*
STATISTICS
Statistics
*Statistics is
the area of science that deals with
collection, organization, analysis, and interpretation
of data.
*A
collection of numerical information is called
statistics.
*Because many aspects of engineering practice involve
working with data, obviously some knowledge of
statistics is important to an engineer.
•Specifically, statistical techniques can be a powerful aid in
designing new products and systems, improving existing
designs, and improving production process.
*the
methods of statistics
allow
scientists
and
engineers to design valid
experiments and to draw
reliable conclusions from
the data they produce
Basic Terms in Statistics
* Population
- Entire collection of individuals which are characteristic being studied.
* Sample
- A portion, or part of the population interest.
* Variable
- Characteristics of the individuals within the population.
* Observation
- Value of variable for an element.
* Data Set
- A collection of observation on one or more variables.
Collecting Data
*Direct observation
The simplest method of obtaining data.
Advantage: relatively inexpensive
Disadvantage: difficult to produce useful information since it
does not consider all aspects regarding the issues.
*Experiments
More expensive methods but better way to produce data
Data produced are called experimental
Surveys
Most familiar methods of data collection
Depends on the response rate
Personal Interview
Has the advantage of having higher expected
response rate
Fewer incorrect respondents.
Grouped Data Vs Ungrouped Data
*Grouped data - Data that has been organized into
groups (into a frequency distribution).
*Ungrouped data - Data that has not been organized
into groups. Also called as raw data.
*
Graphical Data Presentation
* Data can be summarized or presented in two ways:
1. Tabular
2. Charts/graphs.
* The presentations usually depends on the type (nature) of data
whether the data is in qualitative (such as gender and ethnic
group) or quantitative (such as income and CGPA).
Data Presentation of Qualitative Data
* Tabular presentation for qualitative data is usually in the
*
form of frequency table that is a table represents the
number of times the observation occurs in the data.
*Qualitative :- characteristic being studied is nonnumeric.
Examples:- gender, religious affiliation or eye color.
The most popular charts for qualitative data are:
1. bar chart/column chart;
2. pie chart; and
3. line chart.
Types of Graph
Qualitative Data
Example 1.1:
frequency table
Observation Frequency
Malay
33
Chinese
9
Indian
6
Others
2
*Bar Chart: used to display the frequency distribution in the
graphical form.
Example 1.2:
*Pie Chart: used to display the frequency distribution. It
displays the ratio of the observations
Malay
Chinese
Indian
Others
Example 1.3 :
Jan
10
Feb
7
Mar
5
Apr
10
May
39
Jun
7
Jul
260
Aug
316
Sep
142
Oct
11
Nov
4
Dec
9
*Line chart: used to display the trend of observations. It is a
very popular display for the data which represent time.
Example 1.4
Data Presentation Of Quantitative Data
* Tabular presentation for quantitative data is usually
in the form of frequency distribution that is a
table represent the frequency of the observation
that fall inside some specific classes (intervals).
*Quantitative : variable studied are numerically. Examples:balanced in accounts, ages of students, the life of an
automobiles batteries such as 42 months).
* Frequency distribution: A grouping of data into mutually
exclusive classes showing the number of observations in
each class.
* There are few graphs available for the
graphical presentation of the quantitative data.
The most popular graphs are:
1. histogram;
2. frequency polygon; and
3. ogive.
Example 1.5: Frequency Distribution
Weight (Rounded decimal point)
Frequency
60-62
5
63-65
18
66-68
42
69-71
27
72-74
8
*Histogram: Looks like the bar chart except that
the horizontal axis represent the data which
is quantitative in nature. There is no gap between
the bars.
Example 1.6:
*Frequency Polygon: looks like the line chart except that the
horizontal axis represent the class mark of the data which is
