1. CHAPTER 1 EQT 271 (part 1) BASIC STATISTICS

2. Basic Statistics 1.1Statistics in Engineering 1.2Collecting Engineering Data 1.3Data Presentation and Summary 1.4Probability Distributions - Discrete Probability Distribution - Continuous Probability Distribution 1.5Sampling Distributions of the Mean and Proportion

3. 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.

4.  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.

5. 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.

6. 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

7. 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.

8. 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.

9. 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).

10. 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.

11. Types of Graph Qualitative Data Bar chart Pie chart Line chart

12. Frequency table  Bar Chart: used to display the frequency distribution in the graphical form. Observation Frequency Malay33 Chinese9 Indian6 Others2

13.  Pie 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. JanFebMarAprMayJunJulAugSepOctNovDec 1075 3972603161421149

14. 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.

15. 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.

16. Frequency Distribution Weight (Rounded decimal point) Frequency 60-62 5 63-65 18 66-68 42 69-71 27 72-74 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.

17.  Frequency 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.

18. Constructing Frequency Distribution When summarizing large quantities of raw data, it is often useful to distribute the data into classes. Table 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: Weight of 100 male students in XYZ university

19. 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 the 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 fourth class Frequency of the third class. Upper limit of the fifth class Table 1.: Weight of 100 male students in XYZ university

20. 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 = - 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.

21. Example 1: From Table 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

22. 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 2: 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

23. Exercise 1: Given a raw data as below: 2727272827242528 2628262831302626 a)How many classes that you recommend? b)What is the class interval? c)Build a frequency distribution table. d)What is the lower boundary for the first class?

24. HOW 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 -Sturge’s 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 class 7.Draw histogram

25. Exercise 2: The data below represent the waiting time (in minutes) taken by 30 customers at one local bank. 2531203022323728 2923352529352927 2332313224352135 352233243943 1.Construct a frequency distribution and cumulative frequency distribution table. 2.Draw a histogram

26. Data Summary Summary statistics are used to summarize a set of observations. a)Measures of Central Tendency  Mean  Median  Mode b)Measures of Dispersion  Range  Variance  Standard deviation c)Measures of Position  Z scores  Percentiles  Quartiles  Outliers

27. a)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:

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

29. Example 3 (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

30. Solution : Weight (Class Interval Frequency, f 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 Class Mark, x 61 64 67 70 73 fx 305 1152 2814 1890 584 6745 100

31. 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 n 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:

32. Example 4 (Ungrouped data): n is odd Find the median for data 4,6,3,1,2,5,7 ( n = 7) Rearrange the data : 1,2,3,4,5,6,7 (median = (7+1)/2=4 th place) Median = 4 n is even Find the median for data 4,6,3,2,5,7 (n = 6) Rearrange the data : 2,3,4,5,6,7 Median = (4+5)/2 = 4.5

33. Example 5 (Grouped Data): The sample median for frequency distribution as in example 3 Solution: Weight (Class Interval Frequency, f 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 Median class 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

34. 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

35. Mode for grouped data

36. Example 6 (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

37. Example 7 Find the mode of the sample data below Solution: Weight (Class Interval Frequency, f 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 Total100 Mode class Class Boundary 59.5-62.5 62.5-65.5 65.5-68.5 68.5-71.5 71.5-74.5

38. b)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

39. The variance for the grouped data: For sample or For population or

40. 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

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

42. Example 9 (Grouped data) Find the variance and standard deviation of the sample data below: Weight (Class Interval Frequency, f 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 Total100 Class Mark, x fx 61 64 67 70 73 305 1152 2814 1890 584 6745 3721 4096 4489 4900 5329 18605 73728 188538 132300 42632 455803

43. Exercise 3 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? Or Calculate each possible measure of central tendency and dispersion. Defects (Class Interval) Number of products get defect, f (frequency) 2-61 7-114 12-1610 17-213 22-262

44. Exercise 4 (submit on Friday) 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

45. 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. Empirical Rule Applicable 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,

46. Example 10 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: 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.

47. c)Measures of Position To describe the relative position of a certain data value within the entire set of data.  z scores  Percentiles  Quartiles  Outliers

48. 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

49. The positions are integers Example: 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 3, 4, 4, 5, 6, 8, 8

50. The positions are not integers Example: 5, 12, 10, 4, 6, 3, 8, 14 (n=8) 1.Put them in order: 3, 4, 5, 6, 8, 10, 12, 14 2.Calculate the quartiles 3, 4, 5, 6, 8, 10, 12, 14

51. Example 11 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 5.22 1.144.023.411.850.97 Determine the quartiles.

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

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

54. Checking for outliers by using Quartiles Step 1: Determine the first and third quartiles of data. Step 2: Compute the interquartile range (IQR). Step 3: Determine the fences. Fences serve as cut off points for determining outliers. Step 4: If data value is less than the lower fence or greater than the upper fence, considered outlier.

55. Example 12 2.473.973.944.11 5.22 1.144.023.411.850.97 Determine whether there are outliers in the data set.

56. 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 -2.32625 and not greater than 7.86375, then there are no outliers in the data

57. The Five Number Summary Compute the five-number summary

58. Example 13 2.473.973.944.11 5.22 1.144.023.411.850.97 Compute all the five-number summary.

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

60. BOXPLOT The five-number summary can be used to create a simple graph called a boxplot. Form 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. Lower fence Upper fence Outlier

61. Interpreting Boxplot - symmetric - Skewed left because the tail is to the left - Skewed right because the tail is to the right

62. Characteristics Of Skewed Distributions Mean/Median Versus Skewness

63. TO CONSTRUCT BOXPLOT 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 (*).

64. Example 14 2.473.973.944.11 5.22 1.144.023.411.850.97 -Sketch the boxplot and interpret the shape of the boxplot.

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

66. - The distribution is skewed left