classesRange

i

ic

This class width is rounded up to a convenient number.

3. Lower Limit of the First Class or the Starting PointUse the smallest value in the data set.


Example 8

FORMULA

ξ∆Σ λϖβ

FORMULA

ξ∆Σ λϖβ


The following data give the total home runs hit by all players of each of the 30 Major League Baseball teams during 2004 season.

i) Number of classes, c = 1 + 3.3 log 30 = 1 + 3.3(1.48)

= 5.89 ≈ 6 class

ii) Class width, −>

>≈

242 135

6 17.8

18

i

iii) Starting Point = 135

Table 2.10 : Frequency Distribution for Data of Table 2.9Total Home Runs Tally f

135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242

|||| |||||||||| |||| ||||||||

1025634=∑ 30f

2.3.3 Relative Frequency and Percentage Distributions


FORMULA

ξ∆Σ λϖβ

0

2

4

6

8

10

12

1 Total home runs


•∑

Frequency of that classRelative frequency of a class =

Sum of all frequencies

=

Percentage = (Relative frequency) 100

f

f

(Refer example 8)

Table 2.11: Relative Frequency and Percentage Distributions

Total Home Runs Class BoundariesRelative

Frequency%

135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242

134.5 less than 152.5152.5 less than 170.5170.5 less than 188.5188.5 less than 206.5206.5 less than 224.5224.5 less than 242.5

0.33330.06670.16670.20000.10000.1333

33.33 6.6716.6720.0010.0013.33

Total 1.0 100%

2.3.4 Graphing Grouped Data

a) Histograms

A histogram is a graph in which the class boundaries are marked on the

horizontal axis and either the frequencies, relative frequencies, or percentages

are marked on the vertical axis. The frequencies, relative frequencies or

percentages are represented by the heights of the bars.

In histogram, the bars are drawn adjacent to each other and there is a space

between y axis and the first bar.

(Refer example 8)

Frequency histogram for Table 2.9

b) Polygon

A graph formed by joining the midpoints of the tops of successive bars in a

histogram with straight lines is called a polygon.


134.5 152.5 170.5 188.5 206.5 224.5 242.5

Example 9

Example 10


Frequency polygon for Table 2.11

For a very large data set, as the number of classes is increased (and the width of

classes is decreased), the frequency polygon eventually becomes a smooth

curve called a frequency distribution curve or simply a frequency curve.

Frequency distribution curve

c) Shape of Histogram

Same as polygon.

For a very large data set, as the number of classes is increased (and the width

of classes is decreased), the frequency polygon eventually becomes a smooth

curve called a frequency distribution curve or simply a frequency curve.

The most common of shapes are:

(i) Symmetric

(ii) Right skewed

(iii) Left skewed


Example 11

134.5 152.5 170.5 188.5 206.5 224.5 242.5


Symmetric histograms

Right skewed and Left skewed

Describing data using graphs helps us insight into the main characteristics of the

data.

When interpreting a graph, we should be very cautious. We should observe

carefully whether the frequency axis has been truncated or whether any axis has

been unnecessarily shortened or stretched.

2.3.5 Cumulative Frequency Distributions

• A cumulative frequency distribution gives the total number of values that

fall below the upper boundary of each class.


Example 12


Using the frequency distribution of table 2.11,

Total Home Runs

Class Boundaries fCumulative Frequency

135 – 152153 – 170171 – 188189 – 206207 – 224225 – 242

134.5 less than 152.5152.5 less than 170.5170.5 less than 188.5188.5 less than 206.5206.5 less than 224.5224.5 less than 242.5

10 2 5 6 3 4

• Ogive

An ogive is a curve drawn for the cumulative frequency distribution by joining

with straight lines the dots marked above the upper boundaries of classes at

heights equal to the cumulative frequencies of respective classes.

Two type of ogive:

(i) ogive less than(ii) ogive greater than

First, build a table of cumulative frequency.

