Normal Probability Normal Probabilities Departures from Normality

Normal Probability

Betsy Greenberg

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

1 Normal Probability

2 Normal Probabilities

3 Departures from Normality

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Describe the distribution.

Center ≈ 0.1,Spread from -12 to +12 with some low outliers

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Describe the shape of the distribution.

Symmetric, single peaked, bell shapedNormal

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Normal Distribution

The model for symmetric, bell-shaped, unimodal histograms isthe Normal model.

We write N(µ, σ) to represent a Normal model with mean µand standard deviation σ.

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Nomal Approximation to Binomial

The Binomial distribution is used when we’re

counting the number of successesin n independent trialseach with probability p of success

A discrete Binomial model is approximately Normal if weexpect at least 10 successes and 10 failures:

np ≥ 10 and n(1 − p) ≥ 10

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Central Limit Theorem

The probability distribution of a sum of independent randomvariables of comparable variance tends to a normal distributionas the number of summed random variables increases.

Explains why bell-shaped distributions are so common

Observed data are often the accumulation of many smallfactors (e.g., the value of the stock market depends on manyinvestors)

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

The Normal Probability Distribution

Defined by the parameters µ and σ

The mean µ locates the center

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

The Normal Probability Distribution

Defined by the parameters µ and σ

The standard deviation σ controls the spread

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

The 68-95-99.7 Rule (Empirical Rule)

P(µ− σ < x < µ+ σ) = 68%

P(µ− 2σ < x < µ+ 2σ) = 95%

P(µ− 3σ < x < µ+ 3σ) = 99.7%

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Other probabilities

Standardize

z =X − µ

σ

Use the Table of Normal Probabilities

Figure 5.5 on page 216 (DA&DM)Table Z in Appendix D, pages A108-A109 (BS)

Use Excel’s NORM.DIST function

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Example

Suppose a packaging system fills boxes such that the weightsare normally distributed with a µ = 16.3 oz. and σ = 0.2 oz.

The package label states the weight as 16 oz.

What is the standardized value?

z = X−µσ = 16−16.3

0.2 = −1.5

What is the probability that a box has less than 16 oz?

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Standardize and use table

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Use Excel

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Example

Suppose a packaging system fills boxes such that the weightsare normally distributed with a µ = 16.3 oz. and σ = 0.2 oz.

The package label states the weight as 16 oz.

To what weight should the mean of the process be adjusted sothat the chance of an underweight box is only 0.005?

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Departures from Normality

Multimodality: More than one mode suggesting data comefrom distinct groups

Skewness: Lack of symmetry

Outliers: Unusual extreme values

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Departures from Normality

Normal Quantile Plot

Diagnostic scatterplot used to determine the appropriateness of anormal modelIf data track the diagonal line, the data are normally distributed

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Normal Quantile Plot

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Normal Quantile Plot

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Install StatTools

Go tohttp://www.mccombs.utexas.edu/Tech/Computer-Services/COE/Decision-Tools

Login with your UTEID

You must be running Windows and Excel for Windows

On Mac, use Remote Desktop

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Try StatTools

Use the Data Set Manager to define the data set

Make a histogram

Make a boxplot

Make a Normal Quantile plot

Betsy Greenberg McCombs

Statistics and Modeling

Normal Probability Normal Probabilities Departures from Normality

Departures from Normality

Skewness

Measures lack of symmetry.K3 = 0 for normal data.

K3 =z31 + z32 + · · · + z3n

n

Kurtosis

Measures the prevalence ofoutliers.K4 = 0 for normal data.

K4 =z41 + z42 + · · · + z4n

n− 3

StatTools doest nothave −3 in the K4

formulaBetsy Greenberg McCombs

Statistics and Modeling