CHAPTER 1 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 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 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.

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

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.

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.

Graphically data presentation

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 Qualitative Data

Example 1.1: frequency table Bar Chart: used to display the frequency distribution in the graphical form. Example 1.2: Observation Frequency Malay 33 Chinese 9 Indian 6 Others 2

Pie Chart: used to display the frequency distribution 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 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 10 7 5 39 260 316 142 11 4 9

Data Presentation Of Quantitative Data Tabular presentation for quantitative data is usually in the form of frequency distribution that is 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.

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

NUMERICALLY SUMMARIZING DATA

Table 1.1: Weight of 100 male students in XYZ university 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. Weight Frequency 60-62 5 63-65 18 66-68 42 69-71 27 72-74 8 Total 100 Table 1.1: Weight of 100 male students in XYZ university

Class is in first column for frequency distribution table. 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 Weight Frequency 60-62 5 63-65 18 66-68 42 69-71 27 72-74 8 Total 100 Frequency column Variable Frequency of the third class. Third class (Interval Class) Lower Limit of the fifth class Upper limit of the sixth 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. 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.

Weight (Class Interval) Example 1.9: From Table 1.1: Class Boundary Weight (Class Interval) Class Boundary Frequency 60-62 59.5-62.5 5 63-65 62.5-65.5 18 66-68 65.5-68.5 42 69-71 68.5-71.5 27 72-74 71.5-74.5 8 Total 100

Example 1.10: Given a raw data as below: 27 27 27 28 27 24 25 28 26 28 26 28 31 30 26 26 33 28 How many classes that you recommend? What is the class interval? Build a frequency distribution table. 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 Interva;) Number of Students, f Class Boundaries Cumulative Frequency 60-62 5 59.5-62.5 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.

Data summary Measures of Central Tendency Measures of Dispersion Measures of Position

Data Summary Summary 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 Tendency Mean Median Mode Measures of Dispersion Range Variance Standard deviation Measures of Position Z scores Percentiles Quartiles Outliers

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:

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. Weight Frequency 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8

Weight (Class Interval 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

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): The median for data 4,6,3,1,2,5,7 is 4 Rearrange the data : 1,2,3,4,5,6,7 median

Weight (Class Interval Cumulative Frequency, F 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 Cumulative 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

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

Weight (Class Interval Cumulative Frequency, F 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 fx Cumulative 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 23 65 92 100 59.5-62.5 62.5-65.5 65.5-68.5 68.5-71.5 71.5-74.5 Total 6745 Mode class

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)

The positive square root of the variance is the standard deviation 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

Weight (Class Interval Cumulative Frequency, F 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 60-62 63-65 66-68 69-71 72-74 5 18 42 27 8 61 64 67 70 73 305 1152 2814 1890 584 23 65 92 100 59.5-62.5 62.5-65.5 65.5-68.5 68.5-71.5 71.5-74.5 Total 6745

Number of products get defect, f (frequency) 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

Construct a frequency distribution table. 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) Construct a frequency distribution table. Find the mean, variance and standard deviation, mode and median. Construct a histogram. 25 11 15 29 22 10 5 17 21 13 26 16 18 12 9 26 20 16 23 14 19 23 20 16 27 9 21 14

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: Determine the interval Find value of Change the value in step 2 to a percent 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: Determine interval Find 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 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

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.

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

Percentiles Divide a data set which written in ascending order. Often used in order to give the relative standing of a data value. Example:- In Engineering Statistics examination score 68 was at the 75th percentile. Means that approximately 75% of the students score below 68 and 25% are above 68.

Determining the kth Percentile Step 1: Arrange data in ascending order Step 2: Compute i, using formula k- percentile of data n- number of individuals in data set Step 3: a) If i is not an integer, round up to the next highest integer. b) If i is an integer, the kth percentile is the arithmetic mean of the ith and (i+1)st data value.

Example 1.21 The following data represent the number of inches of rain in Chicago during the month of April for 20 randomly years. 2.47 3.97 3.94 4.11 1.14 4.02 3.41 1.85 5.22 0.97 6.14 2.34 3.48 4.77 2.78 4.00 6.28 5.50 7.69 5.79 a) Determine the number of inches of rain correspond to 15th percentile,

Finding the Percentile that Corresponds to a Data Value Step 1: Arrange the data in ascending order. Step 2: Use formula Round this number to the nearest integer. Example 1.22 A metereology said that if the inches rain earn more than 5.22 inches of rain, flood will happen. Determine the percentile that correspond to the 5th highest number of inches of rains with 5.22 inches of rain.

Divide data sets into fourths or four equal parts. Quartiles Divide data sets into fourths or four equal parts. Example 1.23 (Based on data in example 1.21) Determine the quartiles. Smallest data value Largest data value Q1 Q2 Q3 25% of data 25% of data 25% of data 25% of data

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: Determine the first 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.

Example 1. 24 (Based on example 1 Example 1.24 (Based on example 1.21) Determine whether there are outliers in the data set.

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

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 (*).

Example 1.25 (Based on example 1.21) Construct a boxplot.