1. Standard Deviation and the Normal Model
Chapter 6
Standard Deviation and the Normal Model

2. Standard Deviation
Standard deviation is a measure of spread, or variability.
The smaller the standard deviation, the less variability is present in the data.
The larger the standard deviation, the more variability is present in the data.
Standard deviation can be used as a ruler for measuring how an individual compares to a group.

3. To measure how far above or below the mean any given data value is, we find its standardized value, or z-score.
To standardize a value, subtract the mean and divide by the standard deviation.

4. Measure your height in inches
Measure your height in inches. Calculate the standardized value for your height given that the average height for women is 64.5 inches with a standard deviation of 2.5 inches and for men is 69 inches with a standard deviation of 2.5 inches. Are you tall?

5. Suppose the average woman’s shoe size is 8
Suppose the average woman’s shoe size is 8.25 with a standard deviation 1.15 and the average male shoe size is 10 with a standard deviation of 1.5. Do you have big feet? Suppose Sharon wears a size 9 shoe and Andrew wears a size 9. Does Sharon have big feet? Does Andrew?

6. In order to compare values that are measured using different scales, you must first standardize the values. The standardized values have no units and are called z-scores.
Z-scores represent how far the value is above the mean (if positive) or below the mean (if negative).
Ex: z = 1 means the value is one standard deviation above the mean
z = -0.5 means the value is one-half of a standard deviation below the mean
The larger the z-score, the more unusual it is.
Standardized values, because they have no units, are therefore useful when comparing values that are measured on different scales, with different units, or from different populations.

7. HW:
100
#5-9

8. Shifting
Adding a constant to all of the values in a set of data adds the same constant to the measures of center and percentiles. It does not, however, affect the spread.
Example: Add 5 to each value in the given set of data (on the left) to form a new set of data (on the right). Then find the indicated measures of center and spread.

9. The following histograms show a shift from men’s actual weights to kilograms above recommended weight (by subtracting 74 kg from each):

10. Scaling
Multiplying a constant to all of the values in a set of data multiplies the same constant to the measures of center and spread.
Example: Multiply each value in the given set of data (on the left) by 2 to form a new set of data (on the right). Then find the indicated measures of center and spread.

11. The men’s weight data set measured weights in kilograms
The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale the data (multiply by 2.2):

12. By standardizing values, we shift the distribution so that the mean is 0, and rescale it so that the standard deviation is 1. Standardizing does not change the shape of the distribution.

13. HW:
p. 100 #1-4 & Read p

14. The Normal Model is symmetric and bell-shaped.
follows the Rule
About 68% of the values fall within one standard deviation of the mean.
About 95% of the values fall within two standard deviations of the mean.
About 99.7% (almost all) of the values fall within three standard deviations of the mean.

16. The standard Normal model has mean 0 and standard deviation 1.

17. The Normal model is determined by sigma and mu
The Normal model is determined by sigma and mu. We use the Greek letters sigma and mu because this is a model; it does not come from actual data. Sigma and mu are the parameters that specify the model.

18. The larger sigma, the more spread out the normal model appears
The larger sigma, the more spread out the normal model appears. The inflection points occur a distance of sigma on either side of mu.

19. Standardizing Normal Data:

20. To assess normality:
Examine the shape of the histogram or stem- and-leaf plot. A normal model is symmetric about the mean and bell-shaped.
Compare the mean and median. In a Normal model, the mean and median are equal.
Verify that the Rule holds.

21. When we use the Normal model, we are assuming the distribution is Normal.
We cannot check this assumption in practice, so we check the following condition:
Nearly Normal Condition: The shape of the data’s distribution is unimodal and symmetric.
This condition can be checked with a histogram or a Normal probability plot (not responsible for).

22. Finding Normal Percentiles by Hand
When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look it up in a table of Normal percentiles.
Table Z in Appendix D provides us with normal percentiles, but many calculators and statistics computer packages provide these as well.
The AP test provides a normal percentile table (Table A) for students during all portions of the exam. However, most students simply use their calculator.

23. Finding Normal Percentiles by Hand (cont.)
Table Z 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:

24. 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?

25. From Percentiles to Scores: z in Reverse (cont.)
Look in Table Z 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 standard deviations below the mean.

26. Finding Normal Percentiles Using Technology
Many calculators and statistics programs have the ability to find normal percentiles for us.
Enter the lower bound and upper bound as z-scores into your calculator’s normalcdf (cumulative distribution function) and your calculator will compute the percentage between the z-scores.
Enter a percentile into your calculator’s inverse-normal command and it will provide you with the corresponding z-score.

27. What Can Go Wrong?
Don’t use a Normal model when the distribution is not unimodal and symmetric.

28. What Can Go Wrong? (cont.)
Don’t use the mean and standard deviation when outliers are present—the mean and standard deviation can both be distorted by outliers.
Don’t round your results in the middle of a calculation.
Don’t worry about minor differences in results.

29. What have we learned?
The story data can tell may be easier to understand after shifting or rescaling the data.
Shifting data by adding or subtracting the same amount from each value affects measures of center and position but not measures of spread.
Rescaling data by multiplying or dividing every value by a constant changes all the summary statistics—center, position, and spread.

30. What have we learned? (cont.)
We’ve learned the power of standardizing data.
Standardizing uses the SD as a ruler to measure distance from the mean (z- scores).
With z-scores, we can compare values from different distributions or values based on different units. Even if the data does not follow a normal model.
z-scores can identify unusual or surprising values among data.

31. What have we learned? (cont.)
We’ve learned that the Rule can be a useful rule of thumb for understanding distributions:
For data that are unimodal and symmetric, about 68% fall within 1 SD of the mean, 95% fall within 2 SDs of the mean, and 99.7% fall within 3 SDs of the mean.

32. What have we learned? (cont.)
We see the importance of Thinking about whether a method will work:
Normality Assumption: We sometimes work with Normal tables (Table Z). These tables are based on the Normal model.
Data can’t be exactly Normal, so we check the Nearly Normal Condition by making a histogram (is it unimodal, symmetric and free of outliers?) or a normal probability plot (is it straight enough?).

33. AP Tips Probability rubrics usually have three requirements:
Name, Parameters, Calculations
Thus, full credit can be earned like this:
Normal, μ = 110, σ = 25, x = 140; P(z > 1.2) = 11.5% (whether by calculator or table)
However, drawing and shading a curve is also a useful (and sometimes requested) task. Plus it adds further proof that you understand what you’re doing and gives you an easy way to check your work!

34. AP Tips
Your calculator will let you skip the step of finding a z- score. But this is not a good habit. Showing z-scores and knowing what they mean is a crucial skill for this course.