3
GENERAL PROBABILITY
Textbook references:
Berenson et al Basic Business Statistics 5th edition,
Chapter 4 Sections 4.1,4.2
General Probability
4
Learning Objective
Apply simple concepts of probability and
probability distributions to problems in business
decision making
5
What is probability?
• A numerical measure of the chance or likelihood that a
•
•
particular event will occur.
Strictly, it is always a number between 0 & 1.
In practice, it is often quoted as a percentage.
Probability
0
0.2
0.5
1
As a %
0%
20%
50%
100%
6
Comment
impossible event
a 1 in 5 chance
as likely as not
certain event
Probability
Probability - who needs it?
• Important in decision making as it provides a mechanism
to deal with uncertainties associated with future events.
Applications of Probability
• the ‘chance’ that sales will fall if the price rises.
• the ‘likelihood’ that a new assembly method will increase
productivity.
• the ‘odds’ that an investment will be profitable.
• inference about a population from sample data.
7
Some terminology
•
•
A random experiment is a process or course of action
that results in one of a number of possible outcomes.
A random experiment can have one or more steps.
For example, the random experiment of tossing a
balanced coin has one step.
An outcome from a random experiment cannot be
predicted with certainty, for example a Head on a single
toss of a coin or a ‘4’ on a single roll of a die.
8
Some more terminology
The sample space (S) of a random experiment is the list of
all possible outcomes of the experiment.
For example,
• on 1 toss of a coin: sample space is S={H, T}
• on 1 roll of a die: sample space is S = {1, 2, 3, 4, 5, 6}
• inspect a product: sample space is S = {Defective, OK}
• make a sales call: sample space is S = {Sale, No sale}
9
Some more terminology
•
•
•
•
An event is a collection of outcomes - it is a subset of the
sample space (simple event is a single outcome)
For example, the event “an even number occurring on one
roll of a die” consists of the outcomes {2, 4, 6}.
Usually, we use any capital letter (other than S) to denote an
event. For example, A = {2, 4, 6}.
Events can be represented in Venn diagrams, contingency
tables and tree diagrams
S
A
10
Assigning Probabilities
11
Assigning Probabilities
There are 3 ways of assigning probabilities:
1.
2.
3.
Classical /Theoretical Approach
Relative Frequency Approach
Subjective Approach
12
1.
Classical/Theoretical Approach
Probability of an event is based on prior knowledge.
Textbook calls it “a priori classical”
• Applies when outcomes of an experiment are equally likely.
• If n= the number of equally likely outcomes of an experiment,
probability 1/n is assigned to each simple event.
For example, when tossing a balanced coin,
Head and Tail are each assigned probability 1/2.
13
1.
Classical/Theoretical Approach
• To calculate the probability of an event
when outcomes are equally likely,
Number of outcomes in A
P ( A) =
Number of outcomes in S
• Example, roll a die.
A: get an even number
P(A) = 3/6
= 1/2
14
2.
•
•
•
•
Relative Frequency Approach
Used when we don’t know the probabilities in advance;
Outcomes of an experiment are not equally likely.
There is a history of repetitions of the experiment.
The probability of an outcome is estimated by the relative
frequency of the outcome occurring in the past. Based on
observed data not on prior knowledge of the process
For example,
If 600 of the last 1,000 customers
entering a particular department store
have made a purchase, the probability
that any given customer entering the
store will make a purchase is 0.6.
15
3.
Subjective Approach
• Outcomes of the experiment are not equally likely.
• There is no history of repetitions of the experiment.
• Used when classical and relative frequency
approaches are unavailable
• Assignment of probabilities to outcomes is
based on personal judgement or one’s own
evaluation of the situation.
You may, for instance, assign a 90% probability that
you will clinch a business deal tonight.
16
Contingency Tables
Example: Student grades and jobs
Research question: does a part-time job interfere
with study (as-in, affect the chance of an HD)
Conduct a survey to obtain some data on HD and
Jobs.
• Experiment: randomly select a student and record
whether Job or No Job and whether HD or no HD.
• Experiment is repeated “n” times, where “n” is the
no. of students surveyed. Organise results in a
Contingency table
18
Example: Student grades and jobs
• We organise our survey responses into a two-way
frequency (or relative frequency) table – a Contingency
table.
• The row totals are the numbers of students with HD / no
HD.
• The column totals are the numbers working / not
working.
• The total number of students surveyed is n = 80.
Frequency
Job
No Job
Total
HD
8
12
20
Not HD
40
20
60
Total
48
32
80
19
Example: Student grades and jobs
• Convert to relative frequency, and we can read the
probabilities of certain events directly from the table:
Relative frequency
Job
No Job
Total
HD
0.1
0.15
0.25
Not HD
0.5
0.25
0.75
Total
0.6
0.4
1
• Probabilities in the green cells are marginal probabilities.
