1. Maz Jamilah Masnan Institute of Engineering Mathematics Semester I 2015/16 EQT271 ENGINEERING STATISTICS

2. The Four Steps of Engineering Statistics Methods for analyzing data Maz Jamilah Masnan, EQT271, S2 2014/15

3. 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

4. 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.

5.  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.

6. 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

7. 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

8. 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.

9. 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!

10. Man Environment Method Material Machine Sources of variability

11. 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.

12.  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.

13. 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.

14. 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

15. SOURCES OF DATA Primary Data (data collected by the researcher) Examples:- i.Personal Interview ii.Telephone Interview iii.Questionnaire iv.Observations Secondary Data (already collected/ published by someone else) Examples: - From books, magazines, annual report, internet

16. 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.

17. 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 - 35213.20 4 - 613233.50 6 - 913133.25 10 - 126115.48 13 - 15184.57 Total394100.00

18. POPULATION SAMPLE Random variable (X) STATISTICS Descriptive Statistics Inferential Statistics Graphically Numerically 1.Tabular 2.Chart/ graph 1.Measure of Central Tendency 2.Measure of Dispersion 3.Measure of Position Maz Jamilah Masnan, S2 2014/15

20. 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).

21. 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.

22. Types of Graph for Qualitative Data

23. Example 1.1: frequency table  Bar Chart: used to display the frequency distribution in the graphical form. Example 1.2: Observation Frequency Malay33 Chinese9 Indian6 Others2

24.  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 JanFebMarAprMayJunJulAugSepOctNovDec 1075 3972603161421149

25. 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.

26.  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.

27. Example 1.5: Frequency Distribution Weight (Rounded decimal point)Frequency 60-625 63-6518 66-6842 69-7127 72-748  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:

28.  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 :

29. POPULATION SAMPLE Random variable (X) STATISTICS Descriptive Statistics Inferential Statistics Graphically Numerically 1.Tabular 2.Chart/ graph 1.Measure of Central Tendency 2.Measure of Dispersion 3.Measure of Position Maz Jamilah Masnan, S2 2014/15

31. 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. WeightFrequency 60-625 63-6518 66-6842 69-7127 72-748 Total100 Table 1.1: Weight of 100 male students in XYZ university

32.  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. WeightFrequency 60-625 63-6518 66-6842 69-7127 72-748 Total100 Variable Frequency column Third class (Interval Class) Lower Limit of the fifth class Frequency of the third class. Upper limit of the fifth class Table 1.2 : Weight of 100 male students in XYZ university

33.  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 = - 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. 606263 62.5 59.5

34. Example 1.9: From Table 1.1: Class Boundary Weight (Class Interval) Class BoundaryFrequency 60-6259.5-62.55 63-6562.5-65.518 66-6865.5-68.542 69-7168.5-71.527 72-7471.5-74.58 Total100

35. Example 1.10: Given a raw data as below: 2727272827202528 2628262831302626 33 283539 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?

36. 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 60-62559.5-62.5 5 63-651862.5-65.5 5 + 18 = 23 66-684265.5-68.5 23 + 42 = 65 69-712768.5-71.5 65 + 27 =92 72-74871.5-74.5 92 + 8 = 100 100

37. Exercise 1.1 : The data below represent the waiting time (in minutes) taken by 30 customers at one local bank. 2531203022323728 2923352529352927 2332313224352135 352233243943  Construct a frequency distribution and cumulative frequency distribution table.  Construct a histogram.

38. Measures of Central Tendency Measures of Dispersion Measures of Position

39. 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.

41. 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

42. 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: Mean for group data is given by:

43. Example 1.11 (Ungrouped data): Mean for the sets of data 3,5,2,6,5,9,5,2,8,6 Solution :

44. Example 1.12 (Grouped Data): Use the frequency distribution of weights 100 male students in XYZ university, to find the mean. WeightFrequency 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8

45. Solution : Weight (Class Interval Frequency, fClass Mark, x fx 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8

46.  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:

47. 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

48. 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 fxCumulative Frequency, F Class Boundary 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

50.  Mode for grouped data

51. 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 (Class Interval Frequency, f Class Mark, x fxCumulative Frequency, F Class Boundary 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 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 Total1006745 Mode class

52. 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)

53.  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

54. 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

55. Example 1.18 (Grouped data) Find the variance and standard deviation of the sample data below: Weight (Class Interval Frequency, f Class Mark, x fxCumulative Frequency, F Class Boundary 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 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 Total1006745

56. 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) 2-61 7-114 12-1610 17-213 22-262

57. 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. 825111529221051721 2213261618129262016 2314192320162792114

58. 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. i) Chebyshev’s Theorem At 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 interval 2) Find value of 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

59. 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

60. 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 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

61. 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: 95% of the people in the sample are 16 to 64 yrs old.

62. Measures of Position To describe the relative position of a certain data value within the entire set of data.  z scores  Percentiles  Quartiles  Outliers

63. 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.

65. 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. The calculus score of 65 was actually 1.5 standard deviations above the mean 50 The history score of 30 was actually 1.0 standard deviations above the mean 25

66. Exercise: Find the z score for each test, and state which is higher. TestXX barS Mathematics38405 Statistics9410010

67. 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)

68. Quartiles  Divide data sets into fourths or four equal parts. Smallest data value Q1Q2Q3 Largest data value 25% of data 25% of data 25% of data 25% of data

69. 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.

70. Outliers  Extreme observations  Can occur because of the error in measurement of a variable, during data entry or errors in sampling.

71. 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.

72. 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.

73. The Five Number Summary (Boxplots)  Compute the five-number summary Example 1.24 (Based on example 1.20) Compute all five-number summary.

74. Q 2 Median Q3Q3 Q1Q1 MinimumMaximum

75.  Boxplots Step 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 (*).

76. Example 1.23 (Based on example 1.21) Construct a boxplot.

77.  Boxplots Step 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 (*).

78. 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.

79. Q 2 = 14 Q 3 = 20 Q1=9Q1=9 Min = 5Max = 50 A right skewed distribution / positive skewed Lower Fence = 9 -16.5=7.5 Upper Fence = 20+16.5=36.5

81.  By using the mean and standard deviation, we can find the percentage of total observations that fall within the given interval about the mean.

82.  Empirical Rule  Chebyshev’s Theorem (IMPORTANT TERM: AT LEAST)

83. 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)