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Section 5.1: Normal Distributions

Section 5.1: Normal Distributions. (Day 1). Normal Curves. So far, we have used histograms to represent the overall shape of a distribution. Now smooth curves can be used:. Normal Curves. If the curve is symmetric, single peaked, and bell-shaped, it is called a normal curve .

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Section 5.1: Normal Distributions

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  1. Section 5.1: Normal Distributions (Day 1)

  2. Normal Curves • So far, we have used histograms to represent the overall shape of a distribution. Now smooth curves can be used:

  3. Normal Curves • If the curve is symmetric, single peaked, and bell-shaped, it is called a normal curve.

  4. Describing Distributions • Plot the data: usually a histogram or a stem plot. • Look for overall pattern • Shape • Center • Spread • Outliers

  5. Describing Distributions • Choose either 5 number summary or “Mean and Standard Deviation” to describe center and spread of numbers • 5 number summary used when there are outliers and graph is skewed; center is the median. • Mean and Standard Deviation used when there are no outliers and graph is symmetric; center is the mean • Now, if the overall pattern of a large number of observations is so regular, it can be described by a normal curve.

  6. Describing Distributions • The tails of normal curvesfall off quickly. • There are no outlier • There are no outliers. • The mean and median are the same number, located at the center (peak) of graph.

  7. Density Curves • Most histograms show the “counts” of observations in each class by the heights of their bars and therefore by the area of the bars. • (12 = Type A) • Curves show the “proportion” of observations in each region by the area under the curve. The scale of the area under the curve equals 1. This is called a density curve. • (0.45 = Type A)

  8. Density Curves • Median: “Equal-areas” point – half area is to the right, half area is to the left. • Mean: The balance point at which the curve would balance if made of a solid material (see next slide). • Area: ¼ of area under curve is to the left of Quartile 1, ¾ of area under curve is to the left of Quartile 3. (Density curves use areas “to the left”). • Symmetric: Confirms that mean and median are equal. • Skewed: See next slide.

  9. Balance Point for Means

  10. Density Curves • The mean of a skewed distribution is pulled along the long tail (away from the median).

  11. Density Curves • Uniform Distributions (height = 1)

  12. Standard Deviations • If the curve is a normal curve, the standard deviation can be seen by sight. It is the point at which the slope changes on the curve. • A small standard deviation shows a graph which is less spread out, more sharply peaked…

  13. Standard Deviations • Carl Gauss used standard deviations to describe small errors by astronomers and surveyors in repeated careful measurements. A normal curve showing the standard deviations was once referred to as an “error curve”. • The 68-95-99.7 Rule shows the area under the curve which shows 1, 2, and 3 standard deviations to the right and the left of the center of the curve…more accurate than by sight.

  14. Tomorrow… • More about 68-95-99.7 Rule, z-scores, and percentiles… • We will be doing group activities. Please bring your calculators and books!!! • Homework: None… 

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