Download SHF/SEF1124 CHAPTER 3: INTODUCTION TO STATISTICS The

SHF/SEF1124
CHAPTER 3: INTODUCTION TO STATISTICS
The Statistical Process
Statistical POPULATION :
-Collection of data we wish to gather
information about
- Eg: All students of CFS IIUM
Make Inferences :
Determine what the
statistics tell us
about the Population
Sample Statistics :
-Graphic : Eg: Histogram, Ogive,
Frequency Polygon
-Numeric : Eg: Mean, Standard
Deviation
Plan the Investigation:
What? How? Who? Where?
Collect the Sample
SAMPLE:
Data collected from Population
-Eg: Students of Dept. of
Collect
Science
Analyze the Data :
Organize, Describe &
Present them
3.1 Introduction
Statistics:
 A field of study which implies collecting, presenting, analyzing and interpreting data
as a basis for explanation, description and comparison.
 used to analyze the results of surveys and as a tool in scientific research to make
decisions based on controlled experiments.
 Also useful for operations, research, quality control, estimation and prediction.
Population: a collection, or set of individuals or objects or events whose properties are
to be analyzed.
Sample: a group of subjects selected from the population. Sample is a subset of a
population.
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Data: consist a set of recorded observations or values. Any quantity that can have a
number of values is variable. Variables whose values are determined by chance are
called random variables.
Data set: a collection of data values. Each value in the data set is called a data value or a
datum.
Variable: a characteristics or attribute that can assume different values.
A statistical exercise normally consists of 4 stages:
i) Collection of data by counting or measuring.
ii) Ordering and presentation of the data in a convenient form.
iii) Analysis of the collected data.
iv) Interpretation of the results and conclusions formulated.
3.1.1 Two branches of Statistics
STATISTICS
DESCRIPTIVE STATISTICS
INFERENTIAL STATISTICS
Consists of the collection,
organization,
summarization
and presentation of data.
-Describes a situation. Data
presented in the form of charts,
graphs or tables.
-Make
use
of
graphical
techniques
and
numerical
descriptive measures such as
average to summarize and
present the data.
-E.g.: National census conducted
by Malaysian goverment every 5
years or 10 years. The results of
this
census
give
some
information regarding average
age,
income
and
other
characteristics of the Malaysian
population
Consists of generalizing from samples to
populations, performing hypothesis
tests, detemining relationships among
variables and making prediction
- Inferences are made from samples to
populations
-Use probability, that is the chance of an
event occurring.
-The area of inferential statistics called
hypotesis testing is a decision-making
process for evaluating claims about a
population, based on information
obtined from samples.
- E.g.: A researcher may want to know if
a new product of skin lotion containing
aloe vera will reduce the skin problem
on children. For this study, two group of
young children would be selected. One
group would be given the lotion
containing aloe vera and the other would
be given a normal lotion without
containing aloe vera. As aresult is
observed by experts to see the
effectiveness of the new product.
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3.1.2 Variables and Types of Data
LEVEL OF MEASUREMENT
NOMINAL
QUALITATIVE
ORDINAL
TYPES OF DATA
(VARIABLES)
RATIO
CONTINUOUS
QUANTITATIVE
INTERVAL
DISCRETE
 Statisticians gain information about a particular situation by collecting data for
random variables.
 Types of Data (variables)
1) Qualitative variables



Variables that can be placed into distinct categories, according to some
characteristics or attribute.
Nonnumeric categories
E.g.: Gender , color, religion , workplace and etc
2) Quantitative variables


