SP and R.pptx

Statistics & Probability
Chapter 29 & 30

Statistics - Definition
Statistics is a field of study which deals with the collection, analysis
and interpretation of data. It deals with all related aspects starting
from the planning of a study, the data collection and to the design
of experiments. It falls into two categories depending on the
purpose of the study:
– It is called as descriptive statistics when the study is limited to
summarization and description of data.
– If the study is extended to perform estimation, prediction, and/or
generalization of the results then it is called inferential statistics.

Statistics
Following are essential elements for an inferential statistical study.
1. Population denoted as “N” is the envelope of the selected elements of
interest to the decision-maker. Normally the data from the population
becomes very large to manage.
2. Sample denoted as “n” is a subset of data randomly selected from a
population. The analyst would have to carefully select the sample to a
true sample as a distorted sample would yield wrong results.
3. Statistical inference is an estimation, prediction or generalization of the
results from the sample on to the population.
4. Reliability is the measurement of the “goodness” of the inference.

Qualitative Data vs. Quantitative Data
Any data selected for a statistical study can be categorized as:
– Qualitative Data take on values that are names or labels. The color of a
car (e.g., red, green, blue) would be examples of categorical variables.
– Quantitative variables are numerical. They represent a measurable
quantity. For example, when we speak of the population of a city, we are
talking about the number of people in the city - a measurable attribute of
the city.

Quantitative Data - Representation
The Quantitative Data can be described or represented
graphically or numerically.
Numerical methods
– Array, Columns, Rows
– Tabular form
– Case form
Graphical methods
– Stem and leaf plots
– Histograms, Bar chart, Pie chart etc.
– Frequency distribution and relative frequency (f/n)

Data Analysis
Numerical methods for analysis of data are:
– Measures of location (central tendency),
• Mean (average),
• Median, and
• Mode;
– Measures of dispersion,
• Range,
• Variance, and
• Standard deviation;
– Relative standing
• Percentile, and
• Z - score

Organizing Data: Steam and Leaf Plot
• A stem and leaf plot looks something like a bar graph. Each number in the
data is broken down into a stem and a leaf, thus the name. The stem of the
number includes all but the last digit. The leaf of the number will always be
a single digit.
• Example – Making a stem and leaf plot
– A teacher asked 10 of her students how many books they had read in the last 12
months. Their answers were as follows:
– 12, 23, 19, 6, 10, 7, 15, 25, 21, 12
– Prepare a stem and leaf plot for these data.
Steam Leaf
0 6, 7
1 2, 9, 0, 5, 2
2 3, 5, 1

Example – Let us take one example and explain
it all
• An estimator has given the task of
estimating total hours required to
complete 111200 m3 of
excavation for the next project.
He referred 100 previous project
files to study the time taken to
complete 100 m3.
• The company has the given data.
50 54 55 92 61 45 70 65 60 55
55 70 44 91 60 60 75 60 64 63
65 75 55 92 62 65 45 50 55 60
65 80 95 94 64 62 60 65 63 70
70 82 90 95 65 63 95 45 60 65
64 84 86 80 67 64 70 85 75 74
66 80 66 82 69 65 72 72 70 70
67 82 67 83 50 60 74 94 80 72
60 67 55 81 55 70 56 90 90 73
65 66 48 80 45 75 78 84 60 62
*In Minutes

Now Prepare Stem and Leaf Plot for our
example
50 54 55 92 61 45 70 65 60 55
55 70 44 91 60 60 75 60 64 63
65 75 55 92 62 65 45 50 55 60
65 80 95 94 64 62 60 65 63 70
70 82 90 95 65 63 95 45 60 65
64 84 86 80 67 64 70 85 75 74
66 80 66 82 69 65 72 72 70 70
67 82 67 83 50 60 74 94 80 72
60 67 55 81 55 70 56 90 90 73
65 66 48 80 45 75 78 84 60 62

Steam and Leaf
Stem Leaf
4 4 5 5 5 5 8
5 0 0 0 4 5 5 5 5 5 5 5 6
6 0 0 0 0 0 0 0 0 0 0 1 2 2 2 3 3 3 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 7 7 7 7 9
7 0 0 0 0 0 0 0 0 2 2 2 3 4 4 5 5 5 5 8
8 0 0 0 0 0 1 2 2 2 3 4 4 5 6
9 0 0 0 1 2 2 4 4 5 5 5

