Section 5.1: Normal Distributions

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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)

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.
Describing Distributions
• Plot the data: usually a histogram or a stem plot.
• Look for overall pattern
• Shape
• Center
• Outliers
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.
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.
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)
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.
Density Curves
• The mean of a skewed distribution is pulled along the long tail (away from the median).
Density Curves
• Uniform Distributions (height = 1)
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