It is numerical in nature and can be ordered or ranked.
A quantitative variable may be one of two kinds:
 Discrete variable – a variable that can be counted or for which there is a fixed
set of values. Example: the number of children in a family, the number of
students in a class and etc
 Continuous variable – a variable that can be measured on continuous scale ,
the result depending on the precision of the measuring instrument, or the
accuracy of the observer. Continuous variable can assume all values between
any two specific values. Example: temperatures, heights, weights, time taken
and etc.
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 Variables can be classified by how they are categorized, counted or measured. Data/
variables can be classified according to the LEVEL OF MEASUREMENT as follows:
1) Nominal Level Data: - classifies data (persons/objects) into two or more
categories. Whatever the basis for classification, a person can only be in one
category and members of a given category have a common set of characteristics.
 The lowest level of measurement.
 No ranking/order can be placed on the data
 E.g. : Gender (Male / Female) , Type of school (Public / Private),
Height (Tall/Short) , etc
2) Ordinal Level Data:- classifies data into categories that can be ranked; however
precise differences between the ranks do not exist.
 This type of measuring scale puts the data/subjects in order from highest to
lowest, from most to least. It does not indicate how much higher or how
much better. Intervals between ranks are not equal.
 E.g.: Letter grades (A,B,C,D,E,F) ; Man’s build (small, medium, or large)-large
variation exists among the individuals in each class.
3) Interval Level Data:- has all characteristics of a nominal and ordinal scale but in
addition it is based upon predetermined equal interval. It has no true zero point
(ratio between number on the scale are not meaningful). E.g.:
 Achievement test; aptitude tests, IQ test. A one point difference between IQ
test of 110 and an IQ of 111 gives a significant difference.
 The Fahrenheit scale is a clear example of the interval scale of measurement.
Thus, 60 degree Fahrenheit or -10 degrees Fahrenheit represent interval
data. Measurement of Sea Level is another example of an interval scale. With
each of these scales there are direct, measurable quantities with equality of
units. In addition, zero does not represent the absolute lowest value. Rather,
it is point on the scale with numbers both above and below it (for example,
-10degrees Fahrenheit).
4) Ratio Level Data:- possesses all the characteristics of interval scale and in
addition it has a meaningful (true zero point). True ratios exist when the same
variable is measured on two different members of the population.
 The highest, most precise level of measurement.
 E.g.: Weight, number of calls received; height.
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3.1.3 Data collection and Sampling Techniques
 Sampling is the process of selecting a number of individuals for a study in such a way
that the individuals represent the larger group from which they were selected.
 The purpose of sampling is to use a sample to gain information about a population.
 In order to obtain samples that are unbiased, statisticians use 4 basic methods of
sampling:
i) Random Sampling: subjects are selected by random numbers.
ii) Systematic Sampling: Subjects are selected by using every kth number after
the first subject is randomly from 1 through k.
iii) Stratified Sampling: Subjects are selected by dividing up the population into
groups (strata) and subjects within groups are randomly selected.
- E.g.: We divide the population into 5 group then we take the subjects from
each group to become our sample.
iv) Cluster Sampling: Subjects are selected by using an intact group that is
representative of the population.
- E.g.: We divide the population into 5 group then we take 2 groups to
become our sample. That means 2 group of subject represent 5 groups of
subjects.
Exercise:
A ) Classify each set of data as discrete or continuous.
1) The number of suitcases lost by an airline.
2) The height of corn plants.
3) The number of ears of corn produced.
4) The number of green M&M's in a bag.
5) The time it takes for a car battery to die.
6) The production of tomatoes by weight.
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B) Identify the following as nominal level, ordinal level, interval level, or ratio level data.
1) Percentage scores on a Math exam.
2) Letter grades on an English essay.
3) Flavors of yogurt.
4) Instructors classified as: Easy, Difficult or Impossible.
5) Employee evaluations classified as : Excellent, Average, Poor.
6) Religions.
7) Political parties.
8) Commuting times to school.
9) Years (AD) of important historical events.
10) Ages (in years) of statistics students.
11) Ice cream flavor preference.
12) Amount of money in savings accounts.
13) Students classified by their reading ability: Above average, Below average,
Normal.
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3.2
ORGANIZING DATA & PRESENTATION OF DATA
3.2.1 FREQUENCY DISTRIBUTION
A frequency distribution is the organization of raw data in table form, using classes and
frequencies.
There are three types of frequency distribution.