• The values in the red cells are joint probabilities.
• Eg: the probability of a randomly selected student getting an
HD is 0.25
• Eg: the probability that a randomly selected student got HD
and had a job is 0.1
20
Example: Student grades and jobs
• Notation for the events:
Job (J)
HD (H)
เดฅ
Not HD (๐ป)
Total
No Job
P( H and J )
P( H and J )
J
P( H and J )
P( H and J )
P( J )
P(J )
Total
P(H)
P( H )
1
Marginal Probabilities
Joint Probabilities
21
Independent Events
• A and B are said to be independent events if the probability
of event A occurring does not depend on the occurrence of
event B (and vice versa).
• i.e., A is unaffected by B, and vice versa.
• This would mean
•
• and
P(A | B) = P(A)
P(B | A) = P(B).
• (Either both are true or neither is true.)
• i.e., the conditional probabilities of independent events are
equal to their unconditional (marginal) probabilities.
23
Checking for independence
•
Three ways to check independence
1. Determine whether P(A and B) = P(A) × P(B)
2. Determine whether P(A | B) = P(A)
3. Determine whether P(B | A) = P(B)
If any one is true, the others will be.
24
Independent events
• Are good grades (an HD) and part-time work in some way
dependent on each other? (Research question: does a
part-time job interfere with study (as-in, affect the
chance of an HD)
Compare the two probabilities:
P(HโJ)(this is read as Probability of H given J) and P(H)
Or
Compare
P(J โ H) and P(J)
Job
No Job
Total
HD
0.1
0.15
0.25
Not HD
0.5
0.25
0.75
Total
0.6
0.4
1
25
Conditional Probability
• If two events A and B are related, then knowledge that event B has
occurred can be used to “update” the probability of event A.
• P(A | B) is read as probability of A given B and can be calculated as
๐ ๐ด∩๐ต
๐(๐ต)
Job
No Job
Total
HD
0.1
0.15
0.25
Not HD
0.5
0.25
0.75
Total
0.6
0.4
1
• e.g. P(HโJ)= Proportion from J (Job) column who have HD
=
๐(๐ป∩๐ฝ)
๐(๐ฝ)
=
0.1
0.6
= 0.17
• P(H) = 0.25
Since P(HโJ) ≠ P(H)
the events H and J are not independent.
26
Mutually exclusive vs Independent
“Mutually exclusive” and “independent” are very different.
• A and B are mutually exclusive if A occurs, B cannot
(and if B occurs, A cannot)
๏ P(A | B) = P(B | A) = 0
• In other words, P(A∩B) = 0
• If A and B independent:
•
๏ P(A | B) = P(A)
• and
P(B | A) = P(B)
• So unless P(A) = P(B) = 0, A and B can’t be both
mutually exclusive and independent.
In fact, mutual exclusion is a rather extreme form of
dependence: if A occurs then B cannot!
27
Venn Diagrams and Probability
Trees
28
Notation of events
A
B
เดค
Not (๐ต)
Total
P(A∩B)
เดค
P(A∩๐ต)
P(A )
Not A (๐ด)าง
าง
P(๐ด∩B)
าง ๐ต)
เดค
P(๐ด∩
P (๐ด)าง
Total
P(B)
เดค
P(๐ต)
1
Joint Probabilities: for example for joint events:
Marginal
Probabilities: for
example for events:
29
Notation of events
A
B
เดค
Not (๐ต)
Total
P(A∩B)
เดค
P(A∩๐ต)
Not A (๐ด)าง
าง
P(๐ด∩B)
าง ๐ต)
เดค
P(๐ด∩
P (๐ด)าง
P(A )
Total
P(B)
เดค
P(๐ต)
1
P(AUB) = also known as A or B
S
เดค + P(๐ดาง ∩ B)
= P(A ∩ B) + P(A ∩ ๐ต)
= P(A) + P(B) – P(A ∩ B)
“A or B” = Union of two events
[The event that at least one of A and B
occurs] AUB
30
Probability Trees
• A probability tree is a graphical representation of a probability
problem resulting from a multilevel experiment.
• Helps in building a sample space and calculating probabilities
• Building blocks:
– random (chance) nodes – denotes a stage where various outcomes
are possible
– branches – each branch represents one possible outcome of a chance
experiment (trial)
– Branch probabilities for branches from a node are conditional
probabilities, given that the node has been reached, of outcomes
beyond that node.
