Download Statistics in engineering

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
STATISTICS IN ENGINEERING
Statistics is the area of science that deals with collection,
organization, analysis, and interpretation of data.
It also deals with methods and techniques that can be used
to draw conclusions about the characteristics of a large
number of data points, commonly called a population.
By using a smaller subset of the entire data called sample.
Because many aspects of engineering practice involve
working with data, obviously some knowledge of statistics
is important to an engineer.
Specifically, statistical techniques can be a powerful aid in
designing new products and systems, improving existing
designs, and improving production process.
 Engineers apply physical
and chemical laws and
mathematics to design,
develop,
test,
and
supervise
various
products and services.
 Engineers perform tests
to learn how things
behave under stress, and
at what point they might
fail.
 As engineers perform experiments, they collect data that
can be used to explain relationships better and to reveal
information about the quality of products and services
they provide.
Collecting Engineering Data
 Direct observation
The simplest method of obtaining data.
Advantage: relatively inexpensive
Drawbacks: 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
 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.
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).
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.
The most popular charts for qualitative data are:
1. bar chart/column chart;
2. pie chart; and
3. line chart.
 Example:
frequency table
Observation
Malay
Chinese
Indian
Others
Frequency
33
9
6
2
Bar Chart: used to display the frequency distribution in the
graphical form.
 Pie Chart: used to display the frequency distribution. It
displays the ratio of the observations
Malay
Chinese
Indian
Others
 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
Dec
10
7
5
10
39
7
260
316
142
11
4
9
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)
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.
 Frequency Distribution
When summarizing large quantities of raw data, it is often
useful to distribute the data into classes. There are no specifics
rules to determine the classes size and the number of individual
belonging to each class.
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.
 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.
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 Central Tendency
 Mean
Mean of a sample is the sum of the sample data divided by the
total number sample. Mean for ungroup data is given by:
_
x
x1  x2  .......  xn

x
, for n  1,2,..., n or x 
n
n
_
Mean for group data is given xby:
n

x
fx
fx

or
f

f

i 1
n
i 1
i i
i
 Median: The middle value after the data is arranged from the
lowest to the highest value. If the number of data is even, median
is the average of the two middle values.
 Mode: 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.
 Example->
Example: Ungrouped Data
Example: Grouped Data
CGPA(x)
Frequency(f)
fx
2.625
2
5.250
2.875
10
28.750
3.125
15
46.875
3.375
13
43.875
3.625
7
25.375
3.875
3
11.625
Total
50
161.750
n
 Mean:

x
fx
i 1
n
i i
f
i 1

161.75
 3.235
50
i
 Median:(3.125+3.375)/2=3.25
 Mode:3.125
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:

2
(
x

x
)
S2  
n 1
 The variance for the grouped data:
S
2
fx


2
2
nx
n 1
 The positive square root of the variance is the standard
deviation

S
 ( x  x)
n 1
2

 fx
2
2
nx
n 1
 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.
Example: Ungrouped Data
 7 , 6, 8, 5 , 9 ,4, 7 , 7 , 6, 6
 Range = 9-4=5
 Mean =
x

x
 6.5
_
n
 Variance=

S 
2
2
(
x

x
)

n 1
 Standard Deviation=
S
18.5

 2.0556
9

2
(
x

x
)

n 1
 2.0556  1.4337
 Using excel and click data analysis, choose descriptive
statistics to get the data summary result.
Column1
Mean
6.5
Standard Error
Median
Mode
0.453
6.5
7
Standard Deviation
1.434
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
2.056
0.239
0
5
4
9
65
10
Example: Grouped Data
x
4
3
2
1
0
f
10
12
8
6
4
n
Mean, x 
fx
i 1
n
f
i 1
5
Variance 

i 1
 2.45
i
fi xi 2  nx 2
n 1
10  4   12  3  8  2   6 1  4  0   40  2.45 
2

i i
2
 1.69
Std  1.69  1.30
2
2
39
2
2