Transcript Lecture 1 - Portal UniMAP

EQT271
ENGINEERING STATISTICS
Maz Jamilah Masnan
Institute of Engineering Mathematics
Semester I 2015/16
The Four Steps of Engineering Statistics
Methods for analyzing data
Maz Jamilah Masnan, EQT271, S2 2014/15
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 from a
population 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 subjects/individuals which are characteristic being
studied.
 Sample
- A portion, or part of the population interest.
 Random Variable (X)
- Characteristics of the individuals within the population.
 Observation
- Value of variable for an element.
POPULATION
 Data Set
- A collection of observation on one
SAMPLE
or more variables.

X
Example



An engineer is developing a rubber
compound for use in O-rings.
The engineer uses the standard rubber
compound to produce eight O-rings in
a development laboratory and
measures the tensile strength of each
Variable
specimen.
The tensile strengths (in psi) of the
eight O-rings
Observation
1030,1035,1020, 1049, 1028, 1026, 1019, and 1010.
Sample
Variability

There is variability in the tensile strength
measurements.
◦ The variability may even arise from the
measurement errors




Tensile Strength can be modeled as a
random variable.
Tests on the initial specimens show that the
average tensile strength is 1027.1 psi.
The engineer thinks that this may be too
low for the intended applications.
He decides to consider a modified
formulation of rubber in which a Teflon
additive is included.
Statistical thinking


Statistical methods are used to help us
describe and understand variability.
By variability, we mean that successive
observations of a system or phenomenon
do not produce exactly the same result.
Are these gears produced exactly the same size?
NO!
Method
Environment
Material
Man
Sources of
variability
Machine
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 data.

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.
Type of data

Classification for its measurement scale:
◦ Qualititative
 Binary - dichotomous
 Ordinal
 Nominal
◦ Quantitative
 Discrete
 A variable whose values are countable, can assume
only certain values with no intermediate values.
 Continuous
 A variable that can assume any numerical value
over a certain interval or intervals.
Type of data - Examples

Qualitative
◦ Dichotomous - binary
 Gender: male or female.
 Employment status: employment or without employment.
◦ Ordinal
 Socioeconomic level: high, medium, low.
◦ Nominal
 Residency place: center, North, South, East, West.
 Civil status: single, married, widowed, divorced, free union.

Quantitative
◦ Discrete
 Number of offspring, number of houses, cars accidents :
1,2,3,4, ...
◦ Continuous
 Glucose in blood level: 110 mg/dl, 145 mg/dl.
 length, age, height, weight, time
Primary Data
(data collected
by the researcher)
SOURCES
OF DATA
i.
ii.
iii.
iv.
Examples:Personal Interview
Telephone Interview
Questionnaire
Observations
Secondary Data
(already collected/
published by someone else)
Examples:
- From books, magazines,
annual report, internet
Data summary



Generally, we want to show the data in a
summary form.
Number of times that an event occur, is of
our interest, it show us the variable
distribution.
We can generate a frequency list
quantitative or qualitative.
Grouped Data Vs Ungrouped Data


Grouped data - Data that has been organized into groups
(into a frequency distribution).
Age (years)
n
%
<1 - 3
52
13.20
4-6
132
33.50
6-9
131
33.25
10 - 12
61
15.48
13 - 15
18
4.57
Total
394
100.00
Ungrouped data - Data that has not been organized into
groups. Also called as raw data.
1030,1035,1020, 1049, 1028, 1026, 1019, and 1010.
POPULATION
SAMPLE
Random variable
(X)
STATISTICS
Graphically
1. Tabular
2. Chart/
graph
Descriptive
Statistics
Inferential
Statistics
Numerically
1. Measure of Central Tendency
2. Measure of Dispersion
3. Measure of Position
Maz Jamilah Masnan, S2 2014/15
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 for
Qualitative Data
Civil status of women in a community
Free union
9%
Widowed
8%
Single
28%
Divorced
11%
Married
44%
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
Chinese
Indian
Others
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 Jun Jul
39
7 260
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 :

POPULATION
SAMPLE
Random variable
(X)
STATISTICS
Graphically
1. Tabular
2. Chart/
graph
Descriptive
Statistics
Inferential
Statistics
Numerically
1. Measure of Central Tendency
2. Measure of Dispersion
3. Measure of Position
Maz Jamilah Masnan, S2 2014/15
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 fifth 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.
60
62
63
Formula:
59.5
62.5
- 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
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
Interval)
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.
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.
 Three basic
summary statistics are
1. measures of central tendency,
2. measures of dispersion, and
3. measure of position.
Measures in Statistics
Measure of Central Tendency
• MEAN
• MODE
• MEDIAN
Measure of Dispersion
• RANGE
• VARIANCE
• STANDARD DEVIATION
Measure of Position
• QUARTILE
• Z-SCORE
• PERCENTILE
• OUTLIER
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

fj




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
Cumulative
Frequency,
F
 f

 Fj 1 

2
x  Lc
 ?
fj




Class
Boundary

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
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
 1 
xˆ  L  c 
 ?
 1   2 
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:

◦
S
2
( x  x)


n 1

2
2
(
x

x
)
(for sample) S 2  
(for population)
n
The variance for the grouped data:
2
2
fx
 nx
S  
2
n 1
or
S 
2
2
2
fx
nx
2 
S 
n
or
S 
2


( fx) 2
fx 
n
n 1
2
(
fx
)
fx 2  
n
n
2
(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 meanx and standard deviation, we can find the
percentage of total observations that fall within the given
interval about the mean.
i) Chebyshev’s Theorem
1
(1

