Chapter 2: Density Curves and Normal Distributions

Chapter 2: Density Curves Chapter 2: Density Curves and Normal Distributionsand Normal Distributions

Density CurvesDensity Curves

What is the probability of scoring in one of these regions?What do all of the areas of the rectangles add up to be?What do all of the probabilities add up to be?

Density CurvesDensity CurvesImpose a curve over a relative

frequency histogram:

Density CurvesDensity CurvesThe area of the curve should

approximate the area of the histogram.

Both areas should = 1

Density Curves:Density Curves:Properties:

◦Is always on or above the horizontal axis.

◦Has area exactly equal to 1.◦Describes the overall pattern of the

distribution.◦The area under the curve and above

any range of values is the proportion of observations that fall in that range.

Density Curves:Density Curves:Come in many shapes just like

their counterparts histograms:◦Normal Curves: symmetric with the

median and mean at the highest point of the graph.

◦Skewed Left: mean is to the left of the median.

◦Skewed Right: mean is to the right of the median.

Density CurveDensity CurveAn approximately normal curve:

Density CurvesDensity CurvesA skewed left density curve and

skewed right density curve:

Skewed Right Density Curve Skewed Left Density Curve

Density CurvesDensity CurvesThe median is the equal areas point,

the point with half of the area under the curve to its left and the remaining half of the area to its right.

The quartiles do the same thing as the median but splits the data into ¼ s.

The mean is the point at which the curve would balance if it were made of solid material.

Density CurvesDensity CurvesSymmetric curves: the mean and

the median are the same for a symmetric curve.

The mean of a skewed curve is pulled away from the median in the direction of the long tail.

Skewed right – mean to the right of the median.

Skewed left – mean to the left of the median.

Density Curves – Normal Density Curves – Normal DistributionsDistributionsSymmetric, bell-shaped, and

single-peaked.Mean, median, and mode occur

at the peak.The distribution is in terms of

numbers of standard deviations from the mean.

The mean is zero standard deviations from the mean.

Normal DistributionsNormal Distributions

Centered at the mean.Positive standard deviations are data to the right of the mean, data greater than the mean.Negative standard deviations are data to the left of the mean, data less than the mean.

Normal DistributionsNormal Distributions68 – 95 – 99.7 Rule

Approximately 68% of the data falls within one standard deviation to the left and right of the mean.Approximately 95% of the data falls within two standard deviations to the left and right of the mean.Approximately 99.7% of the data falls within three standard deviations to the left and right of the mean,

Normal DistributionsNormal DistributionsThe larger the standard deviation is

the flatter the bell-shaped curve will be.

The smaller the standard deviation the tighter the data will be.

Inflection points - the first standard deviation to the left and right occur at the inflection points of the graph; the place where the graph switches from concave up to concave down.

Normal DistributionsNormal Distributions

Standard Normal Standard Normal DistributionsDistributions

We can standardize data set, so that each data point is represented by number of standard deviations from the mean.

Standard Normal Standard Normal CalculationsCalculations

Standardizing DataStandardizing Data

Standard Normal Standard Normal DistributionsDistributionsThe standard normal distribution

is the normal distribution N(0,1) with mean 0 and standard deviation 1.

Standard Normal Standard Normal DistributionsDistributionsThe standard normal table:

◦Provides the probability of falling within a certain region of the standard normal distribution.

◦The distribution provides the area under the curve from - ∞ to some z – score.

Standard Normal Standard Normal DistributionsDistributions

Standard Normal Standard Normal DistributionsDistributionsFind the area under the curve for

the following inequalities:◦z < 1.52◦z < -1.78◦z > 2.10◦z > -1.23◦-2.10 < z < 1.27◦-1.33 < z < 3.12

Finding Normal Finding Normal ProportionsProportionsStep 1 : State the problem in terms

of the observed value x. Draw a picture of the distribution and shade the area of interest under the curve.

Step 2 : Standardize x to restate the problem in in terms of standard normal variable z. Draw a picture to show the area of interest under the standard normal curve.

Finding Normal Finding Normal ProportionsProportionsStep 3 : Find the required area

under the standard normal curve, using Table and the fact that the total area under the curve is 1.

Step 4 : Write your conclusion in the context of the problem.

Example 2.8Example 2.8The level of cholