Chapter-2-10-11

# Chapter-2-10-11 - Chapter 2 The Normal Distributions...

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Chapter 2: The Normal Distributions

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Section 2.1: Density curves and the Normal Distributions rom chapter 1 we have a strategy for exploring data on a single quantitative variable: lot your data: make a graph, usually a histogram or a stemplot. ook for the overall pattern (shape, center, spread) and for any striking deviations such as outliers. 2
Here is the next step: If the overall pattern of a large number of observations is very regular we can describe it with a smooth curve. This curve is a mathematical model for the distribution and it gives a compact picture of the overall pattern. This is also known as a density curve. 3

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A density curve is a curve that: Is always on or above the horizontal axis Has an area of exactly 1 underneath it A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range. One curve that you might have heard of before is a normal curve. 4
The Median and Mean of a Density curve: The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. 5

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The median and mean are the same location if the density curve is symmetric, and they both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. 6
Since a density curve is an idealized description of the distribution of data, we need to distinguish between the mean and standard deviation of the density curve and the mean and standard deviation computed from the actual observations. The new notation is the Greek letter mu for the mean and the Greek letter sigma for standard deviation. μ σ 7

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Normal curves are curves that are symmetric, single-peaked, and bell- shaped. They are used to describe normal distributions. The center of a normal curve is at its mean and has a standard deviation of . There are roughly 6 widths of standard deviation in a normal curve, 3 on one side of center and 3 on the other side. μ

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Chapter-2-10-11 - Chapter 2 The Normal Distributions...

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