STATISTICS

*CHAPTER 5

Statistics*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.

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 of the individuals within the population.

*Observation

- Value of variable for an element.

*Data Set

- A collection of observation on one or more variables.

Collecting 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

SurveysMost familiar methods of data collectionDepends on the response rate

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

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.

*Graphically data presentation

Graphical Data Presentation

* 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 1.1:

frequency table

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

Example 1.2:

Observation FrequencyMalay 33Chinese9Indian 6Others 2

*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

MalayChineseIndianOthers

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 1.5: Frequency Distribution Weight (Rounded decimal point) Frequency

60-62 5

63-65 18

66-68 42

69-71 27

72-74 8

*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:

*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 :

*NUMERICALLY SUMMARIZING DATA

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.

WeightFrequenc

y60-62 563-65 1866-68 4269-71 2772-74 8Total 100

Table 1.1: Weight of 100 male students in XYZ university

*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.

Weight Frequency60-62 563-65 1866-68 4269-71 2772-74 8Total 100

Variable Frequencycolumn

Third class (Interval Class)

Lower Limit of the fifth class

Frequencyof the third class.

Upper limit of the fifthclass

Table 1.2 : Weight of 100 male students in XYZ university

*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 (Sturge`s Rule)

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.

1 3.3log n

Example 1.9:

From Table 1.1: Class Boundary

Weight (Class

Interval)Class

Boundary Frequency60-62 59.5-62.5 563-65 62.5-65.5 1866-68 65.5-68.5 4269-71 68.5-71.5 2772-74 71.5-74.5 8Total 100

Example 1.10:

Given a raw data as below:

27 27 27 28 27 20 25 28

26 28 26 28 31 30 26 26

33 28 35 39

a) How many classes that you recommend?

b) How many class interval?

c) Build a frequency distribution table.

d) What is the lower boundary for the first class?

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 59.5-62.55

63-65 18 62.5-65.55 + 18 = 23

66-68 42 65.5-68.523 + 42 = 65

69-71 27 68.5-71.565 + 27 =92

72-74 8 71.5-74.592 + 8 = 100

100

Exercise 1 :

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

*Construct a frequency distribution and cumulative frequency distribution table.

*Construct a histogram.

• Measures of Central Tendency

•Measures of Dispersion

•Measures of Position

*Data summary

Data SummarySummary 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

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:

x

n

xxornnfor

n

xxxx n

_21

_

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

f

fxor

f

xfx n

ii

n

iii

1

1

Example 1.11 (Ungrouped data):

Mean for the sets of data 3,5,2,6,5,9,5,2,8,6

Solution :

3 5 2 6 5 9 5 2 8 65.1

10x

Example 1.12 (Grouped Data):

Use the frequency distribution of weights 100 male students in XYZ university, to find the mean. Weight Frequency

60-6263-6566-6869-7172-74

51842278

Solution :

Weight (Class Interval

Frequency, f Class Mark, x

fx

60-6263-6566-6869-7172-74

51842278

?fx

xf

*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:

1

1

2

where

L = the lower class boundary of the median class

c = the size of median class interval

F the sum of frequencies of all classes lower than the median class

the fre

j

j

j

j

fF

x L cf

f

quency of the median class

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

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

60-6263-6566-6869-7172-74

51842278

6164677073

305115228141890584

12 ?j

j

fF

x L cf

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

1

1 2

When data has been grouped in classes and a frequency curveis drawn

to fit the data, the mode is the value of x corresponding to the maximum

point on the curve, that is

ˆ

the lower c

x L c

L

1

2

lass boundary of the modal class

c = the size of the modal class interval

the difference between the modal class frequency and the class before it

the difference between the modal class frequency a

nd the class after it

*the class which has the highest frequency is called the modal class

*Mode for grouped data

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

60-6263-6566-6869-7172-74

51842278

6164677073

305115228141890584

5236592

100

59.5-62.562.5-65.565.5-68.568.5-71.571.5-74.5

Total 100 6745

Mode class

1

1 2

ˆ ?x L c

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)

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 x

Sn

*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

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 2

2

2

( )?

1

( )?

1

x xs

n

x xs

n

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

60-6263-6566-6869-7172-74

51842278

6164677073

305115228141890584

5236592

100

59.5-62.562.5-65.565.5-68.568.5-71.571.5-74.5

Total 100 6745

2x2fx

22

2

( )

?1

fxfx

nSn

2

2

?1

fx n xS

n

Exercise 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

Exercise 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)

a) Construct a frequency distribution 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