1.1 STATISTICS IN ENGINEERING1.2 COLLETING ENGINEERING DATA1.3 DATA SUMMARY & PRESENTATION

Maz Jamilah MasnanInstitute of Engineering Mathematics

Semester I 2015/16

EQT271ENGINEERING STATISTICS

The Four Steps of Engineering Statistics

Gather

Data

• Chapter (1.1 – 1.2)

Summarize

Data

• Chapter 1 (1.3)

Analyze

Data

• Chapter 1 (1.4) – Background for analyzing data• Chapter 2• Chapter 3• Chapter 4• Chapter 5

Reporting Results

• Reporting the analyses results creatively, interestingly and easy to understand (integrated in FYP)

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

CHAPTER 1

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.

the methods of statistics allow scientists and engineers to design valid experiments and to draw reliable conclusions from the data they produce

• Specifically, statistical techniques can be a powerful aid in designing new products and systems, improving existing designs, and improving production process.

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. Data Set- A collection of observation on one or more variables.

POPULATION

SAMPLE

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 specimen.

The tensile strengths (in psi) of the eight O-rings

1030,1035,1020, 1049, 1028, 1026, 1019, and 1010.

Variable

Sample

Observation

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!

Man

Environment

Method

Material

Machine

Sources of variability

Collecting Engineering Data

Direct observationThe simplest method of obtaining data.Advantage: relatively inexpensive.Disadvantage: difficult to produce useful information since it does not consider all aspects regarding the issues.

ExperimentsMore expensive methods but better way to produce data.Data produced are called experimental data.

SurveysMost familiar methods of data collectionDepends on the response rate

Personal InterviewHas the advantage of having higher expected response rateFewer 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

SOURCES OF DATA

Primary Data(data collected

by the researcher)

Examples:-i. Personal Interviewii. Telephone Interviewiii. Questionnaireiv. 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).

Ungrouped data - Data that has not been organized into groups. Also called as raw data.

1030,1035,1020, 1049, 1028, 1026, 1019, and 1010.

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

POPULATION

SAMPLE

Random variable (X)

STATISTICS

Descriptive Statistics

Inferential Statistics

Graphically

Numerically

1. Tabular2. Chart/

graph

1. Measure of Central Tendency2. Measure of Dispersion3. Measure of Position

Maz Jamilah Masnan, S2 2014/15

Graphically data presentation

Graphical Data Presentation

Data can be summarized or presented in two ways:1. Tabular2. 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; and3. line chart.

Types of Graph for Qualitative Data

Civil status of women in a community

Single28%

Married44%

Divorced11%

Widowed8%

Free union9%

Example 1.1:frequency table

Bar Chart: used to display the frequency distribution in the graphical form.

Example 1.2:

Observation FrequencyMalay 33Chinese9Indian 6Others 2

Pie Chart: used to display the frequency distribution. It displays the ratio of the observations

Example 1.3 :

Line chart: used to display the trend of observations. It is a very popular display for the data which represent time.

Example 1.4

Malay

Chinese

Indian

Others

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec10 7 5 10 39 7 260 316 142 11 4 9

Data Presentation Of Quantitative Data

Tabular presentation for quantitative data is usually in the form of frequency distribution that is atable 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; and3. ogive.

Example 1.5: Frequency Distribution Weight (Rounded decimal point) Frequency

60-62 563-65 1866-68 4269-71 2772-74 8

Histogram: Looks like the bar chart except thatthe horizontal axis represent the data whichis quantitative in nature. There is no gap betweenthe 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

Descriptive Statistics

Inferential Statistics

Graphically

Numerically

1. Tabular2. Chart/

graph

1. Measure of Central Tendency2. Measure of Dispersion3. Measure of Position

Maz Jamilah Masnan, S2 2014/15

NUMERICALLY SUMMARIZING DATA

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.

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.

WeightFrequenc

y60-62 563-65 1866-68 4269-71 2772-74 8Total 100

Table 1.1: Weight of 100 male students in XYZ university

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.

Weight Frequency60-62 563-65 1866-68 4269-71 2772-74 8Total 100

Variable Frequencycolumn

Third class (Interval Class)

Lower Limit of the fifth class

Frequencyof the third class.