(Ogive Less Than)Earnings

(RM)Number of students (f) Earnings (RM)

CumulativeFrequency (F)

30 – 39 5 Less than 29.5 040 – 49 6 Less than 39.5 550 – 59 6 Less than 49.5 1160 - 69 3 Less than 59.5 1770 – 79 3 Less than 69.5 2080 - 89 7 Less than 79.5 23

Less than 89.5 30Total 30

Graph Ogive Less Than


Example 13

0

5

10

15

20

25

30

35

29.5 39.5 49.5 59.5 69.5 79.5 89.5

Cu

mu

lati

ve

Fre

qu

ency

Earnings


(Ogive More Than)

Earnings (RM)

Number of students (f) Earnings (RM)

CumulativeFrequency (F)

30 – 39 5 More than 29.5 3040 – 49 6 More than 39.5 2550 – 59 6 More than 49.5 1960 - 69 3 More than 59.5 1370 – 79 3 More than 69.5 1080 - 89 7 More than 79.5 7

More than 89.5 0Total 30

Graph Ogive More Than

2.3.6 Box-Plot

• Describe the analyze data graphically using 5 measurement: smallest

value, first quartile (K1), second quartile (median or K2), third quartile

(K3) and largest value.


0

5

10

15

20

25

30

35

29.5 39.5 49.5 59.5 69.5 79.5 89.5

Earnings

Cumulative Frequency

Example 14


2.4 MEASURES OF CENTRAL TENDENCY

2.4.1 Ungrouped Data Measurement

• Mean

Mean for population data: x

Nµ = ∑

Mean for sample data: x

xn

= ∑

where: x∑ = the sum of all values

N = the population size n = the sample size,

µ = the population mean

x = the sample mean

The following data give the prices (rounded to thousand RM) of five homes sold recently in Sekayang.

158 189 265 127 191

Find the mean sale price for these homes.

Chapter 2: Descriptive Statistics

Smallest value

Largest value

K1 Median K3

Largest value

K1 Median K3

Largest value

K1 Median K3

Smallest value

Smallest value

For symmetry data

For left skewed data

For right skewed data

20

Example 15

FORMULA

ξ∆Σ λϖβ


Solution:

Thus, these five homes were sold for an average price of RM186 thousand @ RM186 000.

The mean has the advantage that its calculation includes each value of the data set.

• Weighted Mean

Used when have different needs.

Weight mean :

w

wxx

w= ∑

∑where w is a weight.

Consider the data of electricity components purchasing from a factory in the table below:

Type Number of component (w) Cost/unit (x)


Example 16

FORMULA

ξ∆Σ λϖβ


12345

1200 50025001000 800

RM3.00RM3.40RM2.80RM2.90RM3.25

Total 6000

Solution:

1200(3) 500(3.4) 2500(2.8) 1000(2.9) 800(3.25)

1200 500 2500 1000 80017800

60002.967

w

wxx

w=

+ + + ++ + + +

∑∑

=

=

=

Mean cost of a unit of the component is RM2.97

• Median

Median is the value of the middle term in a data set that has been

ranked in increasing order.

Procedure for finding the Median

Step 1: Rank the data set in increasing order.

Step 2: Determine the depth (position or location) of the median.

1

2

n +Depth of Median =

Step 3: Determine the value of the Median.

Find the median for the following data:10 5 19 8 3

Solution:

(1) Rank the data in increasing order


Example 17

FORMULA

ξ∆Σ λϖβ


(2) Determine the depth of the Median

1

25 1

23

n +

+

Depth of Median =

=

= (3) Determine the value of the median

Therefore the median is located in third position of the data set.

Hence, the Median for above data =

Find the median for the following data:10 5 19 8 3 15

Solution:

(1) Rank the data in increasing order

(2) Determine the depth of the Median

1

26 1

23.5

n +

+

Depth of Median =

=

=

(3) Determine the value of the Median

Therefore the median is located in the middle of 3rd position and 4th

position of the data set.8 10

92

+= =Median

Hence, the Median for the above data = The median gives the center of a histogram, with half of the data values

to the left of (or, less than) the median and half to the right of (or, more

than) the median.

The advantage of using the median is that it is not influenced by outliers.

