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Basic Statistics
 Statistics in Engineering
 Collecting Engineering Data
 Data Summary and Presentation
 Probability Distributions
- Discrete Probability Distribution
- Continuous Probability Distribution
 Sampling Distributions of the Mean and Proportion

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
Population
- Entire collection of individuals which are characteristic being studied.
 Sample
- A portion, or part of the population interest.
 Variable
- Characteristics which make different values.
 Observation
- Value of variable for an element.
 Data Set
- A collection of observation on one or more variables.


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 - 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.
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:
frequency table
Observation Frequency
Malay
33
Chinese
9
Indian
6
Others
2

Bar Chart: used to display the frequency distribution in the
graphical form.

Pie Chart: used to display the frequency distribution. It
displays the ratio of the observations
Malay
Chinese
Indian
Others

Line chart: used to display the trend of observations. It is
a very popular display for the data which represent time.
Jan
10
Feb
7
Mar
5
Apr
10
May
39
Jun
7
Jul
260
Aug
316
Sep
142
Oct
11
Nov
4
Dec
9
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: Frequency Distribution
CGPA (Class)

Frequency
2.50 - 2.75
2
2.75 - 3.00
10
3.00 - 3.25
15
3.25 - 3.50
13
3.50 - 3.75
7
3.75 - 4.00
3
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.

Frequency Polygon: looks like the line chart except that the
horizontal axis represent the class mark of the data which is
quantitative in nature.

Ogive: line graph with the horizontal axis represent the upper
limit of the class interval while the vertical axis represent the
cummulative frequencies.
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` CGPA.
CGPA (Class)
2.50 - 2.75
2.75 - 3.00
3.00 - 3.25
3.25 - 3.50
3.50 - 3.75
3.75 - 4.00
Total
Frequency
2
10
15
13
7
3
Table 1.1:The Fequency Distribution of
the Students’ CGPA
50
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 : Weekly Earnings of 100 Employees of a Company
Variable
Third class
(Interval Class)
Lower Limit
of the sixth class
Weekly Earnings
(dollars)
Number of
Employees, f
801-1000
9
1001-1200
22
1201-1400
39
1401-1600
15
1601-1800
9
1801-2000
6
Upper limit of the sixth class
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 = n
- Finding Class Width For Interval Class
Approximate class width = (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:
Given a raw data as below:
27 27
27
28
27
26 28
26
28
31
a)
b)
c)
d)
24
30
25
26
How many classes that you recommend?
What is the class interval?
What is the lower boundary for the first class?
Build a frequency distribution table.
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
Weekly
Number of
Class
Class
Cumulative
Earnings
Employees, f
Boundaries
Width
Frequency
(dollars)
(Class Limit)
801-1000
9
800.5 – 1000.5
200
9
1001-1200
22
1000.5 – 1200.5
200
9 + 22 = 31
1201-1400
39
1200.5 – 1400.5
200
31 + 39 = 70
1401-1600
15
1400.5 – 1600.5
200
70 + 15 = 85
1601-1800
9
1600.5 – 1800.5
200
85 + 9 = 94
1801-2000
6
1800.5 – 2000.5
200
94 + 6 = 100
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
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

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:
Table 1.4 : Example 1.12
Cumulative
Frequency
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 of grouped data:
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
)
(
x

x
)
 2
(for sample) S 2  
(for population)
S 
n 1
n

2
The variance for the grouped data:
2
S2 
S
n 1
fx



2
2
fx
  n x
2
2
nx
n
2
(
fx
)
fx 2  

or 2
n
S 
n 1
2
(
fx
)

fx 2 
or

n
S2 
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.


7 , 6, 8, 5 , 9 ,4, 7 , 7 , 6, 6
Range = 9-4=5
 Mean _
x


x


Variance
n
 6.5

S2 
2
(
x

x
)

n 1

18.5
 2.0556
9
Standard Deviation

S
 ( x  x)
n 1
2
 2.0556  1.4337
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, variance and standard deviation.
The following data give the total 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 table.
Find the mean, variance and standard deviation, mode
and median.
Construct a histogram.

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
a) Construct a frequency table.
b) Find the mean, mode, median, variance and standard
deviation..
c) Construct a histogram.
 The
Apollo space program lasted from 1967
until 1972 and included 13 missions. The
missions lasted from as little as 7 hours to as
long as 301 hours. The duration of each flight
is as below. Find mean, median, standard
deviation. (Counted as a population)
9 195
192 147
241
10
301
295
216
142
260
7
244