Download Standard Normal in Excel

Statistics Quick Overview
Copyright by Michael S. Watson, 2012
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
2
Given That 1/3 of the Bag is Of Each Type,
What is the Probability Of……

Getting 1:
33.3%

Getting 2: 33.3% x 33.3% = 11.1%

Getting 3: 33.3% x 33.3% x 33.3% = 3.7%

Getting 4: 1.2%

Getting 5: 0.4%

Getting 6: 0.1%
When did you get
suspicious of my
claim?
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
3
You Formed a Hypothesis….
Proportion of Hersey’s is not 33%
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
4
Hypothesis Testing
H0- Null Hypothesis (everything else)
Ha- Alternative Hypothesis (what you want to prove)
H-0
H-a
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
5
Hypothesis Testing- Candy Example
H0- Null Hypothesis (Is 33%)
Ha- Alternative Hypothesis (Hershey’s Not 33%)
H-0
H-a
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
6
Hypothesis Testing
Reject
Is 33%
H0
Not 33%
Ha
1
Not Reject
2
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
7
Hypothesis Testing
What kind of evidence do we need to Reject the Null?
Reject
Is 33%
H0
Not 33%
Ha
1
Not Reject
2
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
8
Hypothesis Testing
H0- Not Guilty
Ha- Guilty
Why this way? “Innocent until proven guilty”
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
9
Hypothesis Testing
Does this mean Innocent?
Reject
H0
1
Not Reject
2
Not Guilty
Guilty
Ha
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
10
Hypothesis Testing- types of Errors
Trial Finds Defendant …
Defendant Really is….
Guilty
Innocent
Guilty
Not Guilty
What do we do to avoid these errors?
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
11
Basic Statistics– Mean and Standard Deviation
Data Point
Packaging
Example
Tire
Failure
Tire Failure Miles
1
31,603
2
32,586
3
34,394
4
38,954
5
42,503
6
31,754
7
29,459
8
36,157
9
38,559
10
36,478
11
45,809
12
30,981
13
39,355
14
37,406
15
29,545
16
35,975
17
34,867
18
38,878
19
26,031
20
43,564
21
32,852
22
35,589
23
41,458
24
31,989
25
34,576
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
12
Important Attributes

Mean: The average or ‘expected value’ of a distribution.


Variance: A measure of dispersion and volatility.


Denoted by µ (The Greek letter mu)
Denoted by σ2 (Sigma Squared)
Standard deviation: A related measure of dispersion computed as the
square root of the variance.

Denoted by σ (The Greek letter sigma)
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
13
Which Process is More Variable?

Case 1



Case 2




Average: 10,000
Standard Deviation: 3,000
Coefficient of Variation (CV)


Average: 5,000
Standard Deviation: 2,000
Case 3


Average: 50
Standard Deviation: 25
CV = (Standard Deviation) / (Average)
The CV allows you to compare relative variations



Case 1: 50%
Case 2: 40%
Case 3: 30%
Let’s take a look at spreadsheet
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
14
What-If With Packing Variability
Original Case
Less Variability
More Variability
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
15
Strategic Importance of Understanding
Variability (From GE)

1998 GE Letter to Shareholders



Six Sigma program is uncovering “hidden factory” after “hidden factory”
Now realize that “Variability is evil in any customer-touching process.”
2001 Book “Jack”
−
“We got away from averages and focused on variation by tightening what we call ‘span’”
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
16
Probability Distributions

Many things a firm deals with involves quantities that
fluctuate






Sales
Returned items
Items bought by a customer
Time spent by sales clerk with customer
Machine failures
Etc…

One way to summarize these fluctuations is with a
probability distribution

Although “demand” or some variable is random, it still
follows a “Distribution”

A Distribution is a mathematical equation that defines the shape of the
curve that the distribution follows
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
17
Probability Distributions

A probability distribution allows us to compute the chance
that a variable lies within a given range

Examples:




Probability sales are between 10,000 and 50,000
Probability that a customer buys 2 items
Probability that a machine will break down and probability that it will take
more than 2 hours to fix
Probability that lead time will be more than 2 weeks
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
18
Probability Distributions: Types

Probability distributions can be


Discrete: only taking on certain values
Continuous: taking on any value within a range or set of ranges
Examples:

The number of items that a customer buys follows a
discrete probability distribution

The daily sales at a store follows a continuous
probability distribution
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
19
Continuous Distributions
This area represents the
probability that Sales will be
between 20,000 and 30,000
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
20
Normal Distribution

One of the most common distributions in statistics is the
normal distribution

There are actually innumerable normal distributions each
characterized by two parameters:




The mean
The standard deviation
The standard normal has a mean of zero and a standard deviation of one
Why the Normal?




