The Standard Deviation as a Ruler and the Normal Model 1 The 6895997 Rule Empirical Rule 2 In the normal distribution with the mean and the standard deviation Approximately 68 of the observations fall within 1 standard deviation of the mean ID: 658663
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Slide1
Week 3Lecture 2Chapter 5. The Standard Deviation as a Ruler and the Normal Model
1Slide2
The 68-95-99.7% Rule (Empirical Rule)2
In the normal distribution with the mean
and the standard deviation
:
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.Slide3
Normal Distribution and the Empirical RuleFor 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?
3Slide4
Normal Distribution and the Empirical RuleFor every normal distribution, approximatelywhat 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% - 99.7% = 0.30%)4Slide5
Example5
The distribution of heights of women aged 18-24 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.Slide6
Example6
The distribution of heights of women aged 18-24 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, 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.Slide7
Example7
The distribution of heights of women aged 18-24 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, 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.Slide8
Example8
The distribution of heights of women aged 18-24 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.Slide9
Example9
The distribution of heights of women aged 18-24 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, 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.Slide10
Example10
The distribution of heights of women aged 18-24 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, 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.Slide11
Example11
The distribution of heights of women aged 18-24 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.Slide12
Example12
The distribution of heights of women aged 18-24 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, 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.Slide13
Example13
The distribution of heights of women aged 18-24 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, 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.Slide14
Normal Quantile Plots14
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.Slide15
Normal Quantile Plots15
Histogram of course grades from
An approx. normal distribution.
Normal Quantile plot of course grades from an approx. normal distribution.Slide16
Normal Quantile Plots16
Histogram of course grades from
a left skewed distribution.
Normal Quantile plot of course grades from a left skewed distribution.Slide17
Normal Quantile Plots17
Histogram of Tim Horton’s calories from
a right skewed distribution.
Normal Quantile plot of Tim Horton’s calories from a right skewed distribution.