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handout_1_3 - 1.3 Density Curves and Normal Distributions...

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1.3 Density Curves and Normal Distributions Density Curve software can describe distribution by fitting a smooth curve to the data less arbitrary than choosing classes for histogram idealization that pictures the overall pattern of the data but ignores minor irregularities and outliers smooth approximation to the irregular bars of a histogram Density Curve •The curve is always on or above the horizontal axis. •The curve has area exactly 1 underneath it. • The area under the density curve and above any range of values is the proportion of all observations that fall in that range. Figure 1.23a histogram- areas of bars represent either counts or proportions of observations red- got 6.0 or lower (0.303 of all students) red bars make up 0.303 of total area under all the bars adjust the scale so total area of bars is 1- area of red bars will be 0.303 Figure 1.23b red- area under curve to the left of 6.0 scale is adjusted so total area under curve is exactly 1 area under curve= proportion red area proportion= 0.293 (density curve provides good approximation) Measuring center and spread for density curves mode - a peak point of a curve median - the point with half the total area on each side mean - the balance point (point at which curve would balance if made of solid material) Why? The small area in the long right tail tips the curve more than the same area near the center. (Figure 1.25) •symmetric density curve- mean and median are same •skewed distribution pulls mean towards its tail (Figure 1.24)
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Figure 1.23 density curve- exactly symmetric histogram- only approximately symmetric A density curve is an idealized description of a distribution of data. We need to distinguish between the mean and standard deviation of the density curve and the numbers x and s computed from the actual observations.
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