Engineering statistics EQT 271

Chapter 1: Part 1: Engineering statisticsEQT 271Khatijahhusna Abd RaniSchool Of Electrical System Engineering (PPKSE) Semester II 2014/2015

Slide was prepared by Miss Syafawati (with modification)Chapter 1: Basic Statistics

Chapter 2: Statistical Inference

Chapter 3: Analysis of Variance

Chapter 4: Introductory Linear Regression

Chapter 5: Nonparametric StatisticsEQT 271Statistics in Engineering

Collecting Engineering Data

Data summary & Presentation

Probability DistributionDiscrete Probability DistributionContinuous Probability Distribution

Sampling Distributions of the Mean and Proportion

BASIC STATISTICS

STATISTICSStatistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from dataBluman; Pg2

Statistics in EngineeringTo determine the satisfaction of students towards Universities facilities among UniMAP students

Example 1:

Statistics in EngineeringTo investigate the effects of Mobile base station Exposure on Body Temperature of Children's in Perlis

Example 2:

Gather information from data and make conclusions & recommendations

a collection of numerical information is called statisticsStatistics in EngineeringBASIC terms STATISTICSPopulation: Consists of all subjects (human or otherwise) that are being studied)

Sample:A group of subjects selected from a population

Example 1: undergraduates students in UniMAPExample 2: Children's in PerlisPopulationsampleBluman; Pg3Statistics in EngineeringVariable:Characteristics or attributes that can assume different values

Observation: Value of variable for an element

Data set:A collection of observation on one or more variables.

Example 2: Body temperature, gender, ageDiscussion on Ex1.1QualitativeQuantitativeDiscreteContinuousExamples of measurement scalesNominalOrdinalIntervalRatioZip CodeGenderEye Color (blue, brown, green)NationalityGrade (A, B, C,D)Rating Scale (poor, good, excellent)

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

ExperimentsMore expensive methods but better way to produce dataData produced are called experimentalCoLlecting engineering dataSurveysMost familiar methods of data collectionDepends on the response rate

Personal InterviewHas the advantage of having higher expected response rateFewer incorrect respondents.Dis: High Costs, Need for highly trained interviewers

Collecting Engineering DataGrouped DataUngrouped DataData that has been organized into groups (into a frequency distribution).

When the range of data is largeData that has not been organized into groups. Also called as raw data.

Collecting Engineering Data

Data summary & presentationGraphicalNUmericalWeightFrequency60-62563-651866-684269-712772-748Total100

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).Graphical ???Numerical ???

Data PresentationData Presentation of Qualitative DataTabular 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.

Data PresentationBar Chart: used to display the frequency distribution in the graphical form. Frequencies are shown on the Y-axis and the ethnic group is shown on the X-axis

ObservationsFrequencyMalay33Chinese9Indian6Others2Example 1.1 :Table 1: Frequency table

Example 1.2 :Data PresentationPie Chart: used to display the frequency distribution. It displays the ratio of the observations

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

JanFebMarAprMayJunJulAugSepOctNovDec1075103972603161421149

Example 1.3 :Example 1.4 :Data PresentationData Presentation Of Quantitative DataTabular 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.

Data PresentationThere are few graphs available for the graphical presentation of the quantitative data. The most popular graphs are:1. histogram;2. frequency polygon; and3. ogive.

Data Presentation

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

Example 1.5 :Example 1.6 :WeightFrequency60-62563-651866-684269-712772-748Table 1: Frequency DistributionData PresentationFrequency 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.

Example 1.7 :Example 1.8 :Data Presentation Constructing Frequency DistributionWhen 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.

WeightFrequency60-62563-651866-684269-712772-748Total100Table 1.1: Weight of 100 male students in XYZ university Data PresentationFor 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.

WeightFrequency60-62563-651866-684269-712772-748Total100VariableFrequencycolumnThird class (Interval Class)Lower Limit of the fifth classFrequencyof the third class.Upper limit of the sixth classTable 1.2 : Weight of 100 male students in XYZ university Data PresentationThe class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class. (no gap)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.

Data PresentationExample 1.9:From Table 1.1: Class Boundary

Weight (Class Interval)Class BoundaryFrequency60-6259.5-62.5563-6562.5-65.51866-6865.5-68.54269-7168.5-71.52772-7471.5-74.58Total100Data Presentation Cumulative Frequency DistributionsA 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, fClass BoundariesCumulative Frequency60-62559.5-62.5563-651862.5-65.55 + 18 = 2366-684265.5-68.523 + 42 = 6569-712768.5-71.565 + 27 =9272-74871.5-74.592 + 8 = 100100Data PresentationHow to construct histogram?Prepare the frequency distribution table by:1.Find the minimum and maximum value

2.Decide the number of classes to be included in your frequency distribution table.-Usually 5-20 classes. Too small-may not able to see any pattern OR

-Sturges rule, Number of classes= 1+3.3log n

3.Determine class width, i = (max-min)/num. of class

4.Determine class limit.

5.Find class boundaries and class mid points

6.Count frequency for each classDraw histogram

Exercise 1.1 :

The data below represent the waiting time (in minutes) taken by 30 customers at one local bank.253120302232372829233525293529272332313224352135352233243943

Construct a frequency distribution and cumulative frequency distribution table.Construct a histogram.