Histogram
Frequency Distribution per 100 Cubic Meter
0
4
8
12
16
20
24
28
32
36
40
Minutes
Frequency
40 50 60 70 90
80 100

Frequency Distribution and Relative Frequency
Minutes
X
Frequency
F
Relative
Frequency
Percentage
Frequency
Cumulative
Percentage
40 - 49 6 6/100 = 0.06 6 6
50 - 59 12 12/100 = 0.12 12 18
60 - 69 38 38/100 = 0.38 38 56
70 - 79 19 19/100 = 0.19 19 75
80 - 89 14 14/100 = 0.14 14 89
90 - 99 11 11/100 = 0.11 11 100
Total 100 1.0 100

Numerical methods – Measures of Central
Tendency
• The best way to reduce a set of data and still
retain part of the information, is to summarize
the set with a single value. But how can you
calculate a number that is representative of an
entire list of numbers?
• Measures of central tendency are mean,
median, and mode can help you capture, with a
single number, what is typical of the data.

Measures of Central Tendency
• The mean is the average value of all the data in
the set.
• The median is the value that has exactly half
the data above it and half below it.
• The mode is the value that occurs most
frequently in the set.
• In a normal distribution, mean, median and
mode are identical in value.

Mean
• The mean of a numeric variable is calculated by
adding the values of all observations in a data
set and then dividing that sum by the number
of observations in the set. This provides the
average value of all the data.
• Mean = sum of all the observation values ÷
number of observations
• X = ∑x / n
where x stands for an observed value,
n stands for the number of observations in the data set,
x stands for the sum of all observed x values,
And X stands for the mean value of x.

Mean - Example
• Example 1 – Soccer tournament at Dubai Football Stadium
– UAE hosts a soccer tournament each year. This season, in 10 games, the lead scorer
for the home team scored 7, 5, 0, 7, 8, 5, 5, 4, 1 and 5 goals. What was the mean
score?
• Mean = sum of all the observed values ÷ number of observations
= (7 + 5 + 0 + 7 + 8 + 5 + 5 + 4 + 5 + 1) ÷ 10
= 47 ÷ 10
= 4.7
• The average of 4.7 is not a whole number so it only has meaning in a
statistical sense. In reality, it is impossible to score 4.7 goals, even if you
are a top scorer.

Mean calculation for Frequency Tables
• X = ∑xf / ∑f
• where x stands for an observed value,
– xf stands for the product of an observed value,
multiplied by its frequency,
– ∑ xf stands for the total of all xf values,
– ∑ f stands for the total of all frequencies, and
– X stands for the mean value of x.

Mean for our example
Minutes Midpoint
X
Frequency
F
Total amount
of Mid point
XF
40 - 49 45 6 270
50 - 59 55 12 660
60 - 69 65 38 2470
70 - 79 75 19 1425
80 - 89 85 14 1190
90 - 99 95 11 1045
100 7060
Mean
7060/100 = 70.60

Median
• If observations of a variable are ordered by value, the median value
corresponds to the middle observation in that ordered list.
• The median value corresponds to a cumulative percentage of 50% (i.e., 50%
of the values are below the median and 50% of the values are above the
median).
• The position of the median is n+1 / 2 th value, where n is the number of
values in a set of data.
• In order to calculate the median, the data must be ranked (sorted in
ascending order) first. The median is the number in the middle.