1. Categorical frequency distribution
-for data that can be placed in specific category
Example : The following data represent the color of men’s shirts purchased in the
men’s department of a large department store. Construct a frequency distribution
for the data. (W = White, BL = Blue, BR = Brown, Y =Yellow, G = Gray)
W
W
BL
Y
W
W
W
G
BL
BL
BR
BL
W
G
Y
Y
BR
BL
BR
W
BL
BL
W
G
W
BL
BR
W
BR
BL
W
BL
BL
W
W
W
BL
W
W
BR
Y
BR
BL
BR
G
G
Y
BR
Y
G
(A complete categorical distribution must have class, frequency & percentage column
in the table)
2. Grouped frequency distribution
-when the range of the data is large, the data must be grouped into classes.
Example: The ages of the signers of the Declaration of Independence are shown
below. Construct a frequency distribution for the data using seven classes.
41
54
47
40
39
35
50
37
49
42
70
32
44
52
39
50
40
30
34
69
39
45
33
42
44
63
60
27
42
34
50
42
52
38
36
45
35
43
48
46
31
27
55
63
46
33
60
62
35
46
45
34
53
50
50
Example: The number of calories per serving for selected ready-to-eat cereals is
listed here. Construct a frequency distribution using seven classes.
130 190 140 80
100 120 220 220 110 100
210 130 100 90
210 120 200 120 180 120
190 210 120 200 130 180 260 270 100 160
190 240 80
120 90
190 200 210 190 180
115 210 110 225 190 130
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3. Ungrouped frequency distribution
-when the range of data is small
Example: A survey taken in a restaurant shows the following number of cups of
coffee consumed with each meal. Construct frequency distribution.
0
2
2
1
1
2
3
5
3
2
2
2
1
0
1
2
4
2
0
1
0
1
4
4
2
2
0
1
1
5
Procedure to construct frequency distribution (this procedure is not unique):
1) Determine number of classes which normally 5 – 20
2) Find range = Highest value – lowest value
range
3) The class width should be an odd number. Class width =
and rounding
no. of class
up.
4) Class Limit :
Lower class limit =the lowest value or any number less than the lowest value.
Upper class limit = (Lower class limit + class width) -1
5) Class Boundary: (to separate classes so that there are no gap in the frequency
distribution)
Lower Class Boundary: Lower class limit -0.5
Upper Class Boundary: Upper class limit + 0.5
6) Find frequency and cumulative frequency.
Class width = Upper Class Boundary - Lower Class Boundary
= Lower class limit of one class - Lower class limit of next class
= Upper class limit of one class - Upper class limit of next class
Class Midpoint = (Lower Class Boundary + Upper Class Boundary)/2
= (Upper class limit + Lower class limit)/2
SHF/SEF1124
3.2.2 HISTOGRAMS, FREQUENCY POLYGONS AND OGIVES
Example:
For 108 randomly selected college applicants, the following frequency distribution for
entrance exam scores was obtained.
Class Limit Frequency
90 – 98
6
99 – 107
22
108 – 116
43
117 – 125
28
126 - 134
9
Construct:
1. Histogram
i) x-axis :class boundary
ii) x-axis :class boundary
y-axis : frequency
y-axis : relative frequency
2. Frequency Polygon
i) x-axis :class midpoint
ii) x-axis :class midpoint
y-axis : frequency
y-axis : relative frequency
3. Ogive
i) x-axis : class boundary
ii) x-axis : class boundary
y-axis : cumulative frequency
Relative frequency =
y-axis : cumulative relative frequency
f
f
Cumulative relative frequency =
cumulative frequency
or add the relative frequency in
f
each class to the total relative frequency.
SHF/SEF1124
Note: Graphing
Given the frequency distribution below:
Class Limit
0 – 19
20 – 39
Class Boundary
-0.5 – 19.5
19.5 – 39.5
f
Cf
13
18
13
31
The first value on the x-axis is -0.5 can be drawn as below
OR
-0.5
-0.5 19.5 39.5
19.5 39.5
All graphs must be drawn on the right side of y-axis and omit question on analyzing the
graph in exercise.
Exercise:
1. In a class of 35 students, the following grade distribution was found. Construct a
histogram, frequency polygon and ogive for the data. (A=4, B=3, C=2, D=1, F=0)
Grade Frequency
0
3
1
6
2
9
3
12
4
5
2. Using the histogram shown below. Construct
i) A frequency distribution
ii) A frequency polygon
iii) An ogive
y
7
6
6
5
5
4
3
2
1
3
3
2
1
x
21.5 24.5 27.5 30.5 33.5 36.5 39.5 42.5
Class Boundaries
SHF/SEF1124
3. The number of calories per serving for selected ready-to-eat cereals is listed here.
Construct a histogram, frequency polygon and ogive for the data using relative
frequency.
130 190 140 80
100 120 220 220 110 100
210 130 100 90
210 120 200 120 180 120
190 210 120 200 130 180 260 270 100 160
190 240 80
120 90
190 200 210 190 180
115 210 110 225 190 130
4. Below is a data set for the duration (in minutes) of a random sample of 24 longdistance phone calls:
1
20
10
20
12
23
3
7
18
12
4
5
15
7
29
10
18
10
10
23
4
12
8
6
a) Construct a frequency distribution table for the data using the classes “1 to 5” “6
to 10” etc.
b) Construct a cumulative frequency distribution table and use it to draw up an
ogive.
5. The following table refers to the 2003 average income (in thousand Ringgit) per
year for 20 employees of company A.
Income
(‘000 Frequency
Ringgit)
5 -9
6
10 – 14
3
15 – 19
2
20 – 24
4
25 – 29
3
30 – 34
2
a) Draw the histogram and frequency polygon for the above data.
b) Construct the cumulative frequency table. Hence, draw up an ogive for the above
data.
SHF/SEF1124
3.3
DATA DESCRIPTION
3.3.1 MEASURES OF CENTRAL TENDENCY
 Mean, median and Mode for Ungrouped data

