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Section 5.1: Normal DistributionsPowerPoint Presentation

Section 5.1: Normal Distributions

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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
- Spread
- 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

less spread out,

more sharply

peaked…

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.

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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