• Mode

Mode is the value that occurs with the highest frequency in a data set.


Example 18


1. What is the mode for given data? 77 69 74 81 71 68 74 73

2. What is the mode for given data? 77 69 68 74 81 71 68 74 73

Solution:

1. Mode =

2. Mode =

A major shortcoming of the mode is that a data set may have none or

may have more than one mode.

One advantage of the mode is that it can be calculated for both kinds of

data, quantitative and qualitative.

2.4.2 Grouped Data Measurement

• Mean

Mean for population data:

fxμ =

N∑

Mean for sample data:

∑fxx =n

Where x the midpoint and f is the frequency of a class.

The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean.

Chapter 2: Descriptive Statistics

Number of order f

10 – 1213 – 1516 – 1819 – 21

4122014

n = 50

24

Example 19

Example 20

FORMULA

ξ∆Σ λϖβ


Solution:

Because the data set includes only 50 days, it represents a sample. The value of

fx∑ is calculated in the following table:

Number of order f x fx

10 – 1213 – 1516 – 1819 – 21

4122014

n = 50

The value of mean sample is:

Thus, this mail-order company received an average of 16.64 orders per day during these 50 days.

• Median

Step 1: Construct the cumulative frequency distribution.

Step 2: Decide the class that contain the median.

Class Median is the first class with the value of cumulative frequency is

at least n/2.

Step 3: Find the median by using the following formula:

Based on the grouped data below, find the median:

Time to travel to work Frequency


Median

÷ ÷ ÷

mm

n- F

2= L + if

Where:n = the total frequencyF = the total frequency before class

mediani = the class width

= the lower boundary of the class median

= the frequency of the class median

Example 21

FORMULA

ξ∆Σ λϖβ


1 – 1011 – 2021 – 3031 – 4041 – 50

8141297

Solution:

1st Step: Construct the cumulative frequency distribution

Time to travel to work Frequency Cumulative Frequency

1 – 1011 – 2021 – 3031 – 4041 – 50

8141297

Thus, 25 persons take less than 23 minutes to travel to work and another 25 persons take more than 23 minutes to travel to work.

• Mode

Mode is the value that has the highest frequency in a data set.

For grouped data, class mode (or, modal class) is the class with the

highest frequency.



Formula of mode for grouped data:

Where:

is the lower boundary of class mode

is the difference between the frequency of class mode and the frequency of the class before the class mode

is the difference between the frequency of class mode and the frequency of the class after the class mode

i is the class width

Based on the grouped data below, find the mode

Time to travel to work Frequency1 – 10

11 – 2021 – 3031 – 4041 – 50

8141297

Solution:

Based on the table,

We can also obtain the mode by using the histogram;


÷

Mode 1mo

1 2

Δ= L + i

Δ + Δ

moL

1∆

2∆

Example 22

FORMULA

ξ∆Σ λϖβ


2.4.3 Relationship among Mean, Median & Mode

• As discussed in previous topic, histogram or a frequency distribution curve

can assume either skewed shape or symmetrical shape.

• Knowing the value of mean, median and mode can give us some idea

about the shape of frequency curve.

(1) For a symmetrical histogram and frequency curve with one peak, the

value of the mean, median and mode are identical and they lie at the

center of the distribution.

Mean, median, and mode for a symmetric histogram and frequency distribution curve

(2) For a histogram and a frequency curve skewed to the right, the value of

the mean is the largest that of the mode is the smallest and the value

of the median lies between these two.



Mean, median, and mode for a histogram and frequency distribution curve skewed to the right

(3) For a histogram and a frequency curve skewed to the left, the value of the

mean is the smallest and that of the mode is the largest and the value

of the median lies between these two.

Mean, median, and mode for a histogram and frequency distribution curve skewed to the left

2.5 DISPERSION MEASUREMENT

• The measures of central tendency such as mean, median and mode do not

reveal the whole picture of the distribution of a data set.



• Two data sets with the same mean may have a completely different spreads.

• The variation among the values of observations for one data set may be

much larger or smaller than for the other data set.