Many random variables follow this pattern
When you are doing many samples from unknown distributions, the output
of the samples follow the Normal distribution
When you are dealing with forecast error, it only matters that the forecast
error is normally distributed, not the underlying distribution
Normal is mathematically less complex than others
−
Easily expressed in terms of the mean and standard deviation
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
21
The Normal Distribution: A Bell Curve
The area under this curve
(and all continuous
distributions) is equal to one
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
22
Normal Distribution: Symmetric
So does this half
This half has
an area = 0.50
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
23
Three Normal Distributions
µ=0
µ=1
σ=1
σ=1
µ=0
σ=2
-4
-3
-2
-1
0
1
2
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
3
4
24
Shapes of different Normal curves
Different Normal Curves
M = 50, Std = 4
M = 50, Std = 8
M = 50, Std = 16
0.12
Probability
0.10
0.08
0.06
0.04
0.02
0.00
0
10
20
30
40
50
60
70
80
90
100
Demand
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
25
Normal Distribution Over Time
Demand over Time
M = 50, Std = 4
Actual Demand
100
80
60
40
20
93
97
93
97
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
13
9
5
1
0
Time Period
Demand over Time
M = 50, Std = 16
80
60
40
20
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
13
9
5
0
1
Actual Demand
100
Time Period
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
26
Relationship between demand variability and
service level (1)

Assume that demand for a week has an equal chance of
being any number between 0 and 100.



Is this a Normal distribution?
Average is 50, standard deviation is approximately 30
How much inventory do you need at the beginning of the
week to ensure that you will meet demand 95% of time,
on average
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
27
Relationship between demand variability and
service level (2)
Assume same average demand, with less variation
Normal Distribution (Mean = 50, Std Dev = 8)
Now you
need to
hold
only 63 for
95%
service
level
0.06
0.05
Probability

0.04
0.03
0.02
0.01
0.00
-
10
20
30
40
50
60
70
80
90
100
Demand Value
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
28
The average number of items per customer
A Normal distribution with µ=10, σ=4
0.12
0.1
Area A measures
the probability
that the average is
greater than 14?
0.08
0.06
0.04
0.02
A
0
-5
0
5
10
15
20
25
Typing =1-normdist(14,10,4,true) in Excel returns this probability
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
29
Using Excel
In Excel, you can also click on Insert >>Function>>NORMDIST
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
30
Using Excel (continued)

NORMDIST function provides the area to the LEFT of the value that
you input for “X”

In this case (X=14) that area equals 0.841

We want to measure A which is an area to the right of “X”

Since the total area is equal to one, we know that the area A equals (1
- 0.841) or 0.159
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
31
Inverse Cumulative Normal Distribution
A Normal distribution with µ=10, σ=4
0.12
0.1
0.08
0.06
0.04
0.02
Area = 0.3
0
0
X
What value of X gives an area of 0.3 to its left ?
We’ll use Excel’s NORMINV function to find out.
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
32
Using Excel: NORMINV
In Excel, you can click on Insert >> Function >> NORMINV
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
33
Inverse Cumulative Normal Distribution
A Normal distribution with µ=10, σ=4
0.12
0.1
0.08
0.06
0.04
0.02
Area = 0.3
0
0
7.902
When X=7.902 the area to the left equals 0.30
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
Let’s look at
Tire Example
34
The Standard Normal Distribution
µ=0
σ=1
If X is a Normal
Distribution,
z = (X- µ)/ σ standardizes X
and z follows a standard
normal
z measures the number of
standard deviation away
from the mean
The Standard Normal (with µ=0 and σ=1) is especially
useful. Any normal distribution can be converted into
the Standard Normal distribution.
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
35
Using Excel

Computations for the standard normal distribution in Excel
can be done using the same NORMDIST and NORMINV
functions as before (with µ=0, σ=1)

You can also use the direct functions:


NORMSDIST(z)
NORMSINV(prob)
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
36
Standard Normal in Excel
This function determines the Area under a
standard normal distribution to the left of -0.75
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
37
Inverse Standard Normal in Excel
This function determines the value of z needed to have
an area under a standard normal of .2266 to the left of z
Let’s look at Tire Example
Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book
38