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# Chapter_3 - STAT 2053 Elementary Statistics Chapters 3 The...

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STAT 2053 – Elementary Statistics Chapters 3 – The Normal Distributions 1

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Exploring A Distribution 1. Always plot your data: make a graph, usually a histogram or stem plot. 2. Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers. 3. Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. 2
CHAPTER 3 NOTE: From this point forward, we will talk about histograms that use percentages instead of counts. As more and more observations are recorded, a histogram starts taking a more defined shape. Eventually we can use a curved line to describe the shape. This is called a density curve . 3

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4 Density Curves Always on or above the horizontal axis Has an area of exactly 1 underneath curve Area under the curve and above any range of values is the proportion of all observations that fall in that range Density curves are generalized, ideal shapes of histograms.
5 Density Curves Example: here is a histogram of vocabulary scores of 947 seventh graders. The smooth curve drawn over the histogram is a mathematical model for the distribution.

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6 Density Curves Example: the areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than or equal to 6.0. There are 287 such students who make up this proportion 287/947 is equal to 0.303.
7 Density Curves Example: now the area under the smooth curve to the left of 6.0 is shaded. If the scale is adjusted so the total area under the curve is exactly 1, then this curve is called a density curve . The proportion of the area to the left of 6.0 is now equal to 0.293.

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Because density curves are idealized patterns, a symmetric density curve is exactly symmetric. Therefore, the mean and median of a symmetric density curve are equal. We know that the mean of a skewed distribution is pulled toward the long tail. 8 Density Curves
9 Density Curves The median and mean of a symmetric density curve both lie at the center of symmetry. The median and mean of a right- skewed density curve, where the mean is pulled away from the median toward the long tail.

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10 Density Curves This shows three density curves, each with three points marked on them. At which of these points on each curve do the mean and the median fall?
11 Density Curves This shows three density curves, each with three points marked on them. At which of these points on each curve do the mean and the median fall? For each of the three curves, the highest point(s) of the curves are marked, but are neither the mean nor the median. What do you think it might be?

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Density Curves The density curve is the ideal shape of our distribution. So it’s descriptive statistics (mean, variance, etc.) are not the same as the descriptive statistics of our actual data set!
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