Normal DistributionsNormal Distributions

Fitting a Density Curve to a Fitting a Density Curve to a HistogramHistogram

Skewed right Symmetrical and Bell shaped : Approximately normal

Density CurvesDensity Curves

► A curve through the tops of the histogram bars that A curve through the tops of the histogram bars that smoothes out the irregular ups and downssmoothes out the irregular ups and downs

► Histograms show Histograms show countscounts of observations in each of observations in each class, while Density curves show the class, while Density curves show the proportionproportion of of observations in any regionobservations in any region

► The area under the curve represents this proportion, The area under the curve represents this proportion, so the total area under the curve is equal to so the total area under the curve is equal to 1, or 1, or 100%.100%.

Means and Medians of Density Means and Medians of Density CurvesCurves

► The median is the The median is the equal areasequal areas point – 50% of the observations point – 50% of the observations are on either side of itare on either side of it

► The mean is the The mean is the balancingbalancing point: point: On a On a skewedskewed density curve, the mean is pulled toward the tail density curve, the mean is pulled toward the tail

► Skewed right: Mean is bigger than the medianSkewed right: Mean is bigger than the median► Skewed left: Mean is smaller than the medianSkewed left: Mean is smaller than the median

On a On a symmetricalsymmetrical density curve, the mean and median are equal density curve, the mean and median are equal

Answer the questions as you watch the video Answer the questions as you watch the video http://www.learner.org/courses/againstallodds/unitpages/unit08.html

► All normal curves have what characteristic shape?All normal curves have what characteristic shape?

► A normal distribution can be completely described by which two A normal distribution can be completely described by which two numbers?numbers?

► A normal distribution follows the Empirical Rule. Fill in the percentages below:              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

mean

► 95% of women are between what two heights?95% of women are between what two heights?

► How tall must a woman and man be to join the Boston Beanstalks Club?

Symmetric bell

Man: 6’2” Woman: 5’10”

60.5” and 70.5”

Mean and standard deviation

68%

95%

99.7%

Normal Density CurvesNormal Density Curves

► SymmetricSymmetric ► Bell shapedBell shaped with only with only oneone

peakpeak► Tails fall off quickly, so Tails fall off quickly, so no no

outliersoutliers► Mean and median are Mean and median are equalequal► The point at which the The point at which the

curvature changes is curvature changes is located located oneone standard standard deviation from the meandeviation from the mean

► Normal curves are fully Normal curves are fully described by the mean described by the mean (center) and standard (center) and standard deviation (spread)deviation (spread)

Find the mean and standard Find the mean and standard deviation of the normal curvesdeviation of the normal curves

► There are two normal There are two normal curves graphed at curves graphed at the right, A and B.the right, A and B.

► Which one has the Which one has the greater mean? What greater mean? What is it?is it?

► Which one has the Which one has the greater standard greater standard deviation? What is it?deviation? What is it?

B, about 46, is a greater mean than A, at about 13

B, about 5, is a greater st.dev. than A, at about 3

A B

Barack Obama 47

Presidents’ Ages at InaugurationPresidents’ Ages at Inauguration

► The presidents’ ages at inauguration are The presidents’ ages at inauguration are approximatelyapproximately normal (not normal (not exactlyexactly normal). normal).

► Using the graph at the right, centered at mean 54.6 with Using the graph at the right, centered at mean 54.6 with st.dev 6.2, what percent of the presidents ages were: st.dev 6.2, what percent of the presidents ages were: within 1 standard deviation of the mean?within 1 standard deviation of the mean? within 2 standard deviations of the mean?within 2 standard deviations of the mean? within 3 standard deviations of the mean?within 3 standard deviations of the mean?

(15 + 13)/44 = 64%

(28+6+7)/44 = 93%

100%

Empirical Rule (68-95-99.7) of Empirical Rule (68-95-99.7) of Normal DistributionsNormal Distributions

► In any normal distribution, In any normal distribution, approximately:approximately: 68% of the observations fall 68% of the observations fall

within 1 standard deviation within 1 standard deviation of the meanof the mean

95% of the observations fall 95% of the observations fall within 2 standard within 2 standard deviations of the meandeviations of the mean

99.7% of the observations 99.7% of the observations fall within 3 standard fall within 3 standard deviations of the meandeviations of the mean

SAT Example:SAT Example:SAT scores are approximately normally distributed SAT scores are approximately normally distributed

with a mean of 500 and a standard deviation of 100with a mean of 500 and a standard deviation of 100

► 50% of scores lie below 50% of scores lie below 500500, and , and 50%50% lie above lie above 500500

► 68% of scores fall 68% of scores fall within within 400-600400-600, so , so 34% 34% fall within 500-600fall within 500-600

► What percent of scores What percent of scores fall within 500-700?fall within 500-700?

► What percent of scores What percent of scores fall within 600-700?fall within 600-700?

► What percentile is a What percentile is a student at who scores a student at who scores a 700?700?

½(95) = 47.5%

47.5-34 = 13.5%

50+47.5 = 97.5%tileOr (100-95)/2 = 2.5, and 2.5 + 95 = 97.5%

CD Example: CD Example: The playing times of compact discs are approximately The playing times of compact discs are approximately

normally distributed; the average time in a large collection is normally distributed; the average time in a large collection is 35 minutes, and the standard deviation is 5 minutes.35 minutes, and the standard deviation is 5 minutes.

► The middle 68% of the The middle 68% of the playing times falls between playing times falls between what two times?what two times?

► What percent of the cd’s What percent of the cd’s playing times are between playing times are between 25 and 30 minutes?25 and 30 minutes?

► What percent of the cd’s What percent of the cd’s playing times are below 30?playing times are below 30?

► A playing time of 45 minutes A playing time of 45 minutes corresponds to what corresponds to what percentile of cd playing percentile of cd playing times for this collection?times for this collection?

► A playing time of 40 minutes A playing time of 40 minutes corresponds to what corresponds to what percentile of cd playing percentile of cd playing times for this collection?times for this collection?

20 25 30 35 40 45 50 Playing times

30-40 minutes

13.5%

16%

97.5%

84%

Percents to Memorize!Percents to Memorize!

13.5%2.35%0.15%13.5%2.35%0.15% 34%34%