Upper limit of the fifth class

Table 1.2 : Weight of 100 male students in XYZ university

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 MarkClass midpoint or mark = (Lower Limit + Upper

Limit)/2- Finding The Number of ClassesNumber of classes, c = - Finding Class Width For Interval Classclass 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.

1 3.3log n

60 62 6362.559.5

Example 1.9:From Table 1.1: Class Boundary

Weight (Class

Interval)Class

Boundary Frequency60-62 59.5-62.5 563-65 62.5-65.5 1866-68 65.5-68.5 4269-71 68.5-71.5 2772-74 71.5-74.5 8Total 100

Example 1.10:

Given a raw data as below:27 27 27 28 27 20 25 28 26 28 26 28 31 30 26 26

33 28 35 39

a) How many classes that you recommend?b) How many class interval?c) Build a frequency distribution table.d) What is the lower boundary for the first

class?

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 59.5-62.55

63-65 18 62.5-65.55 + 18 = 23

66-68 42 65.5-68.523 + 42 = 65

69-71 27 68.5-71.565 + 27 =92

72-74 8 71.5-74.592 + 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 2829 23 35 25 29 35 29 2723 32 31 32 24 35 21 3535 22 33 24 39 43

Construct a frequency distribution and cumulative frequency distribution table.

Construct a histogram.

Data summary• 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

MeanMean of a sample is the sum of the sample data divided by the total number sample.

Mean for ungrouped data is given by:

Mean for group data is given by:

x

n

xxornnfor

n

xxxx n

_21

_

,...,2,1,.......

f

fxor

f

xfx n

ii

n

iii

1

1

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 65.1

10x

Example 1.12 (Grouped Data):

Use the frequency distribution of weights 100 male students in XYZ university, to find the mean.

Weight Frequency

60-6263-6566-6869-7172-74

51842278

Solution :

Weight (Class

Interval

Frequency, f Class Mark, x

fx

60-6263-6566-6869-7172-74

51842278

?fx

xf

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:

1

1

2

where

L = the lower class boundary of the median class

c = the size of median class interval

F the sum of frequencies of all classes lower than the median class

the fre

j

j

j

j

fF

x L cf

f

quency 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 1.12Solution:

Weight (Class

Interval

Frequency, f

Class Mark,

x

fx Cumulative Frequency,

F

Class Boundary

60-6263-6566-6869-7172-74

51842278

6164677073

305115228141890584

12 ?j

j

fF

x L cf

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

1

1 2

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

ˆ

the lower c

x L c

L

1

2

lass boundary of the modal class

c = the size of the modal class interval

the difference between the modal class frequency and the class before it

the difference between the modal class frequency a

nd the class after it

*the class which has the highest frequency is called the modal class

Mode for grouped data

Example 1.15 (Ungrouped data)Find the mode for the sets of data 3, 5, 2, 6, 5, 9, 5, 2, 8, 6Mode = number occurring most frequently = 5

Example 1.16 Find the mode of the sample data belowSolution:

Weight (Class

Interval

Frequency, f

Class Mark

, x

fx Cumulative Frequency,

F

Class Boundary

60-6263-6566-6869-7172-74

51842278

6164677073

305115228141890584

5236592

100

59.5-62.562.5-65.565.5-68.568.5-71.571.5-74.5

Total 100 6745

Mode class

1

1 2

ˆ ?x L c

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:

◦ (for sample) (for population)

The variance for the grouped data:

- or (for sample)

- or (for population)

1

)( 22

n

xxS

22

2

1

fx n xS

n

22

2

( )

1

fxfx

nSn

22

2 fx n xS

n

22

2

( )fxfx

nSn

22 ( )x xS

n

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.

The positive square root of the variance is the standard deviation

22 2( )

1 1

x x fx n xS

n n

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

2

2

( )?

1

( )?

1

x xs

n

x xs

n

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 Cumulative Frequency,

F

Class Boundary

60-6263-6566-6869-7172-74

51842278

6164677073

305115228141890584

5236592

100

59.5-62.562.5-65.565.5-68.568.5-71.571.5-74.5

Total 100 6745

2x2fx

22

2

( )

?1

fxfx

nSn

2

2

?1

fx n xS

n

Exercise 1.2

The defects from machine A for a sample of products were organized into the following:

What is the mean, median, mode, variance and standard deviation.