quantitative in nature.
Example 1.7 :
*Ogive: line graph with the horizontal axis represent the upper
limit of the class interval while the vertical axis represent the
cummulative frequencies.
Example 1.8 :
*
Constructing Frequency Distribution
*When summarizing large quantities of raw data, it is often useful to
distribute the data into classes. Table 1.1 shows that the number of
classes for Students` weight.
Weight
60-62
63-65
66-68
69-71
72-74
Total
Frequency
5
18
42
27
8
100
Table 1.1: Weight of 100 male students
in XYZ university
*A frequency distribution for quantitative data lists all the classes and
the number of values that belong to each class.
*Data presented in the form of a frequency distribution are called
grouped data.
* For quantitative data, an interval that includes all the values that fall within two
numbers; the lower and upper class which is called class.
* Class is in first column for frequency distribution table.
*Classes always represent a variable, non-overlapping; each value is belong to one
and only one class.
* The numbers listed in second column are called frequencies, which gives the
number of values that belong to different classes. Frequencies denoted by f.
Table 1.2 : Weight of 100 male students in XYZ university
Variable
Third class
(Interval Class)
Lower Limit
of the fifth class
Weight
60-62
63-65
66-68
69-71
72-74
Total
Frequency
5
18
42
27
8
100
Upper limit of the fifthclass
Frequency
column
Frequency
of the third
class.
*The class boundary is given by the midpoint of the upper
limit of one class and the lower limit of the next class.
*The difference between the two boundaries of a class gives the
class width; also called class size.
Formula:
- Class Midpoint or Mark
Class midpoint or mark = (Lower Limit + Upper Limit)/2
- Finding The Number of Classes
Number of classes, c = 1  3.3log n
- Finding Class Width For Interval Class (Sturge`s Rule)
class width , i = (Largest value – Smallest value)/Number of classes
* Any convenient number that is equal to or less than the smallest values in the
data set can be used as the lower limit of the first class.
Example 1.9:
From Table 1.1: Class Boundary
Weight (Class
Interval)
60-62
63-65
66-68
69-71
72-74
Total
Class
Boundary
59.5-62.5
62.5-65.5
65.5-68.5
68.5-71.5
71.5-74.5
Frequency
5
18
42
27
8
100
Example 1.10:
Given a raw data as below:
27 27
27
28
27
26 28
26
28
31
33 28
35
39
a)
b)
c)
d)
20
30
25
26
How many classes that you recommend?
How many class interval?
Build a frequency distribution table.
What is the lower boundary for the first class?
28
26
Cumulative Frequency Distributions
* A cumulative frequency distribution gives the total number of values that fall
below the upper boundary of each class.
* In cumulative frequency distribution table, each class has the same lower limit
but a different upper limit.
Table 1.3: Class Limit, Class Boundaries, Class Width , Cumulative Frequency
Weight
(Class
Interva;)
Number of
Students, f
Class
Boundaries
Cumulative
Frequency
60-62
5
59.5-62.5
5
63-65
18
62.5-65.5
5 + 18 = 23
66-68
42
65.5-68.5
23 + 42 = 65
69-71
27
68.5-71.5
65 + 27 =92
72-74
8
71.5-74.5
92 + 8 = 100
100
Exercise 1 :
The data below represent the waiting time (in minutes)
taken by 30 customers at one local bank.
25 31
20
30
22
32
37
28
29 23
35
25
29
35
29
27
23
32
31
32
24
35
21
35
35
22
33
24
39
43
* Construct a frequency distribution and cumulative
frequency distribution table.
* Construct a histogram.
*
• Measures of Central Tendency
•Measures of Dispersion
•Measures of Position
Data Summary
Summary statistics are used to summarize a set of observations.
Two basic summary statistics are measures of central tendency and measures of
dispersion.
Measures of Central Tendency
* Mean
* Median
* Mode
Measures of Dispersion
* Range
* Variance
* Standard deviation
Measures of Position
* Z scores
* Percentiles
* Quartiles
* Outliers
Measures of Central Tendency
*Mean
Mean of a sample is the sum of the sample data divided by the
total number sample.
Mean for ungrouped data is given by:
_
x
x1  x2  .......  xn x