Mean (arithmetic average)
Symbol for Sample: X
Symbol for Population: μ
(Syllabus focus on sample formula), Mean, X 
X
n

Median : (the middle point in ordered data set)

- arrange the data in order, ascending or descending
n 1
- select the middle point or use formula T 
, n is number of data.
2
- Then, the median is:
 the value at location T (for odd number of data)
 the average of the value at location T and the value at location (T +1)
(for even number of data)
Mode : the value that occur most often in the data set
Example:
1) The following data are the number of burglaries reported for a specific year for nine
western Pennsylvania universities. Find mean, median and mode.
61, 11, 1, 3, 2, 30, 18, 3, 7
2) Twelve major earthquakes had Richter magnitudes shown here. Find mean, median
and mode.
7.0 , 6.2 , 7.7 , 8.0 , 6.4 , 6.2 , 7.2 , 5.4 , 6.4 , 6.5 , 7.2 , 5.4
3) The number of hospitals for the five largest hospital systems is shown here. Find
mean, median and mode.
340, 75, 123, 259, 151
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 Mean, median and Mode for Ungrouped frequency distribution

Mean, X  

Median :

- find cumulative frequency
f
- Location of median 
2
Mode : the value with the largest frequency
f X
f
Example:
4) A survey taken in a restaurant. This ungrouped frequency distribution of the
number of cups of coffee consumed with each meal was obtained. Find mean,
median and mode.
Number of cups
0
1
2
3
4
5
Frequency
5
8
10
2
3
2
 Mean, median and Mode for Grouped frequency distribution

Mean, X  
f Xm
f
where; X m =class midpoint
(Student must show the working ie. Find midpoint and f X m )

Median :
- find cumulative frequency
- find location of median class 
f
2
f

F 

c
- Median  L   2
f






Where; L=lower boundary of the median class
F = cumulative frequency until the point L (before median class)
f = frequency of the median class
c =class width of median class
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
Mode :
- find location of modal class : class with the largest frequency
- Mode
c
where; L=lower boundary of the modal class
a = different between frequencies of modal class and the class before it.
b= different between frequencies of modal class and the class after it.
c =class width of median class
Example:
5) These numbers of books were read by each of the 28 students in a literature class.
Find mean, median and mode.
Number of books
0–2
3–5
6–8
9 – 11
12 – 14
Frequency
2
6
12
5
3
6) Eighty randomly selected light bulbs were tested to determine their lifetimes (in
hours). This frequency distribution was obtained. Find mean, median and mode.
Class Boundaries
Frequency
52.5 – 63.5
6
63.5 – 74.5
12
74.5 – 85.5
25
85.5 – 96.5
18
96.5 – 107.5
14
107.5 – 118.5
5
SHF/SEF1124
3.3.2 MEASURES OF VARIATION
Variance and Standard deviation (the spread of data set)
80 81 82
Group A
80
81
82
Group B
55
88
100
X =81
Variation, s2 =1
X =81
Variation, s2 =543
55
88
100
Even though the average for both groups is the same, the spread or variation of data in
the Group B larger than Group A.
Population variance , σ2
= (Σ(X -μ)2)/N
Variance
Sample variance , s2
Standard deviation
Population standard deviation , σ
= √(Σ(X -μ)2)/N
=√σ2
Sample standard deviation , s
(Syllabus focus on sample formula)
SHF/SEF1124
Sample variance and standard deviation
 For Ungrouped Data
Variance, s 2
 X  X 

