This document provides an overview of probability, statistics, and their applications in engineering. It defines key probability and statistics concepts like trials, outcomes, random experiments, and frequency distributions. It explains how engineers use statistics and probability to analyze data from tests and experiments to better understand product quality and failure rates. Examples are given of measures of central tendency like mean and median, measures of variation like standard deviation and variance, and the normal distribution curve. Engineering applications include using these analytical techniques to assess results from a class and compare two data histograms.

1. Probability andProbability and
Statistics inStatistics in
EngineeringEngineering
Philip Bedient, Ph.D.Philip Bedient, Ph.D.

2. Probability: Basic IdeasProbability: Basic Ideas
 Terminology:Terminology:

Trial:Trial: each time you repeat aneach time you repeat an
experimentexperiment

Outcome:Outcome: result of an experimentresult of an experiment

Random experiment:Random experiment: one with randomone with random
outcomes (cannot be predicted exactly)outcomes (cannot be predicted exactly)

Relative frequency:Relative frequency: how many times ahow many times a
specific outcome occurs within thespecific outcome occurs within the
entire experiment.entire experiment.

3. Statistics: Basic IdeasStatistics: Basic Ideas
 Statistics is the area of science that deals withStatistics is the area of science that deals with
collection, organization, analysis, andcollection, organization, analysis, and
interpretation of data.interpretation of data.
 It also deals with methods and techniques thatIt also deals with methods and techniques that
can be used to draw conclusions about thecan be used to draw conclusions about the
characteristics of a large number of data points--characteristics of a large number of data points--
commonly called acommonly called a populationpopulation----
 By using a smaller subset of the entire data.By using a smaller subset of the entire data.

4. For Example…For Example…
 You work in a cell phone factory and are askedYou work in a cell phone factory and are asked
to remove cell phones at random off of theto remove cell phones at random off of the
assembly line and turn it on and off.assembly line and turn it on and off.
 Each time you remove a cell phone and turn itEach time you remove a cell phone and turn it
on and off, you are conducting aon and off, you are conducting a randomrandom
experiment.experiment.
 Each time you pick up a phone is aEach time you pick up a phone is a trialtrial and theand the
result is called anresult is called an outcomeoutcome..
 If you check 200 phones, and you find 5 badIf you check 200 phones, and you find 5 bad
phones, thenphones, then
 relative frequencyrelative frequency of failure = 5/200 = 0.025of failure = 5/200 = 0.025

5. Statistics in EngineeringStatistics in Engineering
 Engineers apply physicalEngineers apply physical
and chemical laws andand chemical laws and
mathematics to design,mathematics to design,
develop, test, anddevelop, test, and
supervise varioussupervise various
products and services.products and services.
 Engineers perform testsEngineers perform tests
to learn how thingsto learn how things
behave under stress, andbehave under stress, and
at what point they mightat what point they might
fail.fail.

6. Statistics in EngineeringStatistics in Engineering
 As engineers perform experiments, theyAs engineers perform experiments, they
collect data that can be used to explaincollect data that can be used to explain
relationships better and to revealrelationships better and to reveal
information about the quality of productsinformation about the quality of products
and services they provide.and services they provide.

7. Frequency Distribution:Frequency Distribution:
Scores for an engineering class are as follows: 58, 95, 80,Scores for an engineering class are as follows: 58, 95, 80,
75, 68, 97, 60, 85, 75, 88, 90, 78, 62, 83, 73, 70, 70, 85,75, 68, 97, 60, 85, 75, 88, 90, 78, 62, 83, 73, 70, 70, 85,
65, 75, 53, 62, 56, 72, 7965, 75, 53, 62, 56, 72, 79
To better assess the success of the class, we make aTo better assess the success of the class, we make a
frequency chart:frequency chart:

8. Now the information can be better analyzed.Now the information can be better analyzed.
For example, 3 students did poorly, and 3 didFor example, 3 students did poorly, and 3 did
exceptionally well. We know that 9 studentsexceptionally well. We know that 9 students
were in the average range of 70-79. We can alsowere in the average range of 70-79. We can also
show this data in a freq. histogram (PDF).show this data in a freq. histogram (PDF).
Divide each no. by 26

9. Cumulative FrequencyCumulative Frequency
 The data can be further organized by calculating theThe data can be further organized by calculating the
cumulative frequencycumulative frequency (CDF)(CDF)..
 The cumulative frequency shows the cumulative numberThe cumulative frequency shows the cumulative number
of students with scores up to and including those in theof students with scores up to and including those in the
given range. Usually we normalize the data - divide 26.given range. Usually we normalize the data - divide 26.

10. Measures of Central Tendency &Measures of Central Tendency &
VariationVariation
 Systematic errorsSystematic errors, also called, also called fixed errorsfixed errors, are, are
errors associated with using an inaccurateerrors associated with using an inaccurate
instrument.instrument.

These errors can be detected and avoided by properlyThese errors can be detected and avoided by properly
calibrating instrumentscalibrating instruments
 Random errorsRandom errors are generated by a number ofare generated by a number of
unpredictable variations in a given measurementunpredictable variations in a given measurement
situation.situation.

Mechanical vibrations of instruments or variations inMechanical vibrations of instruments or variations in
line voltage friction or humidity could lead to randomline voltage friction or humidity could lead to random
fluctuations in observations.fluctuations in observations.

