Khatijahhusna Abd Rani School 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 Statistics

Statistics in Engineering Collecting Engineering Data Data summary & Presentation Probability Distribution Discrete Probability Distribution Continuous Probability Distribution Sampling Distributions of the Mean and Proportion

Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data

To determine the satisfaction of students towards Universities facilities among UniMAP students

To investigate the effects of Mobile base station Exposure on Body Temperature of Children's in Perlis Gather information from data and make conclusions & recommendations a collection of numerical information is called statistics

Population: 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 UniMAP Example 2: Children's in Perlis Population sample

Variable: 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, age Qualitative Quantitative Discrete Continuous

Examples of measurement scales NominalOrdinalIntervalRatio Zip Code Gender Eye Color (blue, brown, green) Nationality Grade (A, B, C,D) Rating Scale (poor, good, excellent) Temperature IQ Height Weight Age Time Salary

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. Dis: High Costs, Need for highly trained interviewers

Grouped DataUngrouped Data Data that has been organized into groups (into a frequency distribution). When the range of data is large Data that has not been organized into groups. Also called as raw data.

WeightFrequency Total100

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.

Bar 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 ObservationsFrequency Malay33 Chinese9 Indian6 Others2 Table 1: Frequency table

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

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.

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. WeightFrequency Table 1: Frequency Distribution

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.

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 sixth class 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. (no gap) 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.

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

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 = = = =

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 Draw histogram

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.

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:

Example 1.13 (Ungrouped data): n is odd The median for data 4,6,3,1,2,5,7 (n=7) (median=(7+1/2=4 th place) Rearrange the data : 1,2,3,4,5,6,7 n is even 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 median

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) (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

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 ) 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,

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: 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 scores Percentiles Quartiles Outliers

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: 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 is 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

Exercise 1.4 Find the z score for each test, and state which is higher Test Mathematics38405 Statistics

Quartiles Divide the distribution into four equal groups, denoted by Q₁, Q₂, Q₃. Note that Q₁is the same as the 25 th percentile Q₂ is the same as the 50 th percentile or median Q₃ corresponds to the 75 th percetile

Odd number of observations 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 Q1Q2Q3

Even number of observations: Positions are not integers Example: 5, 12, 10, 4, 6, 3, 8, 14 (n=8) 1. Put them in order: 2. Calculate the quartiles Q3Q1 Q2 3, 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 Determine the quartiles. Exercise 1.5

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.

Determine whether there are outliers in the data set

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 and not greater than , then there are no outliers in the data

The Five Number Summary; Boxplots Compute the five-number summary Example 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

BOXPLOT The 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

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 (*).

Sketch the boxplot and interpret the shape of the boxplot

 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 (*).