Median - Example 1 – Raw data (discrete
variables)
• Imagine that a top running athlete in a typical 200-metre training session runs in the
following times: 26.1, 25.6, 25.7, 25.2 and 25.0 seconds.
• How would you calculate his median time?
– First, the values are put in ascending order: 25.0, 25.2, 25.6, 25.7, 26.1. Then, using the
following formula, figure out which value is the middle value. Remember that n represents the
number of values in the data set.
• Median = (n + 1) ÷ 2th value
= (5 + 1) ÷ 2
= 6 ÷ 2
= 3
• The third value in the data set will be the median. Since 25.6 is the third value, 25.6
seconds would be the median time.
– = 25.6 seconds

Example 2 – Raw data (discrete variables)
• Now, if the runner sprints the sixth 200-metre race in 24.7 seconds, what is the median value now?
• Again, you first put the data in ascending order: 24.7, 25.0, 25.2, 25.6, 25.7, 26.1. Then, you use the
same formula to calculate the median time.
– Median = (n + 1) ÷ 2th value
= (6 + 1) ÷ 2
= 7 ÷ 2
= 3.5
• The median is the 3.5th value in the data set meaning that it lies between the third and fourth
values. Thus, the median is calculated by averaging the two middle values of 25.2 and 25.6. Use the
formula below to get the average value.
– Average = (value below median + value above median) ÷ 2
= (third value + fourth value) ÷ 2
= (25.2 + 25.6) ÷ 2
= 50.8 ÷ 2
= 25.4
• The value 25.4 falls directly between the third and fourth values in this data set, so 25.4 seconds
would be the median time.
Same for the grouped distribution

Median for grouped frequency distribution
Median = L1 + ( (n/2) – Cf ) X C ) / f
L1 - Lower value of the Median class
n - ∑ F (Sum of the frequencies)
Cf – Cumulative frequency of the preceding class
C – Class interval
f – frequency of the Median class
Median class is the class interval which holds the mid-point of the sum of frequencies
(n/2)

Median for grouped frequency distribution
Minutes Frequency
F
End Point
X
Cumulative
Frequency
Percentage Cumulative
Percentage
40 –< 50 6 50 6 6 % 6 %
50 –<60 12 60 18 12 % 18 %
60 –< 70 38 70 56 38 % 56 %
70 –< 80 19 80 75 19 % 75 %
80 –< 90 14 90 89 14 % 89 %
90 –<
100
11 100 100 11 % 100 %

Median for our example
Median = L1 + ( (n/2) – Cf ) X C ) / f
Median = 60 + ( (100/2) – 18 ) X 10 ) / 38
Median = 68.42
It would match within the median class which states the median to be
between 60 and 70

Mode
• In a set of data, the mode is the most frequently
observed data value.
• There may be no mode if no value appears more than
any other.
• There may also be two modes (bimodal), three modes
(trimodal), or four or more modes (multimodal).
• In the case of grouped frequency distributions, the
modal class is the class with the largest frequency.
• Mode = the most frequently observed data value

Measures of dispersion (spread)
• Measures of central tendency attempt to identify
the most representative value in a set of data.
• Mean, median and mode give different
perspectives of a data set's centre, but a data
description is not complete until the spread
variability is also known.
• In fact, the basic numerical description of a data
set requires measures of both centre and spread.
Some methods of measure of spread include
range, quartiles, variance and standard deviation.

Range
• The range is very easy to calculate because it is
simply the difference between the largest and the
smallest observed values in a data set. Thus,
range, including any outliers, is the actual spread
of data.
– Range = difference between highest and lowest
observed values
– The range can be expressed as an interval such as 4–
10, where 4 is the lowest value and 10 is highest.
Often, it is expressed as interval width. For example,
the range of 4–10 can also be expressed as a range of
6.

Quartiles
• The median divides the data into two equal sets.
– The lower quartile is the value of the middle of the first
set, where 25% of the values are smaller than Q1 and
75% are larger. This first quartile takes the notation Q1.
– The upper quartile is the value of the middle of the
second set, where 75% of the values are smaller than
Q3 and 25% are larger. This third quartile takes the
notation Q3.
• It should be noted that the median takes the
notation Q2, the second quartile.

