FARAH ADIBAH ADNAN ENGINEERING MATHEMATICS INSTITUTE (IMK) C HAPTER 1 B ASIC S TATISTICS

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

S TATISTICS IN ENGINEERING  Statistics - area of science that deals with collection, organization, analysis, and interpretation of data.  Statistics - deals with methods and techniques that can be used to draw conclusions about the characteristics of a large number of data points, commonly called a population by using a smaller subset of the entire data called sample.  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

C OLLECTING E NGINEERING D ATA  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.

D ATA P RESENTATION  Data can be categorized into two :- - Qualitative data - qualitative attributes - Quantitative data - quantitative attributes  Two sources of data :- - Primary ( eg. Questionnaire, Telephone Interview) - Secondary (eg. Internet, Annual Report) Data should be summarized in more informative way such as graphical, tables or charts.

D ATA P RESENTATION Data can be summarized or presented in two ways: 1) Tabular 2) Charts/graphs. Data Presentation of Qualitative Data 1) Frequency Distribution Table - represents the number of times the observation occurs in the data. Example: Ethnic Group Observation Frequency Malay33 Chinese9 Indian6 Others2

Bar Chart : Ethnic Group Pie Chart : Gender Line Chart : Number of Sandpipers from Jan 1989 – Dec ) Charts for qualitative data are:

Data Presentation of Quantitative Data 1 ) Frequency Distribution Table – list all classes and the number of values that belong to each class. Weekly Earnings (dollars) (Class Limit) Number of Employees, f Class Boundaries Class Width, c Class Midpoint, x Cumulative Frequency, F – – = – = – = – = – = 100

 This formula will be used to form frequency distribution table, from raw data. Class - an interval that includes all the values that fall within two numbers; the lower and upper class (class limit). Class Boundary - the midpoint of the upper limit of one class and the lower limit of the next class. Class Width/Size/Interval,c - difference between the two boundaries of a class. Formula : C = Upper boundary – Lower Boundary Class Midpoint/Mark, x – formula: (Lower Limit + Upper Limit)/2

How to Form Frequency Distribution Table 1) Decide the number of classes to be used. 2) Determine class width:  When the number of classes are given, Class width =  When the number of classes are not given, Class width = where the number of class =  Don’t forget to always round up to the nearest whole number when dealing with class width/interval.  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: The following data give the total number of iPods sold by a mail order company on each of 30 days. Construct a frequency distribution table. (Hint: 5 number classes). Solution: Number of classes = 5 Class width =

Frequency Distribution Table Class IntervalFrequency, f – – –

2) Graph for quantitative data are: Polygon : Student’s CGPAHistogram: Student’s CGPA Ogive: Student’s CGPA

D ATA S UMMARY Summary statistics are used to summarize a set of observations. Two basic summary statistics are 1) Measures of central tendency - Mean - Median - Mode 2) Measures of dispersion - Range - Variance - Standard deviation

M EASURES OF C ENTRAL T ENDENCY 1) Mean,( )  Mean of a sample ( ) or population ( ) is the sum of the sample data divided by the total number sample.  Mean for ungroup data is given by: Sample: Population:  Mean for group data is given by: Sample: Population: where f = class frequency; x = class mark (mid point)

Example: 1) Find the mean for the set of data 4, 6, 3, 1, 2, 5, 7. Solution: 2) Find the mean of the frequency distribution table below.

Solution: Therefore, the mean of frequency distribution above is: (x) (f)

2) Median, ( )  Median is the middle value of a set of observations arranged in ascending order and normally is denoted by ( ).  Median for ungrouped data: - The median depends on the number of observations in the data,. - If is odd, then the median is the th observation of the ordered observations / middle value. - If is even, then the median is the average of the 2 middle values ( th observation and the th observation).

 Median for grouped data / frequency of distribution. The median of frequency distribution is defined by: where, = the lower class boundary of the median class; = the size of the median class interval; = the sum of frequencies of all classes lower than the median class; = the frequency of the median class.

Example: 1) Find the median for the set of data 4, 6, 3, 1, 2, 5, 7, 3. Solution: Arrange in order of magnitude : 1,2,3,3,4,5,6,7. As n = 8 (even), the median is the mean of the 4th and 5th value. Therefore, the median is 3.5 2) Find the median of the frequency distribution table below.

Solution: To determine median class: So, the median class falls in class 3.00 – Cumulative Frequency

3) Mode, ( )  The mode of a set of observations is the observation with the highest frequency and is usually denoted by ( ). Sometimes mode can also be used to describe the qualitative data. *Note:  If data set with only 1 value that occur with the highest frequency, therefore it has 1 mode and it is called unimodal data.  If data set has 2 measurements with highest frequency, therefore it has 2 modes and known as bimodal data.  If data set has more than 2 measurements with highest frequency, so the data set contains more than 2 modes and said to be multimodal data.

 For ungrouped data: - Defined as the value which occurs most frequent. Example: The mode for data 4,6,3,1,2,5,7,3 is 3.  For grouped data When data has been grouped into classes and a frequency curve is drawn to fit the data, the mode is the value of corresponding to the maximum point on the curve.

- Determining the mode using formula. where = the lower class boundary of the modal class; = the size of the modal class interval; = the difference between the modal class frequency and the class before it;and = the difference between the modal class frequency and the class after it. Note: The class which has the highest frequency is called the modal class.

Example: Find mode of the frequency distribution table below. Solution:

M EASURES OF D ISPERSION  The measure of dispersion/spread is the degree to which a set of data tends to spread around the average value.  It shows whether data will set is focused around the mean or scattered.  The common measures of dispersion are: 1) Range 2) Variance 3) Standard deviation  The standard deviation actually is the square root of the variance.  The sample variance is denoted by s 2 and the sample standard deviation is denoted by s.

1) Range  Simplest measure of dispersion.  Apply for both group & ungroup data. Ungroup data: Formula: Range = Largest value – Smallest value Group data: Formula: Range = Largest value (class limit) – Smallest value (class limit) Example: Solution: Range = Largest Value – Smallest Value = 267, 277 – 49, 651 = 217, 626 square miles. StateTotal Area (square miles) Arkansas53,182 Louisiana49,651 Oklahoma69,903 Texas267, 277

2) Variance, ( )  Measures the variability in a set of data.  The variance for the ungrouped data: Sample: Population:  The variance for the grouped data: Sample: Population:

Example: The variance for grouped data : Solution: CGPA (Class)Frequency, f Class Mark, xfxfx Total

2) Standard Deviation, ( )  The positive square root of the variance is the standard deviation.  A larger value of the standard deviation – the values of the data set are spread relatively large from the mean.  A lower value of the standard deviation – the values of the data set are spread relatively small from the mean.  The standard deviation for the ungrouped data: Sample: Population:

 The standard deviation for grouped data: Sample: Population: Example: From previous example.

E XAMPLE : U SING E XCEL

EXERCISE The final results in business statistics of 40 students are recorded as below a) Present the data in frequency distribution table. b) Construct a histogram c) Calculate mean, median, mode, variance and std deviation