– terminal nodes: possible final outcome
• Final probabilities of events on the terminal nodes are calculated as
a product of branch probabilities on the path from the initial node to
the terminal node
• All this is easier in an example.
31
Hypothetical example from Lecture Videos
This is a hypothetical example:
Suppose someone claims:
– If a student has a job, then their probability of getting an HD is one
in 20, then
– P(HโJ) = 1/20 = 0.05
– Therefore, given that a student has job, there is a 5% chance of
obtaining an HD.
– If a student does not have a job, then their probability of getting an
HD is one in 5
– P(Hโ๐ฝ)าง = 1/5 = 0.20
– 90% of students have jobs
32
Probability Trees
Calculate joint probabilities at the end points:
Start at the origin, follow path along the branches multiplying
J and
probabilities on the way.
H
J
0.9
H
0.05
0.95
Origin
H
P( J and H )
= P( J ) ๏ด P( H | J )
= 0.9 ๏ด 0.05
J and H
J and H
J
0.1
H
0.2
H
0.8
J and H
33
Probability Trees
Find the following probabilities:
P(J∩H) = 0.045
เดฅ = 0.855 + 0.08 = 0.935
P(๐ป)
เดฅ = 0.0855/0.935 = 0.914
P(Jโ๐ป)
P ( J and H )
= 0.9 * 0.05 = 0.045
P ( J and H )
0.9 * 0.95 = 0.855
Origin
P ( J and H )
= 0.1* 0.2 = 0.02
P ( J and H )
= 0.1* 0.8 = 0.08
34
Contingency Tables vs Trees
When do you use tables and when do you use trees?
– Joint and marginal probabilities can be used to
build up a table
– Conditional probabilities can be used to build up a
tree
So it depends on which probabilities you know.
35
Fundamentals of Probability
• Probability is the link between descriptive statistics
and inferential statistics. It is a numerical value that
represents the chance, likelihood, possibility that
an event will occur (always between 0 and 1)
• Event – Each possible outcome of a variable
36
Contingency Table Example
37
Contingency Tables
Using Elecmart.xlsx, create a pivot/contingency table with
1. Gender in the Row Labels, Region in the Column Labels
2. Gender in the ‘∑ Values’ window.
Using the Pivot function in Excel, we end up with the following
contingency table
Count of Gender Region
Gender
MidWest NorthEast South
Female
43
62
63
Male
28
53
30
Grand Total
71
115
93
38
West
66
55
121
Grand Total
234
166
400
Marginal Probability
Count of Gender
Gender
Region
MidWest NorthEast
South
West
Grand Total
Female
43
62
63
66
234
Male
28
53
30
55
166
Grand Total
71
115
93
121
Lets use the count contingency table to review probabilities:
400
The probability that a randomly selected sales person is female:
๐๐๐
P(Female) =
= 0.585
๐๐๐
This is called a marginal probability because the relevant number for it occurs
in a column at the edge of the contingency table.
39
Conditional Probability
Count of Gender
Region
Gender
MidWest
NorthEast
South
West
Grand Total
Female
43
62
63
66
234
Male
28
53
30
55
166
Grand Total
71
115
93
121
400
The probability that a randomly selected sales person is female, given that the person is in
the Midwest:
๐๐
P(FemaleโMidWest) = = 0.606
๐๐
The statement “Given that the person is in the Midwest” tells you to restrict attention to the Midwest
column. So now we’re only looking at the 71 people in the Midwest, and determining the proportion
of those that are female.
If two events A and B are related, then knowledge that event B has occurred can be used to
“update” the probability of event A.
The notation P(A | B) is used to represent the probability of A occurring, given that B has already
occurred. This is a conditional probability
“A | B” is read “A given B”.
40
Further analysis of Pivot Tables
Alternatively, we can use the Pivot function in Excel and present the table in
three different formats:
1. % of Grand Total
The % of Grand Total table can be constructed from the table of raw
counts by dividing all the values in the table by the Grand Total value of
observations
2. % of Column
The % of Column can be constructed from the table of raw counts by
dividing the value inside the cells by that Column’s total.
3. % of Row
The % of Row table can be constructed from the table of raw counts by
dividing the value inside the cells by that Row’s total.
41
Different formats of Pivot Tables
% Grand Total
% Column Total
% Row Total
Please note that values in probability tables typically display probabilities as a proportion and not
percentages. Pivot tables, on the other hand display values as percentages.
42
% of Column Total (Conditional Percentages)
Conditional probability
The circled percentage value is conditional on the column.
P(FemaleโMidwest) = 0.606
Given that the randomly selected customer comes from MidWest, there is a 60.6%
chance that the customer is a female.