2
n 1
where; X =individual value
Standard deviation, s  s 2 
 X  X 
2
n 1
X =sample mean
n = sample size
OR
  X 2 

 X   n 


Variance, s 2 
n 1
2
(Note:
X
2
is not the same as
  X 2 

 X   n 


Standard deviation, s  s 2 
n 1
 X 
2
2
)
Example:
1) The normal daily temperatures (in degrees Fahrenheit) in January for 10 selected
cities are as follows. Find the variance and standard deviation.
50
37
29
54
30
61
47
38
34
61
2) Twelve students were given an arithmetic test and the times (in minutes) to
complete it were
10
9
12
11
8
15
9
7
8
6
12
10
Find the variance and standard deviation.
SHF/SEF1124
 For Grouped Data

Variance, s 2 
  f X 2 
 m 
f X 

 f 

 f 1
2
m

Standard deviation, s  s 
2
  f X 2 
 m 
f X m2  

 f 

 f 1
(Students must show the working ie. Find f X m and f X m2 )
Example:
3) In a class of 29 students, this distribution of quiz scores was recorded. Find variance
and standard deviation.
Grade
0–2
3–5
6–8
9 – 11
12 – 14
Frequency
1
3
5
14
6
4) Eighty randomly selected light bulbs were tested to determine their lifetimes (in
hours). This frequency distribution was obtained. Find variance and standard
deviation.
Class Boundaries
Frequency
52.5 – 63.5
6
63.5 – 74.5
12
74.5 – 85.5
25
85.5 – 96.5
18
96.5 – 107.5
14
107.5 – 118.5
5
5) These data represent the scores (in words per minute) of 25 typists on a speed test.
Find variance and standard deviation.
Class limit
54 – 58
59 – 63
64 – 68
69 – 73
74 – 78
79 – 83
84 – 88
Frequency
2
5
8
0
4
5
1
SHF/SEF1124
3.3.3 MEASURES OF POSITION
Standard scores, percentiles, deciles and quartiles are used to locate the relative position
of the data value in the data set.
 Standard score / z-score
The z-score represent the number of standard deviations the data value is above or
below the mean.
z
X X
s
 if the z score is positive, the score is above the mean
 if the z score is negative, the score is below the mean
Example:
1) Let data set : 65 , 70 , 75 ,80 , 85
65
70
X -2s
75
X -s
z= -2 z= -1
80
85
X +2s X +s
X
z= 0 z= 1
For data value 83: z 
; X =75 , s =5
z= 2
83  75
 1.6
5
2) Test marks are shown here. On which test she perform better?
Math marks:
65
50
45
; X =53.3 , s=10.4
Biology marks:
80
75
70
; X =75 , s=5
zM 
65  53.3
 1.122
10.4
zB 
75  75
0
5
zM  zB , the relative position in math class is higher than her the relative
position in biology class. She performs better in math paper than biology paper.
(the marks that she get from biology paper is more than mathematics paper but we
cannot compare the marks directly because the papers are different i.e. number of
question, standard of questions and so on, that is why we have to compare the relative
position)
SHF/SEF1124
 Quartiles, deciles and percentile
For Ungrouped data
 Quartiles: divide the distribution into four group Q1 , Q2 , Q3
Smallest data
25%
Q1
Q2
25%
Q3
25%
Largest data
25%
Median
 arrange the data in order
 Find location of quartiles, c 
nq
where ; n = total number of values
4
q =quartile
i) If c is not whole number, round up to the next whole number
ii) If c is a whole number, take average of cth and (c+1)th
Example:
1) The weights in pounds in the data set. Find Q1 , Q2 , Q3.
16
18
22
19
3
21
17
2) The test score in the data set. Find Q1 , Q2 , Q3.
42
35
28
12
47
50
20
49
 Deciles: divide the distribution into 10 groups
Smallest data D1
D2
D3
D4
D5
D6
D7
D8 D9 Largest data
10% 10% 10% 10% 10% 10% 10% 10% 10%
Median
 arrange the data in order
 Find location of quartiles, c 
nd
where ; n = total number of values
10
d =decile
iii) If c is not whole number, round up to the next whole number
iv) If c is a whole number, take average of cth and (c+1)th
SHF/SEF1124
Example:
1) (from previous example) Find D5.
16
18
22
19
3
21
17
2) (from previous example)Find D7.
42
35
28
12
47
50
49
20
 Percentiles: divide the distribution into 100 equal groups
Smallest data P1
P2
P3
P97
P98 P99 Largest data
10% 10% 10% 10% 10% 10% 10% 10% 10%
D1 , D2, D3, … , D9 correspond to P10 , P20, P30, … , P90
Q1 , Q2 , Q3 correspond to P25 , P50, P75
Median = Q2 = D5 = P50
 arrange the data in order
 Find location of quartiles, c 
n p
where ; n = total number of values
100
p =percentile
v) If c is not whole number, round up to the next whole number
vi) If c is a whole number, take average of cth and (c+1)th
Example:
1) (from previous example) Find P33.
16
18
22
19
3
21
17
2) (from previous example)Find P60.
42
35
28
12
47
50
49
Finding percentile corresponding to given value, X
Percentile 
 number of values below X   0.5 100%
total number of values
Example of data set :
1
Find percentile for 4.
3  0.5
Percentile 
100%  70%
5
(round off the answer)
1
3
4
P70 = 4
5
20
SHF/SEF1124
Example:
2) (from previous example)Find the percentile rank for each test score in the data set.
42
35
28
12
47
50
49
(Data value 47 = P64 but previously when we want to find P60 the data value is 47b too.
So actually P60 closer to P64 which is data value 47)
For Grouped Data
METHOD 1: (USE PERCENTILE GRAPH)
x-axis: class boundaries
y-axis: relative cumulative frequency (percentage)
Cumulative relative frequency (%) =
cumulative frequency
100%
f
Graph:
i) percentile graph
Relative cumulative frequency (%)
100
25
P25
ii) Ogive using relative frequency
(iii) Ogive
Relative cumulative frequency
Cumulative Frequency
1.0
0.25
75
P25
18.75
P25
25% x 75 =18.75
SHF/SEF1124
METHOD 2: (USE FORMULA)
 n

