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Normal Distributions Fitting a Density Curve to a Histogram Skewed right Symmetrical and Bell shaped : Approximately normal Density Curves ► ► ► A curve through the tops of the histogram bars that smoothes out the irregular ups and downs Histograms show counts of observations in each class, while Density curves show the proportion of observations in any region The area under the curve represents this proportion, so the total area under the curve is equal to 1, or 100%. Means and Medians of Density Curves ► ► The median is the equal areas point – 50% of the observations are on either side of it The mean is the balancing point: On a skewed density curve, the mean is pulled toward the tail Skewed right: Mean is bigger than the median ► Skewed left: Mean is smaller than the median ► On a symmetrical density curve, the mean and median are equal Answer the questions as you watch the video http://www.learner.org/courses/againstallodds/unitpages/unit08.html ► All normal curves have what characteristic shape? Symmetric bell ► A normal distribution can be completely described by which two numbers? Mean and standard deviation ► A normal distribution follows the Empirical Rule. Fill in the percentages below: ► 68% 95% 99.7% mean of the values lie within 1 standard deviation of the mean of the values lie within about 2 standard deviations of the mean (or almost all) of the values lie within 3 standard deviations of the 95% of women are between what two heights? 60.5” and 70.5” ► How tall must a woman and man be to join the Boston Beanstalks Club? Man: 6’2” Woman: 5’10” Normal Density Curves ► ► ► ► ► ► Symmetric Bell shaped with only one peak Tails fall off quickly, so no outliers Mean and median are equal The point at which the curvature changes is located one standard deviation from the mean Normal curves are fully described by the mean (center) and standard deviation (spread) Find the mean and standard deviation of the normal curves ► There are two normal curves graphed at the right, A and B. ► Which one has the greater mean? What is it? B, about 46, is a greater mean than A, at about 13 ► Which one has the greater standard deviation? What is it? B, about 5, is a greater st.dev. than A, at about 3 A B Barack Obama 47 Presidents’ Ages at Inauguration The presidents’ ages at inauguration are approximately normal (not exactly normal). ► Using the graph at the right, centered at mean 54.6 with st.dev 6.2, what percent of the presidents ages were: ► within 1 standard deviation of the mean? (15 + 13)/44 = 64% within 2 standard deviations of the mean? (28+6+7)/44 = 93% within 3 standard deviations of the mean? 100% Empirical Rule (68-95-99.7) of Normal Distributions ► In any normal distribution, approximately: 68% of the observations fall within 1 standard deviation of the mean 95% of the observations fall within 2 standard deviations of the mean 99.7% of the observations fall within 3 standard deviations of the mean SAT Example: SAT scores are approximately normally distributed with a mean of 500 and a standard deviation of 100 50% of scores lie below 500, and 50% lie above 500 ► 68% of scores fall within 400-600, so 34% fall within 500-600 ► What percent of scores fall within 500-700? ½(95) = 47.5% ► What percent of scores fall within 600-700? 47.5-34 = 13.5% ► What percentile is a student at who scores a 700? 50+47.5 = 97.5%tile ► Or (100-95)/2 = 2.5, and 2.5 + 95 = 97.5% CD Example: The playing times of compact discs are approximately normally distributed; the average time in a large collection is 35 minutes, and the standard deviation is 5 minutes. ► ► ► ► ► The middle 68% of the playing times falls between what two times? 30-40 minutes What percent of the cd’s playing times are between 25 and 30 minutes? 13.5% What percent of the cd’s playing times are below 30? 16% A playing time of 45 minutes corresponds to what percentile of cd playing times for this collection? 97.5% A playing time of 40 minutes corresponds to what percentile of cd playing times for this collection? 84% 20 25 30 35 40 45 50 Playing times Percents to Memorize! 0.15% 2.35% 13.5% 34% 34% 13.5% 2.35% 0.15%