OR
Approximately 60.6% of the customers from the MidWest are Females.
43
% of Column Total (Conditional Percentages)
Marginal Probability:
The circled percentage value is can be interpreted as below:
• P(Female) = 0.585
There is a 58.5% chance that a randomly selected customer is Female.
OR
Approximately 58.5% of the customers are Females.
44
% of Row Total (Conditional Percentages)
The above percentage values is conditional on the row, i.e. the person comes from a
particular row and not any other row.
“A randomly selected person from the particular row” or “given that the person is from
the particular row”
P(NorthEastโMale) = 0.319
Given that the randomly selected customer is a male, there is a 31.9% chance that he is
from the NorthEast.
OR
Approximately 31.9% of the Male customers are from the NorthEast.
45
% of Grand Total (Marginal & Joint Percentages)
MARGINAL Probabilities: (shaded in green), we focus on only one of the
characteristics of interest. Example:
P(MidWest)= 0.178
Approximately 17.8% of all customers are from the MidWest
P(Females)= 0.585
Approximately 58.5% of all customers are Females
JOINT Probabilities: (shaded in yellow), we focus on more than one characteristic of
interest. Example:
P(MidWest ∩ Male) = 0.07
Approximately 7% of the customers are from the MidWest AND Male
P(West ∩ Female) = 0.165
Approximately 16.5% of the customers are from the West AND Female
46
% of Grand Total (Conditional Probability)
Conditional probabilities can also be obtained from this table of %
Grand Total. However, we will need to apply a formula. We know from
before that
P(FemaleโMidWest) = 0.606
Probability formula: P(AโB)=
๐(๐ด∩๐ต)
๐(๐ต)
Therefore P(Female โ Midwest) =
๐(๐น๐๐๐๐๐ ∩ ๐๐๐๐ค๐๐ ๐ก)
๐(๐๐๐๐ค๐๐ ๐ก)
47
=
10.8
17.8
= ๐. ๐๐๐
Worked Examples
48
Worked Examples
Question 1: Events A and B are independent if and only if
A.
B.
C.
D.
P(A|B) = P(A) P(B|A)
P(A|B) = P(A)
P(B|A) = P(A) P(B|A)
None of the above
49
Question 2: Refer to the table of probabilities below which
shows municipal waste collected. Given the waste was
glass what is the probability that it was not recycled?
Municipal Waste Collected (millions of tons)
Paper Aluminium Glass Plastic Other Total
Recycled
26.5
1.1
3
0.7
13.7
45
Not recycled 51.3
1.7
10.7
19.3
78.7 161.7
Total
77.8
2.8
13.7
20
92.4 206.7
A. 78.1%
B. 21.9%
C. 5.2%
D. None of the above
50
Question 3: The following pivot table was produced from
Elecmart shopping company. Which of the following is the
correct interpretation of the value 46.09%
Count of Gender
Region
MidWest
NorthEast
South
West
Grand Total
Gender
Female
60.56%
53.91%
67.74%
54.55%
58.50%
Male
39.44%
46.09%
32.26%
45.45%
41.50%
Grand Total
100.00%
100.00%
100.00%
100.00%
100.00%
A. 46.09% of shoppers are Male and from the NorthEast region
B. 46.09% of Male shoppers are from the NorthEast region
C. 46.09% of shoppers from the NorthEast region are Male
51
Worked Examples
52
Contingency Tables
Using Elecmart.xlsx, create a pivot table with Gender in the Row
Labels, Region in the Column Labels, Gender in the ‘∑ Values’ window.
Count of Gender Region
Gender
MidWest NorthEast South
Female
43
62
63
Male
28
53
30
Grand Total
71
115
93
53
West
66
55
121
Grand Total
234
166
400
Pivot Tables and Probability
Count of Gender
Region
Gender
MidWest NorthEast South
West
Female
43
62
63
66
Male
28
53
30
55
Grand Total
71
115
93
121
Grand Total
234
166
400
MARGINAL PROBABILITY:
The probability that a randomly selected customer is female:
P(Female)
234
=
400
= 0.585
This is called a marginal probability because the relevant number for it
occurs in a column at the edge of the contingency table.