Interquartile
• The interquartile range is another range used as a
measure of the spread. The difference between upper
and lower quartiles (Q3–Q1), which is called the
interquartile range, also indicates the dispersion of a
data set.
• The interquartile range spans 50% of a data set, and
eliminates the influence of outliers because, in effect,
the highest and lowest quarters are removed.
• Interquartile range =difference between upper
quartile (Q3) and lower quartile (Q1)

Quartile Example
Data 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36
Ordered Data 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49
Median 41
Upper Quartile 43
Lower Quartile 15
Interquartile 28

Exercise
• A year ago, Angela began working at a computer store. Her supervisor
asked her to keep a record of the number of sales she made each month.
• The following data set is a list of her sales for the last 12 months:
– 34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37
• Use Angela's sales records to find:
– a) the median
b) the range
c) the upper and lower quartiles
d) the interquartile range

Answer
• a) The values in ascending order are:
1, 11, 15, 19, 20, 24, 28, 34, 37, 47, 50, 57.
Median = (12 + 1) ÷ 2
= 6.5th value
= (6th + 7th observations) ÷ 2
= (24 + 28) ÷ 2
= 26
• b) Range = difference between the highest and lowest values
= 57 - 1
= 56

Answer
• c) Lower quartile = value of middle of first half of data Q1
= the median of 1, 11, 15, 19, 20, 24
= (3rd + 4th observations) ÷ 2
= (15 + 19) ÷ 2
= 17
Upper quartile = value of middle of second half of data Q3
= the median of 28, 34, 37, 47, 50, 57
= (3rd + 4th observations) ÷ 2
= (37 + 47) ÷ 2
= 42
• d) Interquartile range = Q3–Q1
= 42 - 17
= 25

Semi-quartile range
• The semi-quartile range is another measure of
spread. It is calculated as one half the
difference between the 75th percentile (often
called Q3) and the 25th percentile (Q1). The
formula for semi-quartile range is:
• (Q3–Q1) ÷ 2.

Variance and Standard Deviation
• Unlike range and quartiles, the variance
combines all the values in a data set to
produce a measure of spread.
• The variance (symbolized by S2) and standard
deviation (the square root of the variance,
symbolized by S) are the most commonly used
measures of spread.

Variance Calculation
• Variance is a measure of how spread out a data set is. It
is calculated as the average squared deviation of each
number from the mean of a data set. For example, for
the numbers 1, 2, and 3 the mean is 2 and the variance
is 0.667.
• [(1 - 2)2 + (2 - 2)2 + (3 - 2)2] ÷ 3 = 0.667
• [squaring deviation from the mean] ÷ number of
observations = variance
• Variance (S2) = average squared deviation of values
from mean

Standard Deviation
• Taking the square root of the variance gives us
the units used in the original scale and this is
the standard deviation.
• Standard deviation (S) = square root of the
variance
• Standard deviation is the measure of spread
most commonly used in statistical practice
when the mean is used to calculate central
tendency.

Standard Deviation (Sigma Level)
• If X = mean, S = standard deviation and x = a
value in the data set, then
• about 68% of the data lie in the interval: X- S
< x < X+ S.
• about 95% of the data lie in the interval: X-
2S < x < X+ 2S.
• about 99% of the data lie in the interval: X- 3S
< x < X+ 3S.

Example
• Standard deviation
• A hen lays eight eggs. Each egg was weighed
and recorded as follows:
• 60 g, 56 g, 6l g, 68 g, 51 g, 53 g, 69 g, 54 g
– a) First, calculate the mean:
– Mean = 472/8
– =59

Example
Weight (X) ( X – X ) ( X – X )2
60 1 1
56 -3 9
61 2 4
68 9 81
51 -8 64
53 -6 36
69 10 100
54 -5 25
472 320

Excavation Example – Grouped frequency
Hours Midpoint (X) Frequency (F) XF ( X - X ) (X - X)2 (X-X)2F
40 - 50 45 6 270 -25.6 655.36 3932.16
50 - 60 55 12 660 -15.6 243.36 2920.32
60 - 70 65 38 2470 -5.6 31.36 1191.68
70 - 80 75 19 1425 4.4 19.36 367.84
80 - 90 85 14 1190 14.4 207.36 2903.04
90 - 100 95 11 1045 24.4 595.36 6548.96
Total 100 7060 17864
X 70.6 S.D 13.37

Accuracy Level
• Assuming the frequency distribution is approximately
normal, calculate the interval within which 99% of the
previous example's observations would be expected to
occur.
• X- 3S < x < X+ 3S.
• 70.6 – (3x13.37) < X < 70.6 + (3x13.37)
• 30.49 < X < 110.71
• This means there is a 99% certainty that excavation of
100 M3 will take some where between 30.49 to 110.71
minutes.

