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# Section 5.1: Normal Distributions - PowerPoint PPT Presentation

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

(Day 1)

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

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

• Plot the data: usually a histogram or a stem plot.

• Look for overall pattern

• Shape

• Center

• Outliers

• 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.

• 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.

• 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)

• 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.

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

• Uniform Distributions (height = 1)

• 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

more sharply

peaked…

• 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.

• 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… 