54
Pivot Tables and Probability
Count of Gender
Region
Gender
MidWest NorthEast South
West
Female
43
62
63
66
Male
28
53
30
55
Grand Total
71
115
93
121
Grand Total
234
166
400
CONDITIONAL PROBABILITY:
The probability that a randomly selected customer is female, given that the
person is in the Midwest:
P(FemaleโMidWest)
๐(๐น๐๐๐๐๐ ∩ ๐๐๐๐๐๐ ๐ก)
=
๐(๐๐๐๐๐๐ ๐ก)
43
=
= 0.606
71
The notation P(A | B) is used to represent the probability of A occurring, given that B
has already occurred. This is a conditional probability
55
Further analysis of Pivot Tables
Alternatively, we can use the Pivot function in Excel and present the table in
three different formats:
1. % of Grand Total
The % of Grand Total table can be constructed from the table of raw
counts by dividing all the values in the table by the Grand Total value of
observations
2. % of Column
The % of Column can be constructed from the table of raw counts by
dividing the value inside the cells by that Column’s total.
3. % of Row
The % of Row table can be constructed from the table of raw counts by
dividing the value inside the cells by that Row’s total.
56
Different formats of Pivot Tables
% Grand Total
% Column Total
% Row Total
Please note that values in probability tables typically display probabilities as a proportion and not
percentages. Pivot tables, on the other hand display values as percentages.
57
% of Grand Total (Marginal & Joint Percentages)
MARGINAL Probabilities:
P(MidWest)= 0.178
Approximately 17.8% of all customers are from the MidWest
P(Females)= 0.585
Approximately 58.5% of all customers are Females
OR There is a 58.5% chance that a randomly selected customer is Female.
JOINT Probabilitieswe focus on more than one characteristic of interest
P(MidWest ∩ Male) = 0.07
Approximately 7% of the customers are from the MidWest AND Male
P(West ∩ Female) = 0.165
Approximately 16.5% of the customers are from the West AND Female
58
% of Grand Total (Conditional Probability)
Conditional probabilities can also be obtained from this table of % Grand
Total.
However, we will need to apply a formula.
Probability formula: P(AโB)=
๐(๐ด∩๐ต)
๐(๐ต)
P(FemaleโMidWest)
=
๐(๐น๐๐๐๐๐ ∩ ๐๐๐๐๐๐ ๐ก)
๐(๐๐๐๐๐๐ ๐ก)
=
10.8
17.8
= 0.6067
59
% of Column Total (Conditional Percentages)
Conditional probability
Is the circled percentage value conditional on the column/row? What is the probability
statement here?
P(FemaleโMidWest) = 0.606
Interpretation:
Given that the randomly selected customer comes from MidWest, there is a 60.6%
chance that the customer is a female.
OR
Approximately 60.6% of the customers from the MidWest are Females.
60
% of Row Total (Conditional Percentages)
The above percentage values are conditional on specific rows, i.e. the person comes
from a particular row and not any other row.
EXAMPLE:
Given that the randomly selected customer is a male, what is the probability that he is
from the NorthEast?
P(NorthEastโMale) = 0.319
Approximately 31.9% of the Male customers are from the NorthEast.
61
Continuous Probability Distributions
Textbook references:
Berenson et al Basic Business Statistics 5th
edition, Chapter 6 Sections 6.1-6.2
Types of Data/Variables
Nominal
Categorical
(Qualitative)
Ordinal
Data/Variables
Discrete (Counts)
Numerical
(Quantitative)
Continuous
(Measurements)
Discrete Random Variables – from last week
For example, toss a coin 3 times.
X: number of heads in 3 tosses
X = 0, 1, 2 or 3.
We can calculate P[X = a particular value]
Continuous Random Variables
A continuous random variable
• has an “uncountably infinite” number of values
• can assume any value in the interval
A continuous random variable
• For example survey women’s heights.
X: height of randomly selected woman
Height is an example of a continuous random variable.
Continuous Random Variables
X may take any value within a feasible interval
However, since an interval contains an infinite number of values
It IS sensible to consider
• P[X lies within a range]
• eg, P[161.5 < X < 162.5]
It is NOT useful to consider
• P[X = a particular value]
• eg, P[X = 162.0000 ... cm] = 0
Continuous Distributions
The Probability Density Function
Survey of women’s heights. Organise the data into classes of width
5cm, and plot the corresponding relative frequency.
180
175
Height (cm)
170
165
160
155
150
Relative
Frequency
Continuous Distributions –The Probability Density Function
Height (cm)
18
17
170
165
160
155
150
5
0
f (x)
Now reduce class width to 2.5cm
Continuous Distributions –The Probability Density Function
8
Then, even further, to 0.5 cm
Probability density function
With the reduction in class width, a smooth curve f(x) provides an
increasingly better approximation to the shape of the histogram.
f(x) is called the probability density function (pdf).
f(x)
x
Checklist of Topics
This video– continuous random variable and a
continuous distribution.
Normal distribution
Standard normal distribution, standardization,
reading standard normal tables
Examples on how to read the standard normal tables
Probability Density Function (pdf)
f(x): probability density function.
