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The Standard Deviation as a Ruler
Week 3 Lecture 2 Chapter 5. The Standard Deviation as a Ruler and the Normal Model
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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.
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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?
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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%)
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Normal Quantile Plots Histogram of course grades from An approx. normal distribution. Normal Quantile plot of course grades from an approx. normal distribution.
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Normal Quantile Plots Histogram of course grades from a left skewed distribution. Normal Quantile plot of course grades from a left skewed distribution.
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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.
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