1. Chapter 7: Normal Probability Distributions
April 17

2. In Chapter 7: 7.1 Normal Distributions
4/20/2017
In Chapter 7:
7.1 Normal Distributions
7.2 Determining Normal Probabilities
7.3 Finding Values That Correspond to Normal Probabilities
7.4 Assessing Departures from Normality
Basic Biostat

3. §7.1: Normal Distributions
Normal random variables are the most common type of continuous random variable
First described de Moivre in 1733
Laplace elaborated the mathematics in 1812
Describe some (not all) natural phenomena
More importantly, describe the behavior of means

4. Normal Probability Density Function
Recall the continuous random variables are described with smooth probability density functions (pdfs) – Ch 5
Normal pdfs are recognized by their familiar bell-shape
This is the age distribution of a pediatric population. The overlying curve represents its Normal pdf model

5. Area Under the Curve
The darker bars of the histogram correspond to ages less than or equal to 9 (~40% of observations)
This darker area under the curve also corresponds to ages less than 9 (~40% of the total area)

6. Parameters μ and σ Normal pdfs are a family of distributions
Family members identified by parameters μ (mean) and σ (standard deviation)
μ controls location
σ controls spread

7. Mean and Standard Deviation of Normal Density
Chapter 7
4/20/2017
Mean and Standard Deviation of Normal Density σ μ
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8. Standard Deviation σ
Points of inflections (where the slopes of the curve begins to level) occur one σ below and above μ
Practice sketching Normal curves to feel inflection points
Practice labeling the horizontal axis of curves with standard deviation markers (figure)

9. Means and Standard Deviations
Chapter 7
4/20/2017
Means and Standard Deviations
The mean and standard deviation from data sets are denoted “xbar” and s
The mean and standard deviation parameters from the Normal distributions are μ and σ
These means and standard deviations are related, but are not the same thing
Basic Biostat

10. 68-95-99.7 Rule for Normal Distributions
Chapter 7
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Rule for Normal Distributions
68% of the AUC falls within ±1σ of μ
95% of the AUC falls within ±2σ of μ
99.7% of the AUC falls within ±3σ of μ
Basic Biostat

11. Chapter 7
4/20/2017
Example: Rule
Wechsler adult intelligence scores are Normally distributed with μ = 100 and σ = 15; X ~ N(100, 15). Using the rule:
68% of scores fall in μ ± σ = 100 ± 15 = 85 to 115
95% of scores fall in μ ± 2σ = 100 ± (2)(15) = 70 to 130
99.7% of scores in μ ± 3σ = 100 ± (3)(15) = 55 to 145
Basic Biostat

12. Symmetry in the Tails
Because of the Normal curve is symmetrical and the total AUC adds to 1…
… we can determine the AUC in tails, e.g., Because 95% of curve is in μ ± 2σ, 2.5% is in each tail beyond μ ± 2σ
95%

13. Chapter 7
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Example: Male Height
Male height is approximately Normal with μ = 70.0˝ and σ = 2.8˝
Because of the rule, 68% of population is in the range 70.0˝  2.8˝ = 67.2 ˝ to 72.8˝
Because the total AUC adds to 100%, 32% are in the tails below 67.2˝ and above 72.8˝
Because of symmetry, half of this 32% (i.e., 16%) is below 67.2˝ and 16% is above 72.8˝
Basic Biostat

14. Example: Male Height 64% 16% 16% 70 67.2 72.8 Chapter 7 4/20/2017
Basic Biostat

15. Reexpression of Non-Normal Variables
Many biostatistical variables are not Normal
We can reexpress non-Normal variables with a mathematical transformation to make them more Normal
Example of mathematical transforms include logarithms, exponents, square roots, and so on
Let us review the logarithmic transformation

16. Logarithms Logarithms are exponents of their base
There are two main logarithmic bases
common log10 (base 10)
natural ln (base e)
Landmarks:
log10(1) = 0 (because 100 = 1)
log10(10) = 1 (because 101 = 10)