Measures how densely probability is concentrated in a particular range of X
values
Probability is ‘spread’ over the whole range of possible X values
in some places, thinly
(low density).
in some places, thickly (high density).
The intervals of X values which are more likely to occur are shown by the regions
of the graph where the probability density function f(x) is larger.
The probability that random variable X lies within a particular interval can be
obtained as the corresponding area under the curve.
Continuous Distributions & Probability Density Functions
A continuous distribution has a continuum of possible values,
[Unlike a discrete distribution with its set of discrete values.]
• The total probability of 1 is spread over this continuum.
[Instead of assigning a probability to each individual value.]
Probability Distributions
There are many continuous probability distributions, e.g.,
f(x)
f(x)
Normal
Uniform
x
f(x)
We will focus on the
normal distribution.
x
f(x)
Exponential
Chi-square
x
x
The Normal Distribution
Most important probability distribution in Statistics because
•
many data sets have a histogram that is well described by
the normal distribution,
e.g. IQ scores, heights, weights etc.
•
it underlies much of statistical inference.
Properties of the Normal distribution
• Symmetric, unimodal.
โข mean = median.
โข the modal class is in the region of the mean(median).
•
•
The distribution is completely defined by two parameters: mean μ and
variance σ2.
Often denoted
X~N(μ,๐2)
Read: X is normally distributed
with mean μ and variance σ2.
There is an entire ‘family’ of normal distributions, each with its own mean μ
and standard deviation σ.
The Normal Curve
Mean
Standard deviation
Probability from Probability Density Function
Probability that X falls in some range = the proportion of the
corresponding area under the curve.
f(x)
i.e., P( a < X < b )
= area under curve
between a and b.
Total Area = 1
a
b
f(x) ≠ p(X = x).
x
The Normal Distribution- finding probabilities
We are interested in a range of values on the horizontal axis and the
corresponding area between the curve and the horizontal axis.
NB: the probability that X equals an individual value is 0.
Standardizing: Z-Values
There are many normal distributions, one for each pair μ and σ.
Any random variable (X) can be standardised or converted to another random
variable Z with the same distributional shape, but with μ = 0 and σ2 = 1.
(Hence, σ = 1, too.)
The standard normal distribution or Z distribution.
Z~N(0,1)
Why standardize?
To compare on the same scale variables which have different means and/or
standard deviations. We can use Excel or tables (but will have to standardise).
Standardizing X: the “Z” transform
To standardise random variable X, which has E(X) = μ and Var(X) = σ², and
get standardised random variable Z:
• subtract μ and divide by σ.
Z has E(Z) = 0, Var(Z) = 12 and StDev(Z) = 1.
To do the “reverse transformation”
• return from standardised variable Z to X:
Standard Normal Transformation
• Standardising transforms X ∼ N(μ, σ2) to the standard normal (Z)
distribution preserves areas Z~N(0,1)
μ = E(X)
σ² =
Var(X)
Z ∼ N(0,1)
X ∼ N(μ,σ2)
μ =0
σ2 = 1
0
z values :
above the mean (0) are positive & below the mean (0) are negative.
Standard Normal Transformation
To find the probability between two X values, find the difference of two
cumulative probabilities.
P(a ≤ X ≤ b)
= P(X ≤ b) – P(X ≤ a)
Standardised Normal Distribution
If X is distributed normally with mean of 100 and standard deviation of 50, the
Z value for X = 200 is:
๐=
๐=
๐−๐
๐
200 −100
50
29
= 2.0
This Z-value reflects the fact that X = 200 is 2 standard deviations above the
mean of 100.
•
•
The shape of the distribution is the same, but the scale is different.
We can express any problem in the original units (X) or in standardised units (Z).
