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CHAPTER 1

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.

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CHAPTER 1

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  1. 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

  2. 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.

  3. Because many aspects of engineering practice involve working with data, obviously some knowledge of statistics is important to an engineer.

  4. 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

  5. BASICTERMS 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. 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

  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. 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

  12. ObservationFrequency Malay 33 Chinese 9 Indian 6 Others 2 Example: frequency table • Bar Chart: used to display the frequency distribution in the graphical form.

  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.

  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. Example: Frequency Distribution CGPA (Class) Frequency 2.50 - 2.75 2 2.75 - 3.00 10 3.00 - 3.25 15 3.25 - 3.50 13 3.50 - 3.75 7 3.75 - 4.00 3 • 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 1.1 shows that the number of classes for Students` CGPA. • 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. Table 1.1:The Fequency Distribution of the Students’ CGPA

  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 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 : Weekly Earnings of 100 Employees of a Company Frequency column Variable Frequency of the third class. Third class (Interval Class) Lower Limit of the sixth class Upper limit of the sixth class

  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 Approximate class width = (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: Given a raw data as below: 27 27 27 28 27 24 25 28 26 28 26 28 31 30 26 26 • How many classes that you recommend? • What is the class interval? • What is the lower boundary for the first class? • Build a frequency distribution table.

  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 1.3: Class Limit, Class Boundaries, Class Width , Cumulative Frequency

  23. 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

  24. 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:

  25. Median of ungrouped data: The median depends on the number of observations in the data, n . If nis 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:

  26. Table 1.4 : Example 1.12 Cumulative Frequency

  27. 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 of grouped data:

  28. 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)

  29. 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.

  30. Example: Ungrouped Data • 7 , 6, 8, 5 , 9 ,4, 7 , 7 , 6, 6 • Range = 9-4=5 • Mean • Variance • Standard Deviation

  31. Example: grouped Data The defects from machine A for a sample of products were organized into the following: What is the mean, variance and standard deviation.

  32. Exercise 1 The following data give the total number of iPads sold by a mail order company on each of 30 days. (Hint : 5 number of classes) • Construct a frequency 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

  33. Exercise 2 • 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 a) Construct a frequency table. b) Find the mean, mode, median, variance and standard deviation.. c) Construct a histogram.

  34. EXERCISE 3 • The Apollo space program lasted from 1967 until 1972 and included 13 missions. The missions lasted from as little as 7 hours to as long as 301 hours. The duration of each flight is as below. Find mean, median, standard deviation. (Counted as a population) 9 195 241 301 216 260 7 244 192 147 10 295 142

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