1. The Standard Deviation as a Ruler
Week 3
Lecture 2 Chapter 5.
The Standard Deviation as a Ruler
and
the Normal Model

2. The 68-95-99.7% Rule (Empirical Rule)
In the normal distribution with the mean 𝝁 and the standard deviation 𝝈:
Approximately 68% of the observations fall within 1 standard deviation of the mean.
Approximately 95% of the observations fall within 2 standard deviation of the mean.
Approximately 99.7% or almost all of the observations fall within 3 standard deviation of the mean.

3. Normal Distribution and the Empirical Rule
For every normal distribution, approximately what percentage is outside:
1 standard deviation of the mean?
2 standard deviation of the mean?
3 standard deviation of the mean?

4. Normal Distribution and the Empirical Rule
For every normal distribution, approximately
what proportion is outside:
1 standard deviation of the mean?
Approx. 32% (100% - 68% = 32%)
2 standard deviation of the mean?
Approx. 5% (100% - 95% = 5%)
3 standard deviation of the mean?
Approx. 0.30% (100% % = 0.30%)

5. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
Approximately, the middle 68% of women are between: __________ to __________ inches tall.

6. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
The middle 68% is within 1 SD of the mean: 64.5 ± 1 (2.5)
= (64.5 – 2.5, )
= (62, 67)
The other 32% have heights outside the range from 62 to 67.
Approx.__________ of women are taller than 67.
Approx.__________ of women have heights below 62.

7. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
The middle 68% is within 1 SD of the mean: 64.5 ± 1 (2.5)
= (64.5 – 2.5, )
= (62, 67)
The other 32% have heights outside the range from 62 to 67.
Approx. 16% of women are taller than 67.
Approx. 16% of women have heights below 62.

8. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
Approximately, the middle 95% of women are between: __________ to __________ inches tall.

9. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
The middle 95% is within 2 SD of the mean: 64.5 ± 2 (2.5)
= (64.5 – 5, )
= (59.5, 69.5)
The other 5% have heights outside the range from 59.5 to 69.5.
Approx.__________ of women are taller than 69.5.
Approx.__________ of women have heights below 59.5.

10. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
The middle 95% is within 2 SD of the mean: 64.5 ± 2 (2.5)
= (64.5 – 5, )
= (59.5, 69.5)
The other 5% have heights outside the range from 59.5 to 69.5.
Approx. 2.5% of women are taller than 69.5.
Approx. 2.5% of women have heights below 59.5.

11. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
Approximately, the middle 99.7% of women are between: __________ to __________ inches tall.

12. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
The middle 99.7% is within 3 SD of the mean: 64.5 ± 3 (2.5)
= (64.5 – 7.5, )
= (57, 72)
The other 0.3% have heights outside the range from 57 to 72.
Approx.__________ of women are taller than 72.
Approx.__________ of women have heights below 57.

13. Example
The distribution of heights of women aged is approx. normal with mean 64.5 (in inches) and standard deviation 2.5 (in inches).
The middle 99.7% is within 3 SD of the mean: 64.5 ± 3 (2.5)
= (64.5 – 7.5, )
= (57, 72)
The other 0.3% have heights outside the range from 57 to 72.
Approx. 0.15% of women are taller than 72.
Approx. 0.15% of women have heights below 57.

14. Normal Quantile Plots
A histogram or stem-and-leaf plot can reveal distinctly non-normal features of a distribution.
If the stem-and-leaf plot or histogram appears roughly symmetric and unimodal, we use another graph, the normal quantile plot as a better way of judging the adequacy of a normal model.
If the points on a normal quantile plot lie close to a straight line, the plot indicated that the data are normal.
Outliers appear as points that are far away from the overall pattern of the plot.

15. Normal Quantile Plots
Histogram of course grades from
An approx. normal distribution.
Normal Quantile plot of course grades from an approx. normal distribution.

16. Normal Quantile Plots
Histogram of course grades from
a left skewed distribution.
Normal Quantile plot of course grades from a left skewed distribution.

17. Normal Quantile Plots
Histogram of Tim Horton’s calories from
a right skewed distribution.
Normal Quantile plot of Tim Horton’s calories from a right skewed distribution.