Table 1a: Standard Normal Distribution
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
–0.1
–0.2
–0.3
–0.4
–0.5
–0.6
–0.7
–0.8
–0.9
–1.0
–1.1
–1.2
–1.3
–1.4
–1.5
–1.6
–1.7
–1.8
–1.9
–2.0
–2.1
–2.2
–2.3
–2.4
–2.5
–2.6
–2.7
–2.8
–2.9
–3.0
–3.1
–3.2
–3.3
–3.4
0.5000
0.4602
0.4207
0.3821
0.3446
0.3085
0.2743
0.2420
0.2119
0.1841
0.1587
0.1357
0.1151
0.0968
0.0808
0.0668
0.0548
0.0446
0.0359
0.0287
0.0228
0.0179
0.0139
0.0107
0.0082
0.0062
0.0047
0.0035
0.0026
0.0019
0.0013
0.0010
0.0007
0.0005
0.0003
0.4960
0.4562
0.4168
0.3783
0.3409
0.3050
0.2709
0.2389
0.2090
0.1814
0.1562
0.1335
0.1131
0.0951
0.0793
0.0655
0.0537
0.0436
0.0351
0.0281
0.0222
0.0174
0.0136
0.0104
0.0080
0.0060
0.0045
0.0034
0.0025
0.0018
0.0013
0.0009
0.0007
0.0005
0.0003
0.4920
0.4522
0.4129
0.3745
0.3372
0.3015
0.2676
0.2358
0.2061
0.1788
0.1539
0.1314
0.1112
0.0934
0.0778
0.0643
0.0526
0.0427
0.0344
0.0274
0.0217
0.0170
0.0132
0.0102
0.0078
0.0059
0.0044
0.0033
0.0024
0.0018
0.0013
0.0009
0.0006
0.0005
0.0003
0.4880
0.4483
0.4090
0.3707
0.3336
0.2981
0.2643
0.2327
0.2033
0.1762
0.1515
0.1292
0.1093
0.0918
0.0764
0.0630
0.0516
0.0418
0.0336
0.0268
0.0212
0.0166
0.0129
0.0099
0.0075
0.0057
0.0043
0.0032
0.0023
0.0017
0.0012
0.0009
0.0006
0.0004
0.0003
0.4840
0.4443
0.4052
0.3669
0.3300
0.2946
0.2611
0.2296
0.2005
0.1736
0.1492
0.1271
0.1075
0.0901
0.0749
0.0618
0.0505
0.0409
0.0329
0.0262
0.0207
0.0162
0.0125
0.0096
0.0073
0.0055
0.0041
0.0031
0.0023
0.0016
0.0012
0.0008
0.0006
0.0004
0.0003
0.4801
0.4404
0.4013
0.3632
0.3264
0.2912
0.2578
0.2266
0.1977
0.1711
0.1469
0.1251
0.1056
0.0885
0.0735
0.0606
0.0495
0.0401
0.0322
0.0256
0.0202
0.0158
0.0122
0.0094
0.0071
0.0054
0.0040
0.0030
0.0022
0.0016
0.0011
0.0008
0.0006
0.0004
0.0003
0.4761
0.4364
0.3974
0.3594
0.3228
0.2877
0.2546
0.2236
0.1949
0.1685
0.1446
0.1230
0.1038
0.0869
0.0721
0.0594
0.0485
0.0392
0.0314
0.0250
0.0197
0.0154
0.0119
0.0091
0.0069
0.0052
0.0039
0.0029
0.0021
0.0015
0.0011
0.0008
0.0006
0.0004
0.0003
0.4721
0.4325
0.3936
0.3557
0.3192
0.2843
0.2514
0.2206
0.1922
0.1660
0.1423
0.1210
0.1020
0.0853
0.0708
0.0582
0.0475
0.0384
0.0307
0.0244
0.0192
0.0150
0.0116
0.0089
0.0068
0.0051
0.0038
0.0028
0.0021
0.0015
0.0011
0.0008
0.0005
0.0004
0.0003
0.4681
0.4286
0.3897
0.3520
0.3156
0.2810
0.2483
0.2177
0.1894
0.1635
0.1401
0.1190
0.1003
0.0838
0.0694
0.0571
0.0465
0.0375
0.0301
0.0239
0.0188
0.0146
0.0113
0.0087
0.0066
0.0049
0.0037
0.0027
0.0020
0.0014
0.0010
0.0007
0.0005
0.0004
0.0003
0.4641
0.4247
0.3859
0.3483
0.3121
0.2776
0.2451
0.2148
0.1867
0.1611
0.1379
0.1170
0.0985
0.0823
0.0681
0.0559
0.0455
0.0367
0.0294
0.0233
0.0183
0.0143
0.0110
0.0084
0.0064
0.0048
0.0036
0.0026
0.0019
0.0014
0.0010
0.0007
0.0005
0.0003
0.0002
Table 1b: Standard Normal Distribution (cont’d)
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
31
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
Table 1b: Standard Normal Distribution (cont’d)
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
32