2.5.1 Ungrouped Data Measurement

• Range

RANGE = Largest value – Smallest value

Find the range of production for this data set,

Solution:

Range = Largest value – Smallest value = 267 277 – 49 651 = 217 626

Disadvantages:

o being influenced by outliers.o based on two values only. All other values in a data set are ignored.

• Variance and Standard Deviation

Standard deviation is the most used measure of dispersion.

A Standard Deviation value tells how closely the values of a data set

clustered around the mean.


Example 23

FORMULA

ξ∆Σ λϖβ


Lower value of standard deviation indicates that the data set value are

spread over relatively smaller range around the mean.

Larger value of data set indicates that the data set value are spread

over relatively larger around the mean (far from mean).

Standard deviation is obtained the positive root of the variance:

Variance for population:

( )

NN

xx∑ ∑−

=

2

2

2σ

Variance for sample:

( )

1

2

2

2

−

−=

∑ ∑n

n

xx

s

Standard Deviation for population:2σσ =

Standard Deviation for sample:2ss =

Let x denote the total production (in unit) of company

Company ProductionABCDE

6293

1267534

Find the variance and standard deviation,

Solution:

Company Production (x) x2

ABCDE

6293

1267534


Example 24

FORMULA

ξ∆Σ λϖβ

FORMULA

ξ∆Σ λϖβ


390

The properties of variance and standard deviation:

o The standard deviation is a measure of variation of all values from the

mean.

o The value of the variance and the standard deviation are never

negative. Also, larger values of variance or standard deviation indicate

greater amounts of variation.

o The value of s can increase dramatically with the inclusion of one or

more outliers.

o The measurement units of variance are always the square of the

measurement units of the original data while the units of standard

deviation are the same as the units of the original data values.

2.5.2 Grouped Data Measurement

• Range


Range = Upper bound of last class – Lower bound of first classFORMULA

ξ∆Σ λϖβ


Class Frequency

41 – 5051 – 6061 – 7071 – 8081 – 9091 - 100

1 3 71310 6

Total 40

Upper bound of last class = 100.5Lower bound of first class = 40.5Range = 100.5 – 40.5 = 60

• Variance and Standard Deviation

Variance for population:

( ) 2

2

2−

σ =

∑∑fx

fxN

N

Variance for sample:

( ) 2

2

2

1

−=

−

∑∑fx

fxns

n

Standard Deviation:

Population: 2σσ =

Sample: 2ss =

Find the variance and standard deviation for the following data:

No. of order f

10 – 1213 – 1516 – 1819 – 21

4122014


Example 25

FORMULA

ξ∆Σ λϖβ

FORMULA

ξ∆Σ λϖβ


Total n = 50

Solution:

No. of order f x fx fx2

10 – 1213 – 1516 – 1819 – 21

4122014

Total n = 50

Variance,

Standard Deviation,

Thus, the standard deviation of the number of orders received at the office of this mail-order company during the past 50 days is 2.75.

2.5.3 Relative Dispersion Measurement

• To compare two or more distribution that has different unit based on their

dispersion OR

• To compare two or more distribution that has same unit but big different in

their value of mean.

• Also called modified coefficient or coefficient of variation, CV.



)(%100

)(%100

populationx

CV

samplex

sCV

−×

=

−×

=

σ

Given mean and standard deviation of monthly salary for two groups of worker who are working in ABC company- Group 1: 700 & 20 and Group 2 :1070 & 20. Find the CV for every group and determine which group is more dispersed.

Solution:

1

2

20100 2 86

70020

100 1 871070

= × =

= × =

CV % . %

CV % . %

The monthly salary for group 1 worker is more dispersed compared to group 2.

2.6 MEASURE OF POSITION

• Determines the position of a single value in relation to other values in a

sample or a population data set.

