Engineering Applications of Statistics(will be known to us eventually as we go through our syllabus)1. Statistical Process Control

2. Quality Assessment

3. Model Building and Predicting

4. Communicating with and Acting on Experimental Results

5. Assessing Design Reliability

6. Experimental Design

VARIABLE AND DATA

Variable A characteristic of a population or sample that is of

interest to us. Example

In case of Hudson Auto a variable is the average cost of parts used in engine tune-ups.

VARIABLE AND DATA

Data The actual values of the variables Data may be quantitative, qualitative

Example In case of Hudson Auto data are the actual costs of

parts used in 50 engine tune-ups observed:91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76

104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

QUANTITATIVE DATA

Quantitative data may have a ratio scale Data possessing a natural zero point and organized

into measures for which differences are meaningful. Examples: Money, income, sales, profits, losses,

heights of NBA players Quantitative data may also have an interval scale

The distance between numbers is a known, constant size, but the zero value is arbitrary.

Examples: Temperature on the Fahrenheit scale.

Types of Quantitative Data

A) Nominal data: They are numbers that represent arbitrary data.

Example: An engineering school, might use numbers to denote undergraduate majors.

1 For electrical engineering

2 For civil engineering and so on.

Types of Quantitative Data

B) Ordinal data convey ranking in terms of importance, strength, or severity.

Example: a value of 3 corresponds to gentle breezeA value of 6 corresponds to a strong breezeA value of 9 signifies a strong galeThe change in force between a gentle and a strong

sea breeze is not equal to that between a strong gale and a strong breeze, as the numbers themselves would misleadingly indicate.

Types of Quantitative Data

C) Interval data allow only addition and subtraction.

Example: Temperature for which scales are arbitrarily chosen. A 100 degree Fahrenheit day is not twice as hot as 50 degree day since 100 degree/ 50 degree is not a meaningful ratio.

Types of Quantitative Data

D) Ratio Data include times and many physical measurements of size, weight, or strength. The arithmetic operations of addition, subtraction, division, and multiplication are all valid with ratio data.

Most statistical investigations involve arithmetic operations, which limits them to interval or ratio data.

Example: income, sales, profits.

QUALITATIVE DATA

Qualitative data has a nominal scale Data that can only be classified into categories and

cannot be arranged in an ordering scheme. Examples: eye color, gender, marital status,

religious affiliation, etc. Examples: Suppose that the responses to the

marital-status question is recorded as follows:

Single 1 Divorced3

Married 2 Widowed 4

85:222: Chapter 2Describing, Displaying, and Exploring Statistical

Data

Assuming you have collected a data set of your interest

The questions are:How can you make sense out of it?How can you organize and summarize the data set to make it more comprehensible and meaningful?

Graphical techniques for Describing Quantitative DataThe Frequency Distribution

The most popular and traditional graphical method for describing quantitative data is the frequency histogram – often called a frequency distribution.

HISTOGRAM

Consider the following data that shows days to maturity for 40 short-term investments

70 64 99 55 64 89 87 6562 38 67 70 60 69 78 3975 56 71 51 99 68 95 8657 53 47 50 55 81 80 9851 31 63 66 85 79 83 70

HISTOGRAM

First, construct a frequency distribution An arrangement or table that groups data into non-

overlapping intervals called classes and records the number of observations in each class

Approximate number of classes: See Table 2.3, p. 29Number of observation Number of classes

Less than 50 5-750-200 7-9200-500 9-10500-1,000 10-111,000-5,000 11-135,000-50,000 13-17More than 50,000 17-20

HISTOGRAM

Approximate class width is obtained as follows:

classes ofNumber

alueSmallest v-lueLargest va widthclass eApproximat

HISTOGRAM

Classes and counts for the days-to-maturity data

Days toMaturity

TALLY Number ofInvestments

HISTOGRAM

Class relative frequency is obtained as follows:

nsobservatio of number Total

frequency Classfrequency relative Class

HISTOGRAM

0

2

4

6

8

10

12

40 50 60 70 80 90 100

Number of Days to Maturity

Fre

qu

ency

HISTOGRAM

Classes: Categories for grouping data. Frequency: The number of observations that fall in a class. Frequency distribution: A listing of all classes along with their frequencies. Relative frequency: The ratio of the frequency of a class to the total number of observations. Relative-frequency distribution: A listing of all classes along with their relative frequencies.

HISTOGRAM

Lower cut point: The smallest value that can go in a class.

Upper cut point: The smallest value that can go in the next higher class. The upper cut point of a class is the same as the lower cut point of the next higher class.

Midpoint: The middle of a class, obtained by taking the aver age of its lower and upper cut points.

Width: The difference between the upper and lower cut points of a class.

HISTOGRAM

Frequency histogram: A graph that displays the classes onthe horizontal axis and the frequencies of the classes on thevertical axis. The frequency of each class is represented by avertical bar whose height is equal to the frequency of the class.

Relative-frequency histogram: A graph that displays theclasses on the horizontal axis and the relative frequencies ofthe classes on the vertical axis. The relative frequency of eachclass is represented by a vertical bar whose height is equal tothe relative frequency of the class.

RELATIVE FREQUENCY HISTOGRAM

Relative-frequency distribution for the days-to-maturity data

Days toMaturity

Relative Frequency

RELATIVE FREQUENCY HISTOGRAM

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

40 50 60 70 80 90 100

Number of Days to Maturity

Rel

ativ

e F

req

uen

cy

FREQUENCY POLYGON

A frequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes.

The frequencies represent the heights of the midpoints.

FREQUENCY POLYGON

0

2

4

6

8

10

12

35 45 55 65 75 85 95

Number of Days to Maturity

Fre

qu

ency

STEM-AND-LEAF DISPLAY

Here the raw data are arranged tabularly by locating each observation on a ‘tree’ This is done by separating the values into a stem digit and a leaf digit.

The main advantage is that it provides the essential features of the histogram; it can be seen easily by rotating the plot 90 degrees

The disadvantage is that the stem-and-leaf plot becomes cumbersome if the number of observations is large

STEM-AND-LEAF DISPLAY

Diagrams for days-to-maturity data: (a) stem-and-leaf Stem Leaves 3 4 5 6 7 8 9 (a)