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
Find P(Z < 2.0)
Find P(Z < 1.58)
Find P(Z > 1.58)
Table 1b: Standard Normal Distribution (cont’d)
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
33
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
P(0 < Z < 1.96)
Table 1a: Standard Normal Distribution
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
–0.1
–0.2
–0.3
–0.4
–0.5
–0.6
–0.7
–0.8
–0.9
–1.0
–1.1
–1.2
–1.3
–1.4
–1.5
–1.6
–1.7
–1.8
–1.9
–2.0
–2.1
–2.2
–2.3
–2.4
–2.5
–2.6
–2.7
–2.8
–2.9
–3.0
–3.1
–3.2
–3.3
–3.4
0.5000
0.4602
0.4207
0.3821
0.3446
0.3085
0.2743
0.2420
0.2119
0.1841
0.1587
0.1357
0.1151
0.0968
0.0808
0.0668
0.0548
0.0446
0.0359
0.0287
0.0228
0.0179
0.0139
0.0107
0.0082
0.0062
0.0047
0.0035
0.0026
0.0019
0.0013
0.0010
0.0007
0.0005
0.0003
0.4960
0.4562
0.4168
0.3783
0.3409
0.3050
0.2709
0.2389
0.2090
0.1814
0.1562
0.1335
0.1131
0.0951
0.0793
0.0655
0.0537
0.0436
0.0351
0.0281
0.0222
0.0174
0.0136
0.0104
0.0080
0.0060
0.0045
0.0034
0.0025
0.0018
0.0013
0.0009
0.0007
0.0005
0.0003
0.4920
0.4522
0.4129
0.3745
0.3372
0.3015
0.2676
0.2358
0.2061
0.1788
0.1539
0.1314
0.1112
0.0934
0.0778
0.0643
0.0526
0.0427
0.0344
0.0274
0.0217
0.0170
0.0132
0.0102
0.0078
0.0059
0.0044
0.0033
0.0024
0.0018
0.0013
0.0009
0.0006
0.0005
0.0003
0.4880
0.4483
0.4090
0.3707
0.3336
0.2981
0.2643
0.2327
0.2033
0.1762
0.1515
0.1292
0.1093
0.0918
0.0764
0.0630
0.0516
0.0418
0.0336
0.0268
0.0212
0.0166
0.0129
0.0099
0.0075
0.0057
0.0043
0.0032
0.0023
0.0017
0.0012
0.0009
0.0006
0.0004
0.0003
0.4840
0.4443
0.4052
0.3669
0.3300
0.2946
0.2611
0.2296
0.2005
0.1736
0.1492
0.1271
0.1075
0.0901
0.0749
0.0618
0.0505
0.0409
0.0329
0.0262
0.0207
0.0162
0.0125
0.0096
0.0073
0.0055
0.0041
0.0031
0.0023
0.0016
0.0012
0.0008
0.0006
0.0004
0.0003
0.4801
0.4404
0.4013
0.3632
0.3264
0.2912
0.2578
0.2266
0.1977
0.1711
0.1469
0.1251
0.1056
0.0885
0.0735
0.0606
0.0495
0.0401
0.0322
0.0256
0.0202
0.0158
0.0122
0.0094
0.0071
0.0054
0.0040
0.0030
0.0022
0.0016
0.0011
0.0008
0.0006
0.0004
0.0003
0.4761
0.4364
0.3974
0.3594
0.3228
0.2877
0.2546
0.2236
0.1949
0.1685
0.1446
0.1230
0.1038
0.0869
0.0721
0.0594
0.0485
0.0392
0.0314
0.0250
0.0197
0.0154
0.0119
0.0091
0.0069
0.0052
0.0039
0.0029
0.0021
0.0015
0.0011
0.0008
0.0006
0.0004
0.0003
0.4721
0.4325
0.3936
0.3557
0.3192
0.2843
0.2514
0.2206
0.1922
0.1660
0.1423
0.1210
0.1020
0.0853
0.0708
0.0582
0.0475
0.0384
0.0307
0.0244
0.0192
0.0150
0.0116
0.0089
0.0068
0.0051
0.0038
0.0028
0.0021
0.0015
0.0011
0.0008
0.0005
0.0004
0.0003
0.4681
0.4286
0.3897
0.3520
0.3156
0.2810
0.2483
0.2177
0.1894
0.1635
0.1401
0.1190
0.1003
0.0838
0.0694
0.0571
0.0465
0.0375
0.0301
0.0239
0.0188
0.0146
0.0113
0.0087
0.0066
0.0049
0.0037
0.0027
0.0020
0.0014
0.0010
0.0007
0.0005
0.0004
0.0003
0.4641
0.4247
0.3859
0.3483
0.3121
0.2776
0.2451
0.2148
0.1867
0.1611
0.1379
0.1170
0.0985
0.0823
0.0681
0.0559
0.0455
0.0367
0.0294
0.0233
0.0183
0.0143
0.0110
0.0084
0.0064
0.0048
0.0036
0.0026
0.0019
0.0014
0.0010
0.0007
0.0005
0.0003
0.0002
Find P(Z < -1.58)