) of the observations will be in the range of k
At least
2
k
standard deviation from mean.
where k is the positive number exceed 1 or (k>1).
Applicable for any distribution /not normal distribution.
Steps:
1) Determine the interval x  ks
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
 80  ( 2)(5)
2)
Find
1
1
k2
 1

3)
4)
3
4
 (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 1.20
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
Z SCORE
•A standard score or z score tells how many standard deviations
a data value is above or below the mean for a specific distribution
of values.
•If a z score is 0, then the data value is the same as the mean.
•The formula is:
value-mean
z
,
standard deviation
for samples,
X X
z
s
for populations,
z
X 

•Note that if the z score is positive, the score is above the mean.
If the z score is 0, the score is the same as the mean. And if the z
score is negative, the z score is below the mean.
xx
z
s
z
x

1σ
2σ
3σ
Example:
A student scored 65 on a calculus test that had a mean of 50 and standard
deviation of 10. She scored 30 on a history test with a mean of 25 and a
standard deviation of 5. Compare her relative positions on the two tests.
Solution:
Find the z scores.
For calculus:
65  50
z
 1.5
10
The calculus score of 65 was
actually 1.5 standard deviations
above the mean 50
For history:
30  25
z
 1.0
5
The history score of 30 was
actually 1.0 standard deviations
above the mean 25
Since the z score for calculus is larger, her relative position in the
calculus class is higher than her relative position in the history class.
Exercise:
Find the z score for each test, and state
which is higher.
Test
X
X bar
S
Mathematics
38
40
5
Statistics
94
100
10
Percentiles
Percentiles are
position measures
used in educational
and health-related
fields to indicate the
position of an
individual in a group.
Percentiles divide the
data set into 100 equal
groups.
Usually used to observe
growth of child (mass,
height etc)
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
1
x(n  1)th
4
1
Q 2  median  x(n  1)th
2
3
Q3  x(n  1)th
4
Q1 
Example 1.21
Solution
Find Q1, Q2, and Q3 for the
data set
15, 13, 6, 5, 12, 50, 22, 18
Step 1 Arrange the data in order.
5, 6, |12, 13, | 15, 18, | 22, 50
Step 2 Find the median (Q2).
5, 6, 12, 13, 15, 18, 22, 50
↑
MD
Step 3 Find the median of the data values less
than 14.
5, 6, 12, 13
↑
Q1 [So Q1 is 9.]
Step 4 Find the median of the data values greater
than 14.
15, 18, 22, 50
↑
Q3 [ Q3 is 20]
Hence, Q1 =9, Q2 =14, and Q3 =20.
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: Rank the data in increasing order,
Step 2: Determine the first, median 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 1.22
Solution


Check the
following data
set for outliers.
5, 6, 12, 13, 15,
18, 22, 50
The data value 50 is extremely suspect. These are the steps
in checking for an outlier.
Step 1 Find Q1 and Q3. This was done in Example 1.21; Q1
is 9 and Q3 is 20.
Step 2 Find the interquartile range (IQR), which is Q3 & Q1.
IQR = Q3-Q1 = 20-9= 11
Step 3 Multiply this value by 1.5.
1.5(11) 16.5
Step 4 Subtract the value obtained in step 3 from Q1, and add
the value obtained in step 3 to Q3.
9 -16.5=7.5 and 20+16.5=36.5
Step 5 Check the data set for any data values that fall outside
the interval from 7.5 to 36.5. The value 50 is outside this
interval; hence, it can be considered an outlier.
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.
Q2
Median
Minimum
Q1
Maximum
Q3
MINIMUM Q1 M Q3 MAXIMUM
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 Q3 .
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
from Q3 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 1.23
(Based on example 1.21)
Construct a boxplot.
Boxplots
Step 1: Rank the data in increasing order.
Step 2: Determine the quartiles and median.
Step 3: Draw vertical lines at Q1 , M and Q3 .
Step 4: Draw a line from Q1 to the smallest data value.
Draw a line from Q3 to the largest data value.
Step 5: Any data value less than the lower fence or greater
than the upper fence are outliers and mark (*).

Solution
Step 1 Arrange the data in order.
5, 6, |12, 13, | 15, 18, | 22, 50
Step 2 Find the median (Q2).
5, 6, 12, 13, 15, 18, 22, 50
↑
MD [Q2=(13+15)/2=14]
Step 3 Find the median of the data values less than 14.
5, 6, 12, 13
↑
Q1 [So Q1 is 9.]
Step 4 Find the median of the data values greater than 14.
15, 18, 22, 50
↑
Q3 [ Q3 is 20]
Step 5 Draw a scale for the data on the x axis.
Step 6 Locate the lowest value, Q1, median, Q3, and the highest value on the scale.
Step 7 Draw a box around Q1 and Q3, draw a vertical line through the median, and
connect the upper value and the lower value to the box.
A right skewed distribution /
positive skewed
Min = 5
Q1
=
9
Q2
=
14
Lower Fence  Q1  1.5( IQR)
Upper Fence  Q3  1.5( IQR)
Q3
Max = 50
=
20
Lower Fence = 9 -16.5=7.5
Upper Fence = 20+16.5=36.5

By using the mean and standard deviation, we can
find the percentage of total observations that fall
within the given interval about the mean.
 Empirical Rule
 Chebyshev’s Theorem
(IMPORTANT TERM: AT LEAST)
Applicable for a symmetric bell shaped distribution /
normal distribution.
There are 3 rules:
i. 68% of the observations lie in the interval
(mean ±SD)
ii. 95% of the observations lie in the interval
(mean ±2SD)
iii. 99.7% of the observations lie in the interval
(mean ±3SD)
IQR  Q3  Q1
Lower Fence  Q1  1.5( IQR)
Upper Fence  Q3  1.5( IQR)