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Chapter 3

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Sociology and Anthropology
SOAN 3120
Michelle Dumas

Chapter 3: The Normal Distributions Density Curves  Exploring a distribution: 1. Always plot data: make a graph, usually a histogram or a stemplot 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 the spread 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve  Our eyes respond to the areas of bars in a histogram, the bar areas represent proportions of the observations  In moving from histogram bars to a smooth curve, we make a specific choice: we adjust the scale of the graph so that the total area under the curve is exactly 1  The total area represents the proportion 1, that is, all the observations  We can then interpret areas under the curve as proportions of the observations  The curve is now a density curve  A density curve is a curve that: o Is always on or above the horizontal axis, and o Has area exactly 1 underneath it o A density curve describes the overall pattern of distribution. The area under the curve and above any range of values is the portion of all observations that fall in that range  Density curves like distributions come in many shapes o Both show the overall shape and the bumps in the long tail  A density curve is often a good description of the overall pattern of a distribution  Outliers which are deviations from the overall pattern, are not described by the curve  Of course no real set of data is exactly described y a density curve, the curve is an idealized description that is easy to use and accurate enough for practical use Describing Density Curves  Our measures of center and spread apply to density curves as well as to actual sets of observations  Areas under the curve show the proportions of the total number of observations  So the median on the density curve is the equal areas point, the point with half the area under the curve to its left and the reaming half the area to its right  You can roughly divide the area under the curve into four equal parts to get the quartiles o One fourth of the area under the curve is to the left of the first quartile o Three fourths of the area is to the left of the third quartile  The median of a symmetric density curve is exactly at its center  There are mathematical ways to find the median when the curve is skewed o The median of a skewed right density curve show the mean is pulled way from the median toward the long tail  The mean density curve is the balance point, at which the curve would balance if made of solid material  The symmetric curve balances at its center because the two sides are identical, the mean and median of a symmetric density curve are equal  The mean of a skewed distribution is pulled toward the long tail  The usual notion for the mean of a density curve is  We write the standard deviation of a density curve as  We can roughly located the by eye, as the balance point, but there is no easy way to locate the standard deviation by eye for density curves Normal Distributions  Normal curves describe distributions called normal distributions  Normal distributions are special and not all “normal” in the sense of being average  We capitalize Normal to remind you that these curves are special  They illustrate important facts: o All normal curves have the same overall shape: symmetric, single peaked and bell shaped o Any specific normal curve is completely described by giving its mean and standard deviation o The mean is located at the center of the symmetric curve and is the same as the median. Changing mean without changing the standard deviation moves the normal curve along the horizontal axis without changing its spread o The standard deviation controls the spread of a normal curve. Curves with larger deviations are more spread out  The standard deviation is the natural measure of spread for the Normal distributions  We can also locate by eye on a Normal curve o The standard deviation is the distance from the center to the change of curvature points on either side  Why are Normal distributions important in statistics? 1. Normal distributions are good descriptions for some distributions of real data  scores on takes taken by many people SAT exams, and repeated careful measurements of the same quantity, and characteristics of biologi
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