17. Example: Logarithmic Re-expression
Chapter 7
4/20/2017
Example: Logarithmic Re-expression
Prostate specific antigen (PSA) not Normal in 60 year olds but the ln(PSA) is approximately Normal with μ = −0.3 and σ = 0.8
95% of ln(PSA) falls in μ ± 2σ = −0.3 ± (2)(0.8) = −1.9 to 1.3
Thus, 2.5% are above ln(PSA) 1.3; take anti-log of 1.3: e1.3 = 3.67
Since only 2.5% of population has values greater than 3.67 → use this as cut-point for suspiciously high results
Basic Biostat

18. §7.2: Determining Normal Probabilities
To determine a Normal probability when the value does not fall directly on a ±1σ, ±2σ, or ±3σ landmark, follow this procedure:
1. State the problem
2. Standardize the value (z score)
3. Sketch and shade the curve
4. Use Table B to determine the probability

19. Example: Normal Probability Step 1. Statement of Problem
Chapter 7
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Example: Normal Probability Step 1. Statement of Problem
We want to determine the percentage of human gestations that are less than 40 weeks in length
We know that uncomplicated human pregnancy from conception to birth is approximately Normally distributed with μ = 39 weeks and σ = 2 weeks. [Note: clinicians measure gestation from last menstrual period to birth, which adds 2 weeks to the μ.]
Let X represent human gestation: X ~ N(39, 2)
Statement of the problem: Pr(X ≤ 40) = ?
Basic Biostat

20. Standard Normal (Z) Variable
Standard Normal variable ≡ a Normal random variable with μ = 0 and σ = 0
Called “Z variables”
Notation: Z ~ N(0,1)
Use Table B to look up cumulative probabilities
Part of Table B shown on next slide…

21. Example: A Standard Normal (Z) variable with a value of 1
Example: A Standard Normal (Z) variable with a value of 1.96 has a cumulative probability of

22. Normal Probability Step 2. Standardize
To standardize, subtract μ and divide by σ.
The z-score tells you how the number of σ-units the value falls above or below μ

23. Steps 3 & 4. Sketch and Use Table B
3. Sketch and label axes
4. Use Table B to lookup Pr(Z ≤ 0.5) =

24. Probabilities Between Two Points
Let a represent the lower boundary and b represent the upper boundary of a range:
Pr(a ≤ Z ≤ b) = Pr(Z ≤ b) − Pr(Z ≤ a)
Use of this concept will be demonstrate in class and on HW exercises.

25. §7.3 Finding Values Corresponding to Normal Probabilities
State the problem.
Use Table B to look up the z-percentile value.
Sketch
4. Unstandardize with this formula

26. Looking up the z percentile value
Use Table B to look up the z percentile value, i.e., the z score for the probability in questions
Look inside the table for the entry closest to the associated cumulative probability.
Then trace the z score to the row and column labels.

27. Suppose you wanted the 97. 5th percentile z score
Suppose you wanted the 97.5th percentile z score. Look inside the table for Then trace the z score to the margins.
Notation: Let zp represents the z score with cumulative probability p, e.g., z.975 = 1.96

28. Finding Normal Values - Example
Chapter 7
4/20/2017
Finding Normal Values - Example
Suppose we want to know what gestational length is less than 97.5% of all gestations?
Step 1. State the problem!
Let X represent gestations length
Prior problem established X ~ N(39, 2)
We want the gestation length that is shorter than .975 of all gestations. This is equivalent to the gestation that is longer than.025 of gestations.
Basic Biostat

29. Chapter 7
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Example, cont.
Step 2. Use Table B to look up the z value. Table B lists only “left tails”.
“less than 97.5%” (right tail) = “greater than 2.5%” (left tail).
z lookup in table shows z.025 = −1.96 z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
–1.9
.0287
.0281
.0274
.0268
.0262
.0256
.0250
.0244
.0239
.0233
Basic Biostat

30. 3. Sketch
4. Unstandardize
“The 2.5th percentile gestation is 35 weeks.”

31. 7.4 Assessing Departures from Normality
The best way to assess Normality is graphically
Approximately Normal histogram
Normal “Q-Q” Plot of same distribution
A Normal distribution will adhere to a diagonal line on the Q-Q plot

32. A negative skew will show an upward curve on the Q-Q plot

33. A positive skew will show an downward curve on the Q-Q plot

34. Same data as previous slide but with logarithmic transform
A mathematical transform can Normalize a skew

35. Leptokurtotic
A leptokurtotic distribution (skinny tails) will show an S-shape on the Q-Q plot