1. Basic Practice of Statistics - 3rd Edition The Normal Distributions
Chapter 3
The Normal Distributions
BPS - 3rd Ed.
Chapter 3
Chapter 3

2. Basic Practice of Statistics - 3rd Edition
Density Curves
Here is a histogram of vocabulary scores of 947 seventh graders
The smooth curve drawn over the histogram is a mathematical model for the distribution.
The mathematical model is called the density function.
BPS - 3rd Ed.
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Chapter 3

3. Basic Practice of Statistics - 3rd Edition
Density Curves
The areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than or equal to 6.0.
This proportion is equal to
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Chapter 3
Chapter 3

4. Areas under density curves
Basic Practice of Statistics - 3rd Edition
Areas under density curves
The scale of the Y-axis of the density curve is adjusted so the total area under the curve is 1
The area under the curve to the left of 6.0 is shaded, and is equal to 0.293
This similar to the areas of the shaded bars in the prior slide!
BPS - 3rd Ed.
Chapter 3
Chapter 3

5. Basic Practice of Statistics - 3rd Edition
Density Curves
BPS - 3rd Ed.
Chapter 3
Chapter 3

6. Basic Practice of Statistics - 3rd Edition
Density Curves
There are many types of density curves
We are going to focus on a family of curves called Normal curves
Normal curves are
bell-shaped
not too steep, not too fat
defined means & standard deviations
BPS - 3rd Ed.
Chapter 3
Chapter 3

7. Basic Practice of Statistics - 3rd Edition
Normal Density Curves
The mean and standard deviation computed from actual observations (data) are denoted by and s, respectively.
The mean and standard deviation of the distribution represented by the density curve are denoted by µ (“mu”) and  (“sigma”), respectively.
BPS - 3rd Ed.
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Chapter 3

8. Bell-Shaped Curve: The Normal Distribution
Basic Practice of Statistics - 3rd Edition
Bell-Shaped Curve: The Normal Distribution
standard deviation
mean
BPS - 3rd Ed.
Chapter 3
Chapter 3

9. The Normal Distribution
Basic Practice of Statistics - 3rd Edition
The Normal Distribution
Mean µ defines the center of the curve
Standard deviation  defines the spread
Notation is N(µ,).
BPS - 3rd Ed.
Chapter 3
Chapter 3

10. Practice Drawing Curves!
Symmetrical around μ
Infections points (change in slope, blue arrows) at ± σ
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Chapter 3

11. 68-95-99.7 Rule for Any Normal Curve
Basic Practice of Statistics - 3rd Edition
Rule for Any Normal Curve
68% of the observations fall within one standard deviation of the mean
95% of the observations fall within two standard deviations of the mean
99.7% of the observations fall within three standard deviations of the mean
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12. 68-95-99.7 Rule for Any Normal Curve
Basic Practice of Statistics - 3rd Edition
Rule for Any Normal Curve
68% + -
+2
-2
95%
+3
-3
99.7%
BPS - 3rd Ed.
Chapter 3
Chapter 3

13. 68-95-99.7 Rule for Any Normal Curve
Basic Practice of Statistics - 3rd Edition
Rule for Any Normal Curve
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Chapter 3
Chapter 3

14. Men’s Height Example (NHANES, 1980)
Basic Practice of Statistics - 3rd Edition
Men’s Height Example (NHANES, 1980)
Suppose heights of men follow a Normal distribution with mean = 70.0 inches and standard deviation = 2.8 inches
Shorthand: X ~ N(70, 2.8)
X  the variable
~  “distributed as”
N(μ, σ)  Normal with mean μ and standard deviation σ
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15. Men’s Height Example 68-95-99.7 rule
Basic Practice of Statistics - 3rd Edition
Men’s Height Example rule
If X~N(70, 2.8)
68% between µ   =  2.8 = 67.2 to 72.8
95% between µ  2 =  2(2.8) = 70.0  5.6 = 64.4 to 75.6 inches
99.7% between µ  3 =  3(2.8) = 70.0  8.4 = 61.6 and 78.4 inches
BPS - 3rd Ed.
Chapter 3
Chapter 3

16. NHANES (1980) Height Example
Basic Practice of Statistics - 3rd Edition
NHANES (1980) Height Example
What proportion of men are less than inches tall? (Note: 72.8 is one σ above μ on this distribution) ?
(height)
68%
(by Rule) -1 +1
16%
84%
BPS - 3rd Ed.
Chapter 3
Chapter 3

