03-normal distributions (1)
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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