Measures of relative standing
• Measures of relative standing are numbers which indicate where a particular
value lies in relation to the rest of the values in a set of data or a population.
We'll review just two types of such measures here.
• The first type, standard (Z) scores, are not only useful as descriptive numbers,
but are of fundamental importance in working with the normal distribution.
• The second, percentiles, and related quantities, are primarily used only as
descriptive numbers, but see very wide use in many fields. The notion of a
"percentile" makes the term convenient to use in a variety of technical
contexts as well.

Standard Z Scores
• The standard score = z = x - μ / σ
– Where x is a raw score to be standardized
– σ is the standard deviation of the population
– μ is the mean of the population
• The quantity z represents the distance
between the raw score and the population
mean in units of the standard deviation. z is
negative when the raw score is below the
mean, positive when above.

Excavation Example
• Suppose I want to find where 86 Minutes
stands in my normal distribution curve.
• Z = 86 – 70.7 / 13.37
• Z = 1.14

Random Variable and Probability Distribution
• To understand probability distributions, it is
important to understand variables. random
variables, and some notation.
• A variable is a symbol (A, B, x, y, etc.) that can
take on any of a specified set of values.
• When the value of a variable is the outcome
of a statistical experiment, that variable is a
random variable.

Probability Distributions
• An example will make clear the relationship between
random variables and probability distributions.
• Suppose you flip a coin two times. This simple
statistical experiment can have four possible outcomes:
HH, HT, TH, and TT.
• Now, let the variable X represent the number of Heads
that result from this experiment.
• The variable X can take on the values 0, 1, or 2. In this
example, X is a random variable; because its value is
determined by the outcome of a statistical experiment.

Probability Distribution
• A probability distribution is a
table or an equation that links
each outcome of a statistical
experiment with its probability of
occurrence.
• Consider the coin flip experiment
described in last slide. The table,
which associates each outcome
with its probability, is an example
of a probability distribution.
Number of
Heads
Probability
0 0.25
1 0.50
2 0.25

Cumulative Probability Distribution
• A cumulative probability refers to the probability that
the value of a random variable falls within a specified
range.
• Let us return to the coin flip experiment. If we flip a
coin two times, we might ask: What is the probability
that the coin flips would result in one or fewer heads?
• The answer would be a cumulative probability. It would
be the probability that the coin flip experiment results
in zero heads plus the probability that the experiment
results in one head.
• P(X < 1) = P(X = 0) + P(X = 1) = 0.25 + 0.50 = 0.75

Cumulative Probability Distribution Table
• Like a probability distribution, a cumulative
probability distribution can be represented by
a table or an equation. In the table below, the
cumulative probability refers to the
probability than the random variable X is less
than or equal to x.

Uniform Probability Distribution
• The simplest probability distribution occurs when all of the values of a
random variable occur with equal probability. This probability distribution
is called the uniform distribution.
• Example: Suppose a die is tossed. What is the probability that the die will
land on 6 ?
• Solution: When a die is tossed, there are 6 possible outcomes represented
by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is a random variable (X),
and each outcome is equally likely to occur. Thus, we have a uniform
distribution. Therefore, the P(X = 6) = 1/6.

Discrete and Continuous Probability
Distributions
• If a variable can take on any value between two specified values, it is called a
continuous variable; otherwise, it is called a discrete variable.
• Some examples will clarify the difference between discrete and continuous variables.
– Suppose the fire department mandates that all fire fighters must weigh between 150 and 250
pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire
fighter's weight could take on any value between 150 and 250 pounds.
– Suppose we flip a coin and count the number of heads. The number of heads could be any
integer value between 0 and plus infinity. However, it could not be any number between 0 and
plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be
a discrete variable.
• Just like variables, probability distributions can be classified as discrete or
continuous.

Discrete Probability Distributions
• If a random variable is a discrete variable, its probability distribution is called a
discrete probability distribution.
• Suppose you flip a coin two times. This simple statistical experiment can have four
possible outcomes: HH, HT, TH, and TT.
• Now, let the random variable X represent the number of Heads that result from
this experiment. The random variable X can only take on the values 0, 1, or 2, so it
is a discrete random variable.
• The probability distribution for this statistical experiment appears below.

