1. The Normal Distributions
Chapter 2:
The Normal Distributions

3. Red shaded region represents an approximation of the fraction of scores between 6 and 8
6>scores>8 same as
6<scores>8
Distribution of data can be approximated by a smooth density curve

4. Area A, represents the proportion observations falling between values a and b

5. Median

6. Symmetric density curve.
Area0.25
Area0.25
Area0.25
Area0.25 Q1 Q2

7. Density curve for uniform distribution
1/5 or 0.2 because total area must equal 1
If d = 7 and c =2, what would be the height of the curve?
Area = l x w = (7-2) x 1/5 = 1

8. Normal Distributions
All Normal distributions have this general shape.
m indicates the mean of a density curve.
s indicates the standard deviation of a density curve.

9. Three normal distributions
Differing m results in center of graph at different location on the x axis
Differing s results in varying spread
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10. Inflection points
-1s
+1s
Mean

11. (aka – The Empirical rule)
Normal distributions are abbreviated as ; N(m,s)
The Normal distribution with mean of 0 and standard deviation of 1 is called the standard Normal curve; N(0,1)

14. Example What is the z score for Iowa test score of 3.74? N(6.84,1.55)
z = (x-m)/s
z = (3.74 – 6.84)/1.55
z = -2
This says that a score of 3.74 is 2 standard deviation below the mean

15. Using the 68 – 95 – 99.7 rule to solve problems
Example
What percentage of scores are greater than 5.29?

17. Finding Normal Percentiles by
Table A is the standard Normal table. We have to convert our data to z-scores before using the table.
The figure shows us how to find the area to the left when we have a z-score of 1.80:

18. Using Table A to find the area under the standard normal curve that lies (a) to the left of a specified z-score, (b) to the right of a specified z-score, and (c) between two specified z-scores
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19. Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and  = 50)
What is the probability that the car will go more than 3,100 yards without recharging?

20. Determine the percentage of people having IQs between 115 and 140
P[115< x < 140]
P[( )/16 < z < ( )/16]
P[0.94< z < 2.50]
= –
= = 16.74%
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21. From Percentiles to Scores: z in Reverse
Sometimes we start with areas and need to find the corresponding z-score or even the original data value.
Example: What z-score represents the first quartile in a Normal model?

22. From Percentiles to Scores: z in Reverse
Look in Table A for an area of
The exact area is not there, but is pretty close.
This figure is associated with z = –0.67, so the first quartile is 0.67 standard deviations below the mean.
To unstandardize; solve x = m + zs

23. What score is the 90th percentile for N(504,22)?
Example
What score is the 90th percentile for N(504,22)?
X = zs + m
= 1.28(22) + 504 = m z

25. Methods for Assessing Normality
If the data are normal
A histogram or stem-and-leaf display will look like the normal curve
The mean ± s, 2s and 3s will approximate the empirical rule percentages. (68%,95%,99.7%)
The ratio of the interquartile range to the standard deviation will be about 1.3
A normal probability plot , a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line

26.    Errors per MLB team in 2003 Mean: 106 Standard Deviation: 17
IQR: 22  
22 out of 30: 73%
28 out of 30: 93%
30 out of 30: 100%

27. A normal probability plot is a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis 

28. A skewed distribution might have a histogram and Normal probability plot like this:

29. Khan Academy Videos