• Quartiles

Quartiles are three summary measures that divide ranked data set into four equal parts.

o The 1st quartiles – denoted as Q1

1

4

+1Depth of Q =

n


Example 26

FORMULA

ξ∆Σ λϖβ

FORMULA

ξ∆Σ λϖβ


o The 2nd quartiles – median of a data set or Q2

o The 3rd quartiles – denoted as Q3

3 1

4

+3Depth of Q =

(n )

Table below lists the total revenue for the 11 top tourism company in Malaysia

109.7 79.9 21.2 76.4 80.2 82.1 79.4 89.3 98.0 103.5

86.8

Solution:

Step 1: Arrange the data in increasing order

76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7

121.2

Step 2: Determine the depth for Q1 and Q3

1 11 13

4 4

+ +1Depth of Q = = =

n

( )3 11 13 19

4 4

++3Depth of Q = = =

(n )

Step 3: Determine the Q1 and Q3

76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7

121.2

Q1 = 79.9 ; Q3 = 103.5

Table below list the total revenue for the 12 top tourism company in Malaysia

109.7 79.9 74.1 121.2 76.4 80.2 82.1 79.4 89.3

98.0 103.5 86.8

Solution:

Step 1: Arrange the data in increasing order

74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7

121.2


Example 27

Example 28

FORMULA

ξ∆Σ λϖβ


Step 2: Determine the depth for Q1 and Q3

1 12 13 25

4 4

+ +1Depth of Q = = =

n.

( )3 12 13 19 75

4 4

++3Depth of Q = = =

(n ).

Step 3: Determine the Q1 and Q3

74.1 76.4 79.4 79.9 80.2 82.1 86.8 89.3 98.0 103.5 109.7

121.2

Q1 = 79.4 + 0.25 (79.9 – 79.4) = 79.525

Q3 = 98.0 + 0.75 (103.5 – 98.0) = 102.125

• Interquartile Range

The difference between the third quartile and the first quartile for a data

set.

IQR = Q3 – Q1

By referring to example 28, calculate the IQR.

Solution:

IQR = Q3 – Q1 = 102.125 – 79.525 = 22.62.6.2 Grouped Data Measurement

• Quartiles

From Median, we can get Q1 and Q3 equation as follows:

1

1

1 QQ

n- F4Q L + if

÷

= ÷ ÷


Example 29

FORMULA

ξ∆Σ λϖβ

FORMULA

ξ∆Σ λϖβ


÷

= ÷ ÷

3

3

3 QQ

3n- F

4Q L + if

Refer to example 22, find Q1 and Q3

Solution:

1st Step: Construct the cumulative frequency distribution

Time to travel to work Frequency Cumulative Frequency

1 – 1011 – 2021 – 3031 – 4041 – 50

81412 9 7

822344350

2nd Step: Determine the Q1 and Q3

1

n 50Class Q 12 5

4 4.= = =

Class Q1 is the 2nd class

Therefore,

1

1

14

12 5 810 5 10

14

13 7143

÷

= + ÷ ÷

= + ÷ =

QQ

n- F

Q L if

. - .

.

( )3

3 503nClass Q 37 5

4 4.= = =

Class Q3 is the 4th class


Example 30


Therefore,

3

3

34

37 5 3430 5 10

9

34 3889

÷

= + ÷ ÷

= + ÷ =

QQ

n- F

Q L if

. - .

.

• Interquartile Range

IQR = Q3 – Q1

Refer to example 30, calculate the IQR.

Solution:

IQR = Q3 – Q1 = 34.3889 – 13.7143 = 20.6746

2.7 MEASURE OF SKEWNESS

• To determine the skewness of data (symmetry, left skewed, right skewed)

• Also called Skewness Coefficient or Pearson Coefficient of Skewness

3( )

k k

mean mode mean medians or s

s s

− −= =

If Sk +ve right skewed

If Sk -ve left skewed


Example 31

FORMULA

ξ∆Σ λϖβ


If Sk = 0 symmetry

If Sk takes a value in between (-0.9999, -0.0001) or (0.0001,

0.9999) approximately symmetry.

The duration of cancer patient warded in Hospital Seberang Jaya recorded in a frequency distribution. From the record, the mean is 28 days, median is 25 days and mode is 23 days. Given the standard deviation is 4.2 days.

a. What is the type of distribution?

b. Find the skewness coefficient

Solution:

This distribution is right skewed because the mean is the largest value

( ) ( )

28 2311905

4 2

3 3 28 2521429

4 2

Mean - Mode

OR

Mean - Median

−= = =

−= = =

k

k

S .s .