 100  f  F 
Pn  Ln  
c
f




Example:
This distribution represents the data for weights of fifth-grade boys.
Weights (pounds)
frequency
52.5 – 55.5
9
55.5 – 58.5
12
58.5 – 61.5
17
61.5 – 64.5
22
64.5 – 67.5
15
1) Find the approximate weights corresponding to each percentile given by
constructing a percentile graph.
(i) Q1
(ii) D8
(iii) Median
(iv) P95
2) Find the approximate percentile ranks of the following weights.
(i) 57 pounds
(ii) 64 pounds
(iii) 62 pounds
(iv) 59 pounds
3) Find P63 by using the formula.
SHF/SEF1124
EXERCISE CHAPTER 3
1. What type of sampling is being employed if a country is divided into economic classes
and a sample is chosen from each class to be surveyed?
2. Given a set of data 5,2,8,14,10,5,7,10,m, n where X =7 and mode = 5. Find the
possible values of m and n.
(ans: m=5, n=4 or m =4 , n =5)
3. Find the value that corresponds to the 30th percentile of the following data set:
78
82
86
88
92
97
(ans: P30 =82)
4. Given the variance of the set of 8 data x1 , x2, x3, … , x8 is 5.67. If
X
2
the mean of the data.
 944.96 , find
(ans: 11.09)
5. Find Q3 for the given data set : 18,22,50,15,13,6,5,12
(ans: 20)
6. The number of credits in business courses that eight applicants took is 9, 12, 15, 27,
33, p, 63, 72. Given the value that corresponds to the 75th percentile is 54, find p.
(ans: 45)
7. The mean of 5, 10, 26, 30, 45, 32, x, y is 25 where x and y are constants. If x = 16, find
the median.
(ans: 28)
8. A physician is interested in studying scheduling procedures. She questions 40 patients
concerning the length of time in minutes that they waste past their scheduled appointment
time. The following data are obtained:
60
29
34
25
31
30
6
17
6
50
10
18
38
25
35
36
31
23
12
52
8
27
27
30
42
9
47
31
27
6
45
33
25
37
3
50
53
28
16
19
a) Construct a frequency distribution by using 7 classes (use 3 as lower limit of the first
class)
b) Find the mean, mode and standard deviation.
(ans: 28.15 , 31.3 , 14.63)
c) Draw an ogive by using relative frequency and estimate the median from the graph.