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

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)

a) Construct a frequency distribution table.b) Find the mean, variance and standard deviation,

mode and median. c) Construct a histogram.

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

Rules of Data DispersionBy using the mean and standard deviation, we can find the percentage of total observations that fall within the given interval about the mean.

i) Chebyshev’s TheoremAt least 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:1) Determine the interval2) Find value of3) Change the value in step 2 to a percent4) Write statement: at least the percent of data

found in step 3 is in the interval found in step 1

2

1(1 )

k

x

ksx2

1(1 )

k

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

2) Find

3) Convert into percentage: 4) Conclusion: At least 75% of the data is found in the

interval from 70 to 90

)90,70(

)5)(2(80

ksx

4

32

11

11

2

2

k

%754

3

ii) Empirical RuleApplicable for a symmetric bell shaped distribution / normal distribution.There are 3 rules:i. 68% of the observations lie in the interval ii. 95% of the observations lie in the interval iii. 99.7% of the observations lie in the interval

Formula for k = Distance between mean and each point

standard deviation

),( sxsx

)2,2( sxsx

)3,3( sxsx

Example 1.20The 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:

95% of the people in the sample are 16 to 64 yrs old.

212

2412

1640

k

Measures of Position

To describe the relative position of a certain data value within the entire set of data.

z scoresPercentilesQuartilesOutliers

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:

•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.

value-mean,

standard deviation

for samples,

for populations,

z

X Xz

s

Xz

s

xxz

x

z

σ1

σ2

σ3

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:

For history:

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.

65 501.5

10z

The calculus score of 65 was

actually 1.5 standard deviations above the mean 50

30 251.0

5z

The history score of 30 was

actually 1.0 standard deviations above the mean 25

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 are position measures used in educational and health-related fields to indicate the position of an individual in a group.

Percentiles

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 Q2 Q3

Largest data value

25% of data

25% of data

25% of data

25% of data

11 ( 1)

41

2 ( 1)2

33 ( 1)

4

Q x n th

Q median x n th

Q x n th

Example 1.21

Find Q1, Q2, and Q3 for the data set

15, 13, 6, 5, 12, 50, 22, 18

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

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).

Step 3: Determine the fences. Fences serve as cutoff

points for determining outliers.

Step 4: If data value is less than the lower fence or greater than the upper fence,

considered outlier.

3 1IQR Q Q

1

3

Lower Fence 1.5( )

Upper Fence 1.5( )

Q IQR

Q IQR

Example 1.22

Check the following data set for outliers.

5, 6, 12, 13, 15, 18, 22, 50

Solution

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

Example 1.24

(Based on example 1.20)Compute all five-number summary.

1 3MINIMUM Q Q MAXIMUMM

Q2

Median

Q3Q1

Minimum Maximum

1 3MINIMUM Q Q MAXIMUMM

BoxplotsStep 1: Determine the lower and upper fences:

Step 2: Draw vertical lines at .Step 3: Label the lower and upper fences.Step 4: Draw a line from to the smallest data value that is larger than the lower fence. Draw a line from 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 (*).

1

3

Lower Fence 1.5( )

Upper Fence 1.5( )

Q IQR

Q IQR

1 3, and Q M Q

3Q

1Q

Example 1.23

(Based on example 1.21)Construct a boxplot.

BoxplotsStep 1: Rank the data in increasing order.Step 2: Determine the quartiles and median.Step 3: Draw vertical lines at .Step 4: Draw a line from to the smallest data

value. Draw a line from 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 (*).

1 3, and Q M Q

1Q

3Q

SolutionStep 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.

Q2

=14

Q3

=

20

Q1

=

9

Min = 5 Max = 50

A right skewed distribution / positive skewed

1

3

Lower Fence 1.5( )

Upper Fence 1.5( )

Q IQR

Q IQR

Lower Fence = 9 -16.5=7.5 Upper Fence = 20+16.5=36.5

Rules of Data Dispersion By using the mean and standard deviation,

we can find the percentage of total observations that fall within the given interval about the mean.

Rules of Data Dispersion

Empirical RuleChebyshev’s Theorem

(IMPORTANT TERM: AT LEAST)

Empirical Rule

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)

Empirical Rule

Boxplot

1

3

Lower Fence 1.5( )

Upper Fence 1.5( )

Q IQR

Q IQR

3 1IQR Q Q

Boxplot