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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 .

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normal curves
Normal Curves
  • So far, we have used histograms to represent the overall shape of a distribution. Now smooth curves can be used:
normal curves1
Normal Curves
  • If the curve is symmetric, single peaked, and bell-shaped, it is called a normal curve.
describing distributions
Describing Distributions
  • Plot the data: usually a histogram or a stem plot.
  • Look for overall pattern
    • Shape
    • Center
    • Spread
    • Outliers
describing distributions1
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 distributions2
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
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 curves1
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 curves2
Density Curves
  • The mean of a skewed distribution is pulled along the long tail (away from the median).
density curves3
Density Curves
  • Uniform Distributions (height = 1)
standard deviations
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 deviations1
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
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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