17. Basic Practice of Statistics - 3rd Edition
NHANES Height Example
What proportion of men are less than 68 inches tall? ?
(height values)
How many standard deviations is 68 from 70?
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Chapter 3
Chapter 3

18. Standard Normal (Z) Distribution
Basic Practice of Statistics - 3rd Edition
Standard Normal (Z) Distribution
The Standard Normal distribution has mean 0 and standard deviation 1
We call this a Z distribution: Z~N(0,1)
Any Normal variable x can be turned into a Z variable (standardized) by subtracting μ and dividing by σ:
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19. Basic Practice of Statistics - 3rd Edition
Standardized Scores
How many standard deviations is 68 from μ on X~N(70,2.8)?
z = (x – μ) / σ
= ( ) / = -0.71
The value 68 is 0.71 standard deviations below the mean 70
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Chapter 3
Chapter 3

20. Men’s Height Example (NHANES, 1980)
Basic Practice of Statistics - 3rd Edition
Men’s Height Example (NHANES, 1980)
What proportion of men are less than 68 inches tall?
(height values) ?
(standardized values)
BPS - 3rd Ed.
Chapter 3
Chapter 3

21. Table A in text: Standard Normal Table
Basic Practice of Statistics - 3rd Edition
Table A in text: Standard Normal Table
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Chapter 3
Chapter 3

22. Table A: Standard Normal Probabilities
Basic Practice of Statistics - 3rd Edition
Table A: Standard Normal Probabilities z
.00
.02
0.8
.2119
.2090
.2061
.2420
.2358
0.6
.2743
.2709
.2676
.01
0.7
.2389
BPS - 3rd Ed.
Chapter 3
Chapter 3

23. Men’s Height Example (NHANES, 1980)
Basic Practice of Statistics - 3rd Edition
Men’s Height Example (NHANES, 1980)
What proportion of men are less than 68 inches tall?
(standardized values)
(height values)
.2389
BPS - 3rd Ed.
Chapter 3
Chapter 3

24. Men’s Height Example (NHANES, 1980)
Basic Practice of Statistics - 3rd Edition
Men’s Height Example (NHANES, 1980)
What proportion of men are greater than 68 inches tall?
Area under curve sums to 1, so Pr(X > x) = 1 – Pr(X < x), as shown below:
(standardized values)
(height values)
1.2389 = .7611
.2389
BPS - 3rd Ed.
Chapter 3
Chapter 3

25. Men’s Height Example (NHANES, 1980)
Basic Practice of Statistics - 3rd Edition
Men’s Height Example (NHANES, 1980)
How tall must a man be to place in the lower 10% for men aged 18 to 24?
.10
? (height values)
BPS - 3rd Ed.
Chapter 3
Chapter 3

26. Table A: Standard Normal Table
Basic Practice of Statistics - 3rd Edition
Table A: Standard Normal Table
Use Table A
Look up the closest proportion in the table
Find corresponding standardized score
Solve for X (“un-standardize score”)
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Chapter 3

27. Table A: Standard Normal Proportion
Basic Practice of Statistics - 3rd Edition
Table A: Standard Normal Proportion z
.07
.09
1.3
.0853
.0838
.0823
.1020
.0985
1.1
.1210
.1190
.1170
.08
1.2
.1003
Pr(Z < -1.28) = .1003
BPS - 3rd Ed.
Chapter 3
Chapter 3

28. Men’s Height Example (NHANES, 1980)
Basic Practice of Statistics - 3rd Edition
Men’s Height Example (NHANES, 1980)
How tall must a man be to place in the lower 10% for men aged 18 to 24?
.10
? (height values)
(standardized values)
BPS - 3rd Ed.
Chapter 3
Chapter 3

29. Observed Value for a Standardized Score
Basic Practice of Statistics - 3rd Edition
Observed Value for a Standardized Score
“Unstandardize” z-score to find associated x :
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Chapter 3
Chapter 3

30. Observed Value for a Standardized Score
Basic Practice of Statistics - 3rd Edition
Observed Value for a Standardized Score
x = μ + zσ = 70 + (-1.28 )(2.8) = 70 + (3.58) = 66.42
A man would have to be approximately inches tall or less to place in the lower 10% of the population
BPS - 3rd Ed.
Chapter 3
Chapter 3