Section1.3

# Section1.3 - STOR 155, Section 3 Tuesday, January 19, 2010...

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STOR 155, Section 3 Tuesday, January 19, 2010 IPS6e Section 1.3

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Section 1.3: Density curves and normal distributions Density curves Center and spread for density curves Normal distributions 68-95-99.7 rule for normal distributions Standardizing observations Calculations for normal distributions Using the standard normal table Inverse normal calculations Normal quantile plots
Density curves A density curve is any curve that is always on or above the horizontal axis has an area of 1 beneath it. Density curves are used as smooth approximations to the shapes of histograms. For many types of data, the histogram of a large dataset can be very well approximated by one kind of density curve or another.

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Histograms can be approximated by density curves Area under a density curve is always 1. Any histogram can be scaled so that the areas of its bars add to 1. Then we can look for a density curve that fits it well. 2
Histograms can be approximated by density curves If a histogram is scaled so that its total area is 1, then the area of any bar equals the proportion of the population with values represented by that bar. If the histogram is well approximated by a density curve, then this area is approximately the area under the curve corresponding to that bar. Proportion of scores under 6 equals area of blue bars. Proportion of scores under 6 approximately equals blue area under density curve. How to find areas under density curves? Would need calculus. BUT: For some special density curves (especially Normal curves), there are tables (or software) that give areas.

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Mean and median for density curves If a histogram is approximated by a density curve, then the median is the point that divides the area under the curve in half. the mean is the point at which the graph would balance if it were solid. Given a formula for a density curve, one would use calculus to find the mean and median, as well as the quartiles and the standard deviation. (Not in this course.)
A few kinds of density curves 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 15 20 25 30 35 40 45 0 0.02 0.04 0.06 0.08 0.1 Uniform (find areas by geometry) Triangular (find areas by geometry) ‘Gamma’ (not in this course) Normal Many real- world distributions are well approximated by normal density curves.

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density curves 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 Suppose the histogram of a large dataset is well approximated by the density curve shown (a uniform density curve). What proportion of the scores
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## This note was uploaded on 06/12/2010 for the course STOR 155 taught by Professor Andrewb.nobel during the Spring '08 term at UNC.

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Section1.3 - STOR 155, Section 3 Tuesday, January 19, 2010...

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