Binominal Distribution
• A binomial experiment (also known as a
Bernoulli trial) is a statistical experiment that has
the following properties:
– The experiment consists of n repeated trials.
– Each trial can result in just two possible outcomes. We
call one of these outcomes a success and the other, a
failure.
– The probability of success, denoted by P, is the same
on every trial.
– The trials are independent; that is, the outcome on
one trial does not affect the outcome on other trials.

Example
• Consider the following statistical experiment. You flip a
coin 2 times and count the number of times the coin
lands on heads. This is a binomial experiment because:
– The experiment consists of repeated trials. We flip a coin 2
times.
– Each trial can result in just two possible outcomes - heads
or tails.
– The probability of success is constant - 0.5 on every trial.
– The trials are independent; that is, getting heads on one
trial does not affect whether we get heads on other trials.

• The following notation is helpful, when we talk about binomial probability.
– x: The number of successes that result from the binomial experiment.
– n: The number of trials in the binomial experiment.
– P: The probability of success on an individual trial.
– Q: The probability of failure on an individual trial. (This is equal to 1 - P.)
– B (x; n, P): Binomial probability - the probability that an n-trial binomial experiment
results in exactly x successes, when the probability of success on an individual trial
is P.
– nCr: The number of combinations of n things, taken r at a time.

Binominal Formula
• Binomial Formula. Suppose a binomial
experiment consists of n trials and results in x
successes. If the probability of success on an
individual trial is P, then the binomial
probability is: b(x; n, P) = nCx * Px * (1 - P)n -
x

Example
• Suppose a die is tossed 5 times. What is the
probability of getting exactly 2 fours?
• Solution: This is a binomial experiment in which
the number of trials is equal to 5, the number of
successes is equal to 2, and the probability of
success on a single trial is 1/6 or about 0.167.
Therefore, the binomial probability is:
• b(2; 5, 0.167) = 5C2 * (0.167)2 * (0.833)3
b(2; 5, 0.167) = 0.161

Continuous Probability Distributions
If a random variable is a continuous variable, its
probability distribution is called a continuous
probability distribution.
For example, consider the probability density function
shown in the graph below. Suppose we wanted to
know the probability that the random variable X was
less than or equal to a. The probability that X is less
than or equal to a is equal to the area under the curve
bounded by a and minus infinity - as indicated by the
shaded area.

Normal Distribution
• The normal distribution refers to a family of
continuous probability distributions described by
the normal equation.
• Normal equation. The value of the random
variable Y is:
– Y = [ 1/σ * sqrt(2π) ] * e(x - μ)2/2σ2
• where X is a normal random variable, μ is the
mean, σ is the standard deviation, π is
approximately 3.14159, and e is approximately
2.71828.

Standard Normal Distribution
• The standard normal distribution is a special case of the normal
distribution. It is the distribution that occurs when a normal random
variable has a mean of zero and a standard deviation of one.
• The normal random variable of a standard normal distribution is called a
standard score or a z-score. Every normal random variable X can be
transformed into a z score via the following equation:
– z = (X - μ) / σ
• where X is a normal random variable, μ is the mean mean of X, and σ is
the standard deviation of X.

Standard Normal Distribution Table
• A standard normal distribution table shows a cumulative probability
associated with a particular z-score.
• Table rows show the whole number and tenths place of the z-score. Table
columns show the hundredths place.
• The cumulative probability (often from minus infinity to the z-score)
appears in the cell of the table.
• To find the cumulative probability of a z-score equal to -1.31, cross-
reference the row of the table containing -1.3 with the column containing
0.01. The table shows that the probability that a standard normal random
variable will be less than -1.31 is 0.0951; that is, P(Z < -1.31) = 0.0951.

Example
• Molly earned a score of 940 on a national achievement test. The mean test score
was 850 with a standard deviation of 100. What proportion of students had a
higher score than Molly? (Assume that test scores are normally distributed.)
(A) 0.10
(B) 0.18
(C) 0.50
(D) 0.82
(E) 0.90
• Solution
– The correct answer is B. As part of the solution to this problem, we assume that test scores
are normally distributed. In this way, we use the normal distribution as a model for
measurement. Given an assumption of normality, the solution involves three steps.

