Subbulakshmi Murugappan H/P: 016-4054017 subbulakshmi@unimap.edu.my

Basic StatisticsStatistics in EngineeringCollecting Engineering DataData Summary and PresentationProbability Distributions

- Discrete Probability Distribution- Continuous Probability Distribution

Sampling Distributions of the Mean and Proportion

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.

the methods of statistics allow scientists and engineers to design valid experiments and to draw reliable conclusions from the data they produce

•Specifically, statistical techniques can be a powerful aid in designing new products and systems, improving existing designs, and improving production process.

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.

Direct observationThe simplest method of obtaining data.Advantage: relatively inexpensiveDisadvantage: difficult to produce useful information since it does not consider all aspects regarding the issues.

ExperimentsMore expensive methods but better way to produce dataData produced are called experimental

SurveysMost familiar methods of data collectionDepends on the response rate

Personal InterviewHas the advantage of having higher expected response rateFewer incorrect respondents.

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.

Data can be summarized or presented in two ways:1. Tabular2. 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).

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; and3. line chart.

Types of Graph Qualitative Data

Example:frequency table

Bar Chart: used to display the frequency distribution in the graphical form.

Observation FrequencyMalay 33Chinese9Indian 6Others 2

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. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec10 7 5 10 39 7 260 316 142 11 4 9

Data Presentation Of Quantitative Data

Tabular presentation for quantitative data is usually in the form of frequency distribution that is atable 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; and3. ogive.

Example: Frequency DistributionCGPA (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 thatthe horizontal axis represent the data whichis quantitative in nature. There is no gap betweenthe bars.

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` 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.

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

Total 50

Table 1.1:The Fequency Distribution of the Students’

CGPA

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. Weekly Earnings

(dollars)Number of

Employees, f

801-1000 9

1001-1200 22

1201-1400 39

1401-1600 15

1601-1800 9

1801-2000 6

Variable Frequencycolumn

Third class (Interval Class)

Lower Limit of the sixth class

Frequencyof the third class.

Upper limit of the sixth class

Table 1.2 : Weekly Earnings of 100 Employees of a Company

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 MarkClass midpoint or mark = (Lower Limit + Upper Limit)/2- Finding The Number of ClassesNumber of classes = - Finding Class Width For Interval ClassApproximate 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.

n

Example:

Given a raw data as below:27 27 27 28 27 24 25 2826 28 26 28 31 30 26 26

a) How many classes that you recommend?b) What is the class interval?c) What is the lower boundary for the first class?d) Build a frequency distribution table.

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

Weekly Earnings (dollars)

(Class Limit)

Number of Employees, f

Class Boundaries

Class Width

Cumulative Frequency

801-1000 9 800.5 – 1000.5 200 9

1001-1200 22 1000.5 – 1200.5 200 9 + 22 = 31

1201-1400 39 1200.5 – 1400.5 200 31 + 39 = 70

1401-1600 15 1400.5 – 1600.5 200 70 + 15 = 85

1601-1800 9 1600.5 – 1800.5 200 85 + 9 = 94

1801-2000 6 1800.5 – 2000.5 200 94 + 6 = 100

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 TendencyMeanMedianModeMeasures of DispersionRangeVarianceStandard deviation

MeanMean 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:

x

n

xxornnfor

n

xxxx n

_21

_

,...,2,1,.......

f

fxor

f

xfx n

ii

n

iii

1

1

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:

Cumulative

Frequency

Table 1.4 : Example 1.12

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)

1

)( 22

n

xxS

22

2

1

fx n xS

n

22

2

( )

1

fxfx

nSn

22

2 fx n xS

n

22

2

( )fxfx

nSn

22 ( )x xS

n

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

22 2( )

1 1

x x fx n xS

n n

7 , 6, 8, 5 , 9 ,4, 7 , 7 , 6, 6

Range = 9-4=5 Mean

Variance

Standard Deviation

22 ( ) 18.5

2.05561 9

x xS

n

_

6.5x

xn

2( )2.0556 1.4337

1

x xS

n

The defects from machine A for a sample of products were organized into the following:

What is the mean, 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

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)

a) Construct a frequency table.b) Find the mean, variance and standard deviation,

mode and median. c) Construct a histogram.

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 9 21 14

The data below represent the waiting time (in minutes) taken by 30 customers at one local bank.25 31 20 30 22 32 37 2829 23 35 25 29 35 29 2723 32 31 32 24 35 21 3535 22 33 24 39 43

a) Construct a frequency table.b) Find the mean, mode, median, variance and

standard deviation.. c) Construct a histogram.

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 7244192 147 10 295 142