Statistics Quick Overview

1. Statistics Quick Overview

2. Let’s Start with Candy

3. 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?

4. You Formed a Hypothesis…. Proportion of Hersey’s is not 33%

5. Hypothesis Testing H0- Null Hypothesis (everything else) Ha- Alternative Hypothesis (what you want to prove)

6. Hypothesis Testing- Candy Example H0- Null Hypothesis (Is 33%) Ha- Alternative Hypothesis (Hershey’s Not 33%)

7. Hypothesis Testing Possible Outcomes Reject Not Reject H0 2 1 Is 33% Get this for Free Ha X Not 33%

8. Hypothesis Testing What kind of evidence do we need to Reject the Null? Reject Not Reject H0 2 1 Is 33% Get this for Free Ha X Not 33%

9. Hypothesis Testing Justice Example H0- Not Guilty Ha- Guilty Why this way? “Innocent until proven guilty”

10. Hypothesis Testing Does this mean Innocent? Reject Not Reject H0 2 1 Not Guilty Get this for Free Ha X Guilty

11. Hypothesis Testing- types of Errors Defendant Really is…. Guilty Innocent Guilty Type I Error Trial Finds Defendant … Not Guilty Type II Error What do we do to avoid these errors?

12. Basic Statistics– Mean and Standard Deviation Packaging Example Tire Failure

13. 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)

14. 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

15. What-If With Packing Variability Original Case Less Variability More Variability

16. 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’”

17. 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

18. 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

19. 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

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

21. 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

22. The Normal Distribution: A Bell Curve The area under this curve (and all continuous distributions) is equal to one

23. Normal Distribution: Symmetric So does this half This half has an area = 0.50

24. Three Normal Distributions µ=0 σ=1 µ=1 σ=1 µ=0 σ=2

25. Shapes of different Normal curves

26. Normal Distribution Over Time

27. 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

28. Relationship between demand variability and service level (2) • Assume same average demand, with less variation Now you need to hold only 63 for 95% service level

29. The average number of items per customer A Normal distribution with µ=10, σ=4 Area A measures the probability that the average is greater than 14? A Typing =1-normdist(14,10,4,true)in Excel returns this probability

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

31. 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

32. Inverse Cumulative Normal Distribution 0.12 0.1 0.08 0.06 0.04 0.02 0 0 A Normal distribution with µ=10, σ=4 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.

33. Using Excel: NORMINV In Excel, you can click on Insert >> Function >> NORMINV

34. Inverse Cumulative Normal Distribution 0.12 0.1 0.08 0.06 0.04 0.02 0 0 A Normal distribution with µ=10, σ=4 Area = 0.3 7.902 When X=7.902 the area to the left equals 0.30 Let’s look at Tire Example

35. 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.

36. 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)

37. Standard Normal in Excel This function determines the Area under a standard normal distribution to the left of -0.75

38. 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