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 In Engineering

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 which make different values.
 Observation
- Value of variable for an element.
 Data Set
- A collection of observation on one or more variables.

Collecting Engineering 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
Example 1.3 :

Malay
Chines
e
Indian
Line chart: used to display the trend of observations. It is
a very popular display for the data which represent time.
Example 1.4

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 1.5: Frequency Distribution
Weight (Rounded decimal point)
60-62
63-65
66-68
69-71
72-74

Frequency
5
18
42
27
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 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, c = 1  3.3log n
- Finding Class Width For Interval Class
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
a)
b)
c)
d)
24
30
25
26
How many classes that you recommend?
What is the 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 1.1 :
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.
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
Weight Frequency Class
fx
Cumulative
Solution:
Mode class
(Class
Interval
,f
Mark
,x
60-62
63-65
66-68
69-71
72-74
5
18
42
27
8
61
64
67
70
73
Total
100
 1 
ˆx  L  c 
 ?
 1   2 
Frequency,
F
305
1152
2814
1890
584
6745
5
23
65
92
100
Class
Boundary
59.5-62.5
62.5-65.5
65.5-68.5
68.5-71.5
71.5-74.5
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
S
( x  x)


n 1
2

(for sample)
S
2
( x  x)


2
(for population)
n
The variance for the grouped data:
2
S2 
2
fx
 nx

n 1
fx

S 
2
2
2
nx
n
2
(
fx
)
fx 2  

2
n
S

or
n 1
2
(
fx
)

fx 2 

n
or S 2 
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 1.2
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 1.3
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.
Rules of Data Dispersion
By using the mean x and standard deviation, we can find the
percentage of total observations that fall within the given
interval about the mean.
i) Chebyshev’s1Theorem
At least (1  k 2 ) of the observations will be in the range of k
standard deviation from mean.
where k is the positive number exceed 1 or (k>1).
Applicable for any distribution /not normal distribution.
Steps:
x  ks
1) Determine the interval
1
(1

)
2) Find value of
k2
3) Change the value in step 2 to a percent
4) Write statement: at least the percent of data found in step 3
is in the interval found in step 1
Example 1.19 :
Consider a distribution of test scores that are badly skewed to the
right, with a sample mean of 80 and a sample standard deviation
of 5. If k=2, what is the percentage of the data fall in the interval
from mean?
Solution:
1) Determine interval
x  ks
2)
Find
1
1
k2
 1

3)
4)
3
4
 80  ( 2)(5)
 (70,90)
1
22
3
 75%
4
Convert into percentage:
Conclusion: At least 75% of the data is found in the interval
from 70 to 90
ii) Empirical Rule
Applicable for a symmetric bell shaped distribution / normal
distribution.
There are 3 rules:


i. 68% of the observations lie in the interval ( x  s, x  s )


ii. 95% of the observations lie in the interval ( x  2s, x  2s)
iii. 99.7% of the observations lie in the interval ( x  3s, x  3s )
Formula for k =Distance between mean and each point
standard deviation
Example 1.19
The age distribution of a sample of 5000 persons is bell shaped
with a mean of 40 yrs and a standard deviation of 12 yrs.
Determine the approximate percentage of people who are 16 to
64 yrs old.
Solution:

40  16 
k
12
24

12
2
95% of the people in the sample are 16 to 64 yrs old.
Measures of Position
To describe the relative position of a certain data value
within the entire set of data.
z scores
Percentiles
Quartiles
Outliers
Percentiles
Divide a data set which written in ascending order.
 Often used in order to give the relative standing of a
data value.
 Example:- In Engineering Statistics examination score
68 was at the 75th percentile. Means that
approximately 75% of the students score below 68
and 25% are above 68.


Determining the kth Percentile
Step 1: Arrange data in ascending order
Step 2: Compute i, using formula
 k 
i
n
100


k- percentile of data
n- number of individuals in data set
Step 3: a) If i is not an integer, round up to the next
highest integer.
b) If i is an integer, the kth percentile is the
arithmetic mean of the ith and (i+1)st data
value.
Example 1.20
The following data represent the number of inches of
rain in Chicago during the month of April for 20
randomly years.
2.47
1.14
5.22
3.48
6.28
3.97
4.02
0.97
4.77
5.50
3.94
3.41
6.14
2.78
7.69
4.11
1.85
2.34
4.00
5.79
a) Determine the number of inches of rain correspond to
15th percentile, P15

Finding the Percentile that Corresponds to a Data
Value
Step 1: Arrange the data in ascending order.
Step 2: Use formula
Percentile of x 
Number of data values less than x
100
n
Round this number to the nearest integer.
Example 1.21
Determine the percentile that correspond to the 15th
number of inches of rains with earn 5.22 inches of
rain.
Quartiles

Divide data sets into fourths or four equal parts.
Smallest
data value Q1
25%
of data
Q2
25%
of data
Largest
Q3 data value
25%
of data
25%
of data
Example 1.22
(Based on data in example 1.20)
Determine the quartiles.
Outliers
Extreme observations
 Can occur because of the error in measurement of a
variable, during data entry or errors in sampling.

Checking for outliers by using Quartiles
Step 1: Determine the first and third quartiles of data.
Step 2: Compute the interquartile range (IQR).
IQR  Q3  Q1
Step 3: Determine the fences. Fences serve as cutoff
points for determining outliers.
Lower Fence  Q1  1.5( IQR)
Upper Fence  Q3  1.5( IQR)
Step 4: If data value is less than the lower fence or
greater than the upper fence, considered outlier.
Example 1.23
(Based on example 1.20)
Determine whether there are outliers in the data set.
The Five Number Summary; Boxplots

Compute the five-number summary
MINIMUM Q1 M Q3 MAXIMUM
Example 1.24
(Based on example 1.20)
Compute all five-number summary.
Boxplots
Step 1: Determine the lower and upper fences:

Lower Fence  Q1  1.5( IQR)
Upper Fence  Q3  1.5( IQR)
Step 2: Draw vertical lines at Q1 , M and Q
.3
Step 3: Label the lower and upper fences.
Step 4: Draw a line from Q1 to the smallest data value
that is larger than the lower fence. Draw a
line fromQ3 to the largest data value that is
smaller than the upper fence.
Step 5: Any data value less than the lower fence or
greater than the upper fence are outliers and
mark (*).
Example 1.25
(Based on example 1.20)
Construct a boxplot.