Solution
• First, we transform Molly's test score into a z-score, using the z-score
transformation equation.
z = (X - μ) / σ = (940 - 850) / 100 = 0.90
• Then, using the standard normal distribution table, we find the cumulative
probability associated with the z-score. In this case, we find P(Z < 0.90) =
0.8159.
• Therefore, the P(Z > 0.90) = 1 - P(Z < 0.90) = 1 - 0.8159 = 0.1841.
• Thus, we estimate that 18.41 percent of the students tested had a higher
score than Molly.

What is Risk?
Risk is an unexpected event.
Generally, risk is related to the expected losses which
can be caused by a risky event and to the probability
of this event.
Risk = (probability of an incident) x (impact of the incident)
While almost all risk as seen as threats, risk gives you
opportunities also.
While we manage risk, we try to increase the
likelihood of the opportunities and decrease the
likelihood of threat.

Basic Types of Risk
RISK
Business Risk Pure Risk
Threats Opportunities
Threats
INSURABLE

Behavior under uncertainty [Risk]
• Which one you buy, Brand “A” or Brand “B”?
– Brand A: Life Cycle 5 years, Proven Technology,
Prints 15 PPS
– Brand B: Life Cycle 8 Years, New Technology, Prints
20 PPS

Which category you are in ?
• Risk Aversion [If you select “A”]
– Risk aversion is the reluctance of a person to
accept a decision with an uncertain payoff rather
than another decision with a more certain but
possibly lower expected payoff.
• Risk Prone/Loving/Seeking [If you select “B”]
– A person willingness to take decision with higher
expected pay with lower possibility.

Risk Neutral
• The term risk neutral is used to describe an
individual who cares only about the expected
return of an investment, and not the risk
(variance of outcomes or the potential gains or
losses). A risk-neutral person will neither pay to
avoid risk nor actively take risks.
• Risk neutral is in between risk aversion and risk
seeking.

Risk Tolerance [Utility Function]

Risk Premium
A risk premium is the minimum amount of money
by which the expected return on a risky asset must
exceed the known return on a risk-free asset, in
order to induce an individual to hold the risky
asset rather than the risk-free asset.
So the risk premium is the minimum amount that
an individual or organization is willing to accept as
a compensation against the risk event.

Why understand Risk Management?
• Risk management is the act or practice of
dealing with risk. Since project seldom to
perform activities as per the plan, we need to
focus on unexpected event, which may affect
our project objective.
• Understanding Risk and How to manage it
through structured systematic risk
management will help us to finish with project
with in its competing demand.

Benefits of Risk Management
• Risk management will help the project from
initiating to completion through structured risk
management activities that anticipates, plans,
qualifies, quantifies and monitor and control
risks.
• Proper risk response planning helps the project
management team to expect the unexpected and
have proper strategies to deal with risk which
may arise.
– Increase the probability of positive risk
– Reduce the probability of negative risk

Approach to Risk Management
• Risk management can be accomplished
through:
Planning -> Identification -> Assessment -> Analysis
-> Mitigation

Risk Analysis
Tools for risk analysis includes:
 Decision Tree
 SWOT analysis
 EMV analysis
 Probability Impact Matrix
 Simulation modeling
 Sensitivity analysis

Risk Analysis
Decision Tree: A decision tree is a decision support
tool that uses a tree like graph to represent the
alternate decisions and their possible
consequences, including the probability of the
outcomes, resource costs, and the utility.
It is one way to display an algorithm and helps to
identify a strategy which is most likely to reach a
goal.

Risk Analysis
SWOT Analysis: SWOT Analysis is a useful technique for
understanding the Strengths and Weaknesses, and for identifying
both the Opportunities available and the Threats ahead.
Using the SWOT framework, you can start to craft a strategy that
helps you distinguish yourself from your competitors, so that you
can compete successfully in the market.
The power of the SWOT technique is that, with a little thought, it
can understand your strengths which would help you uncover
opportunities that you are well placed to exploit. And by
understanding the weaknesses of your business, you can manage
and eliminate those threats that would otherwise unaware of.