S .s .

So, from the Sk value this distribution is right skewed.

ADDITIONAL INFORMATION

Use of Standard Deviation

1. Chebyshev’s Theorem

• According to Chebyshev’s Theorem, for any number k greater than 1, at least (1 – 1/k2) of the data values lie within k standard deviations of the mean.

( )%[email protected]

2

11

11

2

2

=

−=

−=k


Example 32


• Thus; for example if k = 2, then

• Therefore, according to Chebyshev’s Theorem, at least 75% of the values of a data set lie within two standard deviation of the mean

2. Empirical Rule

• For a bell-shaped distribution, approximately

1.68%of the observations lie within one standard deviation of the mean.

2.95% of the observations lie within two standard deviations of mean.

3.99.7% of the observations lie within three standard deviations of the mean.

Measure of Position

1. Ungrouped Data - Quartile Deviation

• QD is a mean for Interquartile Range

• It used to compare the dissemination of two data set.

• If the QD value is high, it means that the data is more

disseminated.

Quartile Deviation = Interquartile Range / 2 = (Q3 - Q1) / 2



2. Ungrouped Data – Percentile

Pk = value of the (kn)th term in a ranked set 100

Where: k = the number of percentile n = the sample size

Percentile rank of xi = Number of values than xi X 100 Total number of values in the data set



EXERCISE 2

1. A survey research company asks 100 people how many times they have been to the dentist in the last five years. Their grouped responses appear below.

Number of Visits Number of Responses0 – 4 165 – 9 25

10 – 14 4815 – 19 11

What are the mean and variance of the data?

2. A researcher asked 25 consumers: “How much would you pay for a television adapter that provides Internet access?” Their grouped responses are as follows:

Amount ($) Number of Responses

0 – 99 2100 – 199 2200 – 249 3250 – 299 3300 – 349 6350 – 399 3400 – 499 4500 – 999 2

Calculate the mean, variance, and standard deviation.

3. The following data give the pairs of shoes sold per day by a particular shoe store in the last 20 days.

85 90 89 70 79 80 83 83 75 7689 86 71 76 77 89 70 65 90 86

Calculate thea. mean and interpret the value.

b. median and interpret the value.

c.mode and interpret the value.

d. standard deviation.

4. The followings data shows the information of serving time (in minutes) for 40 customers in a post office:

2.0 4.5 2.5 2.9 4.2 2.9 3.5 2.83.2 2.9 4.0 3.0 3.8 2.5 2.3 3.52.1 3.1 3.6 4.3 4.7 2.6 4.1 3.14.6 2.8 5.1 2.7 2.6 4.4 3.5 3.02.7 3.9 2.9 2.9 2.5 3.7 3.3 2.4

a.Construct a frequency distribution table with 0.5 of class width.



b.Construct a histogram.

c.Calculate the mode and median of the data.

d.Find the mean of serving time.

e.Determine the skewness of the data.

f.Find the first and third quartile value of the data.

g.Determine the value of interquartile range.

5. In a survey for a class of final semester student, a group of data was obtained for the number of text books owned.

Number of students Number of text book owned

129

1115108

553210

Find the average number of text book for the class. Use the weighted mean.

6.The following data represent the ages of 15 people buying lift tickets at a ski area.

15 25 26 17 38 16 60 2130 53 28 40 20 35 31

Calculate the quartile and interquartile range.

7.A student scores 60 on a mathematics test that has a mean of 54 and a standard deviation of 3, and she scores 80 on a history test with a mean of 75 and a standard deviation of 2. On which test did she perform better?

8.The following table gives the distribution of the share’s price for ABC Company which was listed in BSKL in 2005.

Price (RM) Frequency12 – 1415 – 1718 – 2021 – 2324 – 2627 - 29

51425763

Find the mean, median and mode for this data.


Sqqs1013 ch2-a122

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