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.

 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 individuals which are characteristic being studied.  Sample - A portion, or part of the population interest.  Variable - 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.

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.

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 Malay33 Chinese9 Indian6 Others2

 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

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.

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

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 Total100 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. WeightFrequency Total100 Variable Frequency column Third class (Interval Class) Lower Limit of the fifth class Frequency of the third class. Upper limit of the fifthclass 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 Mark Class midpoint or mark = (Lower Limit + Upper Limit)/2 - Finding The Number of Classes Number of classes, i = - Finding Class Width For Interval Class class width, c = (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.

Example 1.9: From Table 1.1: Class Boundary Weight (Class Interval) Class BoundaryFrequency Total100

Example 1.10: Given a raw data as below: 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 Interva;) Number of Students, f Class Boundaries Cumulative Frequency = = = =

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

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. WeightFrequency

Solution : Weight (Class Interval Frequency, fClass Mark, x fx

 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:

Single middle value Averages (The Median) The median is the middle value of a set of data once the data has been ordered. Example 1.13 (a). Ali hit 11 balls in a golf tournament. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives. 85, 125, 130, 65, 100, 70, 75, 50, 140, 95, 70 Median drives = 85 yards 50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 140 Ordered data

Two middle values so take the mean. Averages (The Median) The median is the middle value of a set of data once the data has been ordered. Example 1.13 (b). Ali hit 12 balls at golf tournament. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives. 85, 125, 130, 65, 100, 70, 75, 50, 140, 135, 95, 70 Median drive = 90 yards 50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 135, 140 Ordered data

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

 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, 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 Total 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)

 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

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

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 Total

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)

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

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

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

Example 1.21 The following data are the incomes (in thousand of dollars) for a sample of 12 households. Find the quartiles

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 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.

Example 1.22 (Based on example 1.21) Determine whether there are outliers in the data set.

The Five Number Summary; Boxplots  Compute the five-number summary  Boxplot- shows the center, spread and skewness of the data set with a box and two whiskers.

 Description of Boxplot

Median Lower Quartile Upper Quartile Lowest Value Highest Value Box Whisker Anatomy of a Box and Whisker Diagram.

 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 1.23 (Based on example 1.21) Construct a boxplot.

 Distribution Shape Based upon Boxplot 1. If the median is near the center of the box and each of the horizontal lines is approximately equal length, then the distribution is roughly symmetric. 2. If the median is to the left of the center of the box or the right line is longer than left line, the distribution is skewed right. 3. If the median is to the right of the center of the box or the left line is longer than the right line, the distribution is skewed left.

Finding the median, quartiles and inter-quartile range. 12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10 4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12 Order the data Inter-Quartile Range = 9 - 5½ = 3½ Example 1: Find the median and quartiles for the data below. Lower Quartile = 5½ Q1Q1 Upper Quartile = 9 Q3Q3 Median = 8 Q2Q2

Upper Quartile = 10 Q3Q3 Lower Quartile = 4 Q1Q1 Median = 8 Q2Q2 3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15, Finding the median, quartiles and inter-quartile range. 6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10 Order the data Inter-Quartile Range = = 6 Example 2: Find the median and quartiles for the data below.

2, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15 Median = 8 hours and the inter-quartile range = 9 – 6 = 3 hours. Battery Life: The life of 12 batteries recorded in hours is: 2, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15 Mean = 93/12 = 7.75 hours and the range = 15 – 2 = 13 hours. Discuss the calculations below. The averages are similar but the measures of spread are significantly different since the extreme values of 2 and 15 are not included in the inter-quartile range.

Median Lower Quartile Upper Quartile Lowest Value Highest Value Box Whisker Boys Girls cm Box and Whisker Diagrams. Box plots are useful for comparing two or more sets of data like that shown below for heights of boys and girls in a class. Anatomy of a Box and Whisker Diagram.

Exercise 1.4 The following data give the numbers of computer keyboards assembled at a company for a sample of 25 days. Construct a boxplot and comment the distribution shape of the boxplot

Exercise 1.5 The following data represent the number of grams of fat in breakfast meals offered at McDonalds a) Find mean, median and mode using Sturge`s rule. b) Construct a histogram c)Find the five-number summary. d) Construct a boxplot and comment the shape of distribution of boxplot.