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

3. 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

4. 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.

5. 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

6. 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.

7. 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.

8. Graphically data presentation

9. 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).

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. 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

13. 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

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. Weight (Rounded decimal point) Frequency
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:

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.
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 :

18. NUMERICALLY SUMMARIZING DATA

19. 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

20. 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

21. 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.

22. Weight (Class Interval)
Example 1.9: From Table 1.1: Class Boundary
Weight (Class Interval)
Class Boundary
Frequency
60-62 5
63-65 18
66-68 42
69-71 27
72-74 8
Total
100

23. Example 1.10:
Given a raw data as below:
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?

24. 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
63-65 18
= 23
66-68 42
= 65
69-71 27
=92
72-74 8
= 100
100

25. 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.

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

27. 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

28. 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:

29. Example 1.11 (Ungrouped data): Mean for the sets of data 3,5,2,6,5,9,5,2,8,6 Solution :

30. 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

31. 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

32. 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:

33. 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

34. 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

35. 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

36. Mode for grouped data

37. 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
Total
6745
Mode class

38. 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)

39. 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.

40. 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

41. 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
Total
6745

42. 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

43. 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.

44. 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

45. 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

46. 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

47. 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.

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

49. 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.

50. 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.

51. Example 1.21 The following data represent the number of inches of rain in Chicago during the month of April for 20 randomly years a) Determine the number of inches of rain correspond to 15th percentile,

52. 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.

53. 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

54. Outliers
Extreme observations
Can occur because of the error in measurement of a variable, during data entry or errors in sampling.

55. 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.

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

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

58. 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 (*).

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