Data PresentationData summary Measures of Central TendencyMeasures of DispersionMeasures of PositionData SummarySummary 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 TendencyMeanMedianModeMeasures of DispersionRangeVarianceStandard deviationMeasures of PositionZ scoresPercentilesQuartilesOutliers

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:

Example 1.11 (Ungrouped data):

Mean for the sets of data 3,5,2,6,5,9,5,2,8,6

Solution :

Example 1.12 (Grouped Data):

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

WeightFrequency60-6263-6566-6869-7172-7451842278Solution :

Weight (Class IntervalFrequency, fClass Mark, xFx6460-6263-6566-6869-7172-745184227861646770733051152281418905846745

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:

Example 1.13 (Ungrouped data):

n is oddThe median for data 4,6,3,1,2,5,7 (n=7)(median=(7+1/2=4th place)Rearrange the data : 1,2,3,4,5,6,7

n is evenThe median for data 4,6,3,2,5,7 (n=6)Rearrange the data : 2,3,4,5,6,7Median=(4+5)/2=4.5

medianExample 1.14 (Grouped Data):The sample median for frequency distribution as in example 1.12Solution:

Weight (Class IntervalFrequency, fClass Mark, xfxCumulative Frequency, FClass Boundary60-6263-6566-6869-7172-74518422786164677073305115228141890584523659210059.5-62.562.5-65.565.5-68.568.5-71.571.5-74.5

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 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 IntervalFrequency, fClass Mark, xfxCumulative Frequency, FClass Boundary60-6263-6566-6869-7172-74518422786164677073305115228141890584523659210059.5-62.562.5-65.565.5-68.568.5-71.571.5-74.5Total1006745Mode class

Measures of Dispersion

Range = Largest value smallest valueVariance: 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)

(for population)

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

42Example 1.17 (Ungrouped data)Find the variance and standard deviation of the sample data : 3, 5, 2, 6, 5, 9, 5, 2, 8, 6

Example 1.18 (Grouped data)Find the variance and standard deviation of the sample data below:Weight (Class IntervalFrequency, fClass Mark, xfxCumulative Frequency, FClass Boundary60-6263-6566-6869-7172-74518422786164677073305115228141890584523659210059.5-62.562.5-65.565.5-68.568.5-71.571.5-74.537214096448949005329186057372818853813230042632Total1006745455803

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)2-617-11412-161017-21322-262Exercise 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)

Construct a frequency distribution table.Find the mean, variance and standard deviation, mode and median. Construct a histogram.

25111529221051721

13261618129262016

2314192320162792114 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) Empirical RuleApplicable for a symmetric bell shaped distribution / normal distribution.There are 3 rules:i. 68% of the data will lie within one standard deviation of the mean, ii. 95% of the data will lie within two standard deviation of the mean,

iii. 99.7% of the data will lie within three standard deviation of the mean,

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:

Approximately 68% of the measurements will fall between 28 and 52, approximately 95% of the measurements will fall between 16 and 64 and approximately 99.7% to fall into the interval 4 and 76.

Measures of Position

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

z scoresPercentilesQuartilesOutliers

Z ScoreA 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:

samples, populations,

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.

A student score 65 on calculus test that has a mean of 50 and a standard deviation of 10; she scored 30 on a history test with mean of 25 and a standard deviation of 5. Compare her relative positions on the two test.Solution:First, find the Z scores.

For calculus the z score is

For history the z score isExample 1.19 :

Since the z score for calculus is larger, her relative position in the calculus class is higher than her relative position in the history classThe calculus score of 65 was actually 1.5 standard deviations above the mean 50The history score of 30 was actually 1.0 standard deviations above the mean 25Exercise 1.4Find the z score for each test, and state which is higherTestMathematics38405Statistics9410010

QuartilesDivide the distribution into four equal groups, denoted by Q, Q, Q. Note that Qis the same as the 25th percentileQ is the same as the 50th percentile or medianQ corresponds to the 75th percetile

Odd number of observationsPositions are integersExample: 5, 8, 4, 4, 6, 3, 8 (n=7)

1. Put them in order: 3, 4, 4, 5, 6, 8, 8

2. Calculate the quartiles Example 1.20 :

3, 4, 4, 5, 6, 8, 8

Q1Q2Q3Even number of observations: Positions are not integersExample: 5, 12, 10, 4, 6, 3, 8, 14 (n=8)

1. Put them in order:2. Calculate the quartiles

Q3Q1Q23, 4, 5,6,8,10,12,14

The following data represent the number of inches of rain in Chicago during the month of April for 10 randomly years.

2.473.973.944.11 5.221.144.023.411.850.97

Determine the quartiles.Exercise 1.5Outliers

Extreme observationsCan 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 3: Compute the interquartile range (IQR).

Step 4: Determine the fences. Fences serve as cutoff points for determining outliers.

Step 5: If data value is less than the lower fence or greater than the upper fence, considered outlier.

Determine whether there are outliers in the data set.

2.473.973.944.11 5.221.144.023.411.850.97

Example 1.21 :Solution: 0.97, 1.14, 1.85, 2.47, 3.41, 3.94, 3.97, 4.02, 4.11, 5.22

Since all the data are not less than -1.8825 and not greater than 7.5975, then there are no outliers in the data

The Five Number Summary; Boxplots

Compute the five-number summary

Example 1.24

2.473.973.944.11 5.221.144.023.411.850.97

Compute all five-number summary.

Solution:0.97, 1.14, 1.85, 2.47, 3.41, 3.94, 3.97, 4.02, 4.11, 5.22

BOXPLOTThe five-number summary can be used to create a simple graph called a boxplot.

From the boxplot, you can quickly detect any skewness in the shape of the distribution and see whether there are any outliers in the data set.

Interpreting Boxplot

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 (*).

2.473.973.944.11 5.221.144.023.411.850.97

Sketch the boxplot and interpret the shape of the boxplotExample 1.22 :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 (*).