x
, for n  1,2,..., n or x 
n
n
_
Mean for group data is given by:
n

x
fx
fx

or
f

f

i 1
n
i 1
i i
i
Example 1.11 (Ungrouped data):
Mean for the sets of data 3,5,2,6,5,9,5,2,8,6
Solution :
35 2 6595 28 6
x
 5.1
10
Example 1.12 (Grouped Data):
Use the frequency distribution of weights 100 male
students in XYZ university, to find the mean.
Weight
Frequency
60-62
63-65
66-68
69-71
72-74
5
18
42
27
8
Solution :
Weight
(Class
Interval
Frequency, f
60-62
63-65
66-68
69-71
72-74
5
18
42
27
8
fx

x
?
f
Class
Mark, x
fx
*Median of ungrouped data: The median depends on the
number of observations in the data, n . If n is odd, then the
median is the (n+1)/2 th observation of the ordered observations.
But if is even, then the median is the arithmetic mean of the
n/2 th observation and the (n+1)/2 th observation.
*Median of grouped data:
 f


F

j 1 
2
x  Lc

f


j


where
L = the lower class boundary of the median class
c = the size of median class interval
Fj 1  the sum of frequencies of all classes lower than the median class
f j  the frequency of the median class
Example 1.13 (Ungrouped data):
The median for data 4,6,3,1,2,5,7 is 4
Rearrange the data : 1,2,3,4,5,6,7
median
Example 1.14 (Grouped Data):
The sample median for frequency distribution as in
example 1.12
Solution:
Weight
(Class
Interval
Frequency,
f
Class
Mark,
x
fx
60-62
63-65
66-68
69-71
72-74
5
18
42
27
8
61
64
67
70
73
305
1152
2814
1890
584
 f


F

j 1 
2
x  Lc
?
f


j


Cumulative
Frequency,
F
Class
Boundary
Mode
Mode of ungrouped data: The value with the highest
frequency in a data set.
*It is important to note that there can be more than one
mode and if no number occurs more than once in the set,
then there is no mode for that set of numbers
*Mode for grouped data
When data has been grouped in classes and a frequency curveis drawn
to fit the data, the mode is the value of x corresponding to the maximum
point on the curve, that is
 1 
xˆ  L  c 




2
 1
L  the lower class boundary of the modal class
c = the size of the modal class interval
1  the difference between the modal class frequency and the class before it
 2  the difference between the modal class frequency and the class after it
*the class which has the highest frequency is called the modal class
Example 1.15 (Ungrouped data)
Find the mode for the sets of data 3, 5, 2, 6, 5, 9, 5, 2, 8, 6
Mode = number occurring most frequently = 5
Example 1.16 Find the mode of the sample data below
Solution:
Weight Frequency Class
(Class
,f
Mark
Interval
,x
Mode class
60-62
63-65
66-68
69-71
72-74
5
18
42
27
8
Total
100
 1 
ˆx  L  c 
 ?
 1   2 
61
64
67
70
73
fx
Cumulative
Frequency,
F
Class
Boundary
305
1152
2814
1890
584
5
23
65
92
100
59.5-62.5
62.5-65.5
65.5-68.5
68.5-71.5
71.5-74.5
6745
Measures of Dispersion
*Range = Largest value – smallest value
*Variance: measures the variability (differences) existing in a set
of data.
The variance for the ungrouped data:

* 2  ( x  x)
S 
n 1
2
(for sample)

S2
( x  x)


The variance for the grouped data:
2
S2 
S
2
2
fx
 nx
*n  1
fx

*
2
2
nx
n
2
(
fx
)
fx 2  

2
n
or S 
n 1
2
(
fx
)

fx 2 

n
or S 2 
n
2
(for population)
n
(for sample)
(for population)
 The positive square root of the variance is the standard
deviation

S
 ( x  x)
n 1
2

 fx
2
2
nx
n 1
*A large variance means that the individual scores (data) of
the sample deviate a lot from the mean.
*A small variance indicates the scores (data) deviate little from
the mean.
Example 1.17 (Ungrouped data)
Find the variance and standard deviation of the sample
data : 3, 5, 2, 6, 5, 9, 5, 2, 8, 6
2
(
x

x
)
2
s 
?
n 1
(x  x 2 )
s
?
n 1
Example 1.18 (Grouped data)
Find the variance and standard deviation of the sample
data below:
Weight
(Class
Interval
Frequency,
f
Class
Mark,
x
fx
60-62
63-65
66-68
69-71
72-74
5
18
42
27
8
61
64
67
70
73
305
1152
2814
1890
584
Total
100
S 
2

2
(
fx
)
fx 2  
n
?
n 1
Cumulative
Frequency,
F
Class
Boundary
5
23
65
92
100
59.5-62.5
62.5-65.5
65.5-68.5
68.5-71.5
71.5-74.5
6745
S

2
fx 2  n x
n 1
?
x
2
fx 2
Exercise 2:
The defects from machine A for a sample of products were
organized into the following:
Defects
(Class Interval)
Number of products get
defect, f (frequency)
2-6
1
7-11
4
12-16
10
17-21
3
22-26
2
What is the mean, median, mode, variance and standard
deviation.
Exercise 3:
The following data give the sample number of iPads sold by a mail
order company on each of 30 days. (Hint : 5 number of classes)
8 25
11
15
29
22
10
5
17
21
22 13
26
16
18
12
9
26
20
16
23 14
19
23
20
16
27
9
21
14
a)
b)
c)
Construct a frequency distribution table.
Find the mean, variance and standard deviation, mode and
median.
Construct a histogram.