statistics quick overview

Copyright by Michael S. Watson, 2012

Statistics Quick Overview

Copyright by Michael S. Watson, 2012; Slides from Managerial Statistics book 2

LET’S START WITH CANDY


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?


You Formed a Hypothesis….

Proportion of Hersey’s is not 33%


Hypothesis Testing

H0- Null Hypothesis (everything else)

Ha- Alternative Hypothesis (what you want to prove)

H-0H-a


Hypothesis Testing- Candy Example

H0- Null Hypothesis (Is 33%)

Ha- Alternative Hypothesis (Hershey’s Not 33%)

H-0H-a


Hypothesis Testing

H0

Ha

Reject Not Reject

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1 2

Is 33%

Not 33%


Hypothesis Testing

H0

Ha

Reject Not Reject

Get this for Free

1 2

What kind of evidence do we need to Reject the Null?

Is 33%

Not 33%


Hypothesis Testing

H0- Not Guilty

Ha- Guilty

Why this way? “Innocent until proven guilty”


Hypothesis Testing

H0

Ha

Reject Not Reject

Get this for Free

1 2

Does this mean Innocent?

Not Guilty

Guilty


Hypothesis Testing- types of Errors

Guilty

Not Guilty

Guilty Innocent

Tria

l Fin

ds D

efen

dant

…

Defendant Really is….

What do we do to avoid these errors?


Basic Statistics– Mean and Standard Deviation

Data Point Tire Failure Miles1 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

Packaging Example

Tire Failure


Important Attributes

Mean: The average or ‘expected value’ of a distribution. Denoted by µ (The Greek letter mu)

Variance: A measure of dispersion and volatility. 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)


Which Process is More Variable?

Case 1 Average: 50 Standard Deviation: 25

Case 2 Average: 5,000 Standard Deviation: 2,000

Case 3 Average: 10,000 Standard Deviation: 3,000

Coefficient of Variation (CV) 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


What-If With Packing VariabilityOriginal Case

Less Variability More Variability


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’”


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


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


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


Continuous Distributions

This area represents the probability that Sales will be between 20,000 and 30,000


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


The Normal Distribution: A Bell Curve

The area under this curve (and all continuous

distributions) is equal to one


Normal Distribution: Symmetric

This half has an area = 0.50

So does this half


Three Normal Distributions

-4 -3 -2 -1 0 1 2 3 4

µ=0σ=1

µ=0σ=2

µ=1σ=1


Shapes of different Normal curves

Different Normal Curves

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50 60 70 80 90 100

Demand

Prob

abili

ty

M = 50, Std = 4M = 50, Std = 8M = 50, Std = 16


Normal Distribution Over Time

Demand over Time

0

20

40

60

80

1001 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Time Period

Act

ual D

eman

d

M = 50, Std = 4

Demand over Time

0

20

40

60

80

100

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Time Period

Act

ual D

eman

d

M = 50, Std = 16


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


Normal Distribution (Mean = 50, Std Dev = 8)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

- 10 20 30 40 50 60 70 80 90 100

Demand Value

Prob

abili

ty

Relationship between demand variability and service level (2)

Assume same average demand, with less variation

Now you need to holdonly 63 for 95% service level


The average number of items per customer

0

0.02

0.04

0.06

0.08

0.1

0.12

-5 0 5 10 15 20 25

A

A Normal distribution with µ=10, σ=4

Area A measures the probability

that the average is greater than 14?

Typing =1-normdist(14,10,4,true) in Excel returns this probability


Using Excel

In Excel, you can also click on Insert >>Function>>NORMDIST


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


Inverse Cumulative Normal Distribution


0

0.02

0.04

0.06

0.08

0.1

0.12

0

Area = 0.3

X

What value of X gives an area of 0.3 to its left ?We’ll use Excel’s NORMINV function to find out.


Using Excel: NORMINV

In Excel, you can click on Insert >> Function >> NORMINV


Inverse Cumulative Normal Distribution


0

0.02

0.04

0.06

0.08

0.1

0.12

0

Area = 0.3

7.902

When X=7.902 the area to the left equals 0.30 Let’s look at Tire Example


The Standard Normal Distribution

µ=0σ=1

The Standard Normal (with µ=0 and σ=1) is especially useful. Any normal distribution can be converted into the Standard Normal distribution.

If X is a Normal Distribution,

z = (X- µ)/ σ standardizes X and z follows a standard

normalz measures the number of standard deviation away

from the mean


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)


Standard Normal in Excel

This function determines the Area under a standard normal distribution to the left of -0.75


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

statistics quick overview

Documents

managerial statistics

managerial statistics

candy copyright

elseha alternative hypothesis

defendant defendant

greek letter mu variance

related measure of dispersion

greek letter sigmacopyright