11.  When analyzing data, the mean alone cannot signalWhen analyzing data, the mean alone cannot signal
possible mistakes. There are a number of ways to definepossible mistakes. There are a number of ways to define
the dispersion or spread of data.the dispersion or spread of data.
 You can compute how much each number deviates fromYou can compute how much each number deviates from
the mean, add up all the deviations, and then take theirthe mean, add up all the deviations, and then take their
average as shown in the table below.average as shown in the table below.

12.  As exemplified in Table 19.4, the sum of deviationsAs exemplified in Table 19.4, the sum of deviations
from the mean for any given sample is always zero.from the mean for any given sample is always zero.
This can be verified by considering the following:This can be verified by considering the following:
 WhereWhere xxii represents data points,represents data points, xx is the average,is the average, nn isis
the number of data points, andthe number of data points, and d,d, represents therepresents the
deviation from the average.deviation from the average.
x=
1
n
xi
i=1
n
∑ di=(xi−x)

13. Therefore the average of the deviations from theTherefore the average of the deviations from the
mean of the data set cannot be used to measuremean of the data set cannot be used to measure
the spread of a given data set.the spread of a given data set.
Instead we calculate the average of theInstead we calculate the average of the absoluteabsolute
valuesvalues of deviationsof deviations. (This is shown in the third. (This is shown in the third
column of table 19.4 in your textbook)column of table 19.4 in your textbook)
For group A the mean deviation is 290, and GroupFor group A the mean deviation is 290, and Group
B is 820. We can conclude that Group B is moreB is 820. We can conclude that Group B is more
scattered than A.scattered than A.
di
i=1
n
∑ = xi
i=1
n
∑ − x
i=1
n
∑ di
i=1
n
∑ =nx−nx=0

14. VarianceVariance
 Another way of measuring the data is byAnother way of measuring the data is by
calculating thecalculating the variancevariance..
 Instead of taking the absolute values ofInstead of taking the absolute values of
each deviation, you can just square theeach deviation, you can just square the
deviation and find the means.deviation and find the means.
 (n-1) makes estimate unbiased(n-1) makes estimate unbiased
v= i=1
n
∑ (xi −x)2
n−1

15.  Taking the square root of the varianceTaking the square root of the variance
which results in thewhich results in the standard deviation.standard deviation.
 The standard deviation can also provideThe standard deviation can also provide
information about the relative spread of ainformation about the relative spread of a
data set.data set.
s= i=1
n
∑ (xi −x)2
n−1

16.  The mean for a grouped distribution is calculatedThe mean for a grouped distribution is calculated
from:from:
 WhereWhere
xx = midpoints of a given range= midpoints of a given range
ff = frequency of occurrence of data in the range= frequency of occurrence of data in the range
nn == ∑∑ff = total number of data points= total number of data points
x=
(xf)∑
n

17. The standard deviation for a grouped distribution isThe standard deviation for a grouped distribution is
calculated from:calculated from:
s=
(x−x)2
f∑
n−1

18. Normal DistributionNormal Distribution
 We could use the probability distribution from the figuresWe could use the probability distribution from the figures
below to predict what might happen in the future. (i.e.below to predict what might happen in the future. (i.e.
next year’s students’ performance)next year’s students’ performance)

19. Normal DistributionNormal Distribution
 Any probability distribution with a bell-shapedAny probability distribution with a bell-shaped
curve is called acurve is called a normal distributionnormal distribution..
 The detailed shape of a normal distributionThe detailed shape of a normal distribution
curve is determined by its mean and standardcurve is determined by its mean and standard
deviation values.deviation values.

22. THE NORMAL CURVETHE NORMAL CURVE
 Using Table 19.11, approx. 68% of the data willUsing Table 19.11, approx. 68% of the data will
fall in the interval offall in the interval of -s-s toto ss, one std deviation, one std deviation
 ~ 95% of the data falls between -2~ 95% of the data falls between -2ss to 2to 2ss, and, and
approx all of the data points lie between -3approx all of the data points lie between -3ss to 3to 3ss
 For a standard normal distribution, 68% of theFor a standard normal distribution, 68% of the
data fall in the interval ofdata fall in the interval of zz = -1 to= -1 to zz = 1.= 1.
zi = (xi - x) / s

23. AREAS UNDER THE NORMAL CURVEAREAS UNDER THE NORMAL CURVE
 zz = -2 and= -2 and zz = 2 (two standard deviations below and= 2 (two standard deviations below and
above the mean) each represent 0.4772 of the total areaabove the mean) each represent 0.4772 of the total area
under the curve.under the curve.
 99.7% or almost all of the data points lie between -399.7% or almost all of the data points lie between -3ss
and 3and 3ss..

24. Analysis of Two HistogramsAnalysis of Two Histograms
Graph A is class distribution of numbers 1-10Graph A is class distribution of numbers 1-10
Graph B is class distribution of semester creditsGraph B is class distribution of semester credits
Data for A = 5.64Data for A = 5.64 +/-+/- 2.6 (much greater spread than B)2.6 (much greater spread than B)
Data for B = 15.7Data for B = 15.7 +/-+/- 1.96 (smaller spread)1.96 (smaller spread)
Skew of A = -0.16 and Skew B = 0.146Skew of A = -0.16 and Skew B = 0.146
CV of A = 0.461 and CV of B = 0.125 (CV = SD/Mean)CV of A = 0.461 and CV of B = 0.125 (CV = SD/Mean)
Frequency A
0
1
2
3
4
5
6
7
2 3 4 5 6 7 8 9 10
Frequency B
0
1
2
3
4
5
6
7
8
9
12 13 14 15 16 17 18 19 20