Risk Analysis
EMV Analysis: The Expected Monetary value analysis is a method of calculating
the average outcome when the future is uncertain.
To do this analysis make the best estimate of the probability of the event
occurring, and then multiply this by the amount it will cost you to set things right
if it happens. This gives you a value for the risk:
Risk Value = Probability of Event x Cost of Event
As a simple example, let's say that you've identified a risk that your rent may
increase substantially. You think that there's an 80 percent chance of this
happening within the next year, because your landlord has recently increased
rents for other businesses. If this happens, it will cost your business an extra
$500,000 over the next year.
So the risk value of the rent increase is:
0.80 (Probability of Event) x $500,000 (Cost of Event) = $400,000 (Risk Value)

Risk Analysis
Probability Impact Matrix: It is a method of
qualitative risk analysis which categorize risks by the
impact and their probability of occurrence. These
matrices provide a risk ranking in categories such as
high, medium and low which can be used to prioritize
and allocate resources to manage these risks.
This is used to classify the events as:
 Must Mitigate
 Mitigate
 Perhaps Mitigate
 Accept

Risk Analysis
Sensitivity Analysis: Sensitivity analysis is the
substitution of variables in a risk model to test the
effects of these changes. Tends to answer what if
situation.

Risk Mitigation - Threat
Risk Avoidance: This includes not performing an
activity that could carry risk. Avoidance may seem the
answer to all risks, but avoiding risks also means losing
out on the potential gain that accepting (retaining) the
risk may have allowed. Not entering a business to
avoid the risk of loss also avoids the possibility of
earning profits.
Example of Risk avoidance include:
Cancel the project, Relocate the project, Delay the
activity, Redesign the project etc.

Risk Prevention: This includes such actions to
reduce the risk factors so that the event of risk
event is prevented from occurring, or, if it does,
the severity is reduced.
Example of Risk prevention include:
Security measures, Safety inspections,
Standardization, Policy & procedures, Redesign the
project etc.

Risk Reduction: Risk reduction involves reducing the severity
of the loss or the likelihood of the loss from occurring.
Acknowledging that risks can be positive or negative,
optimizing the risk means finding a balance between
negative risk and the benefit of the operation or activity; and
between risk reduction and effort applied.
Example of Risk reduction include:
Effectively applying HSE Management, any advance
preparation in anticipation of the occurrence of the risk
event.

Risk Transfer: It is a common method of risk mitigation is to
transfer the risk to an organization that is more competent
to manage the risk or willing to assume it. The risk transfer is
usually accomplished by contract.
While deciding the risk transfer ensure the transferee is
technically qualified and financially prepared to accept the
consequences. Project claims are a good example of the
misunderstanding of who is at risk and who accepts the
consequences.
Example of Risk transfer include:
Contracting, sub-contracting, assignments etc.

Risk Hedging: It is a specialized technique for risk
transfer where the risk of price fluctuations is
assumed by a speculator through the purchasing and
selling of futures contracts.
A hedge contract consists of taking an offsetting
position in a related security, such as a futures
contract and the trading of these futures contracts are
covered by an organized commodity exchange.
Example of Risk hedging include:
Future contracts for Steel, Gold, Crude oil, Agricultural
commodities or Forex.

Insurance: It is another form of risk transfer but to the
insuring companies that indemnify parties against
specific losses in return for an agreed premium.
The contract for such transfer is called the insurance
policy and the premium is the amount to be charged
for the extend of insurance coverage or protection
sought. The principle behind the insurance involves
pooling funds from many insured entities to pay for
the losses that some may incur.
Example of Risk insurance include:
Fire insurance, health insurance, Vehicle insurance etc.

Risk Mitigation - Opportunity
Some actions for the opportunity includes:
Exploit: Take maximum advantage of the opportunity.
In order to increase the likely benefit add resources.
Share: Identify another party who is capable of
managing the risk event more effectively consider
formulating a partnership / JV.
Enhance: Consider strengthening the cause to
enhance the effect.
Accept : Continue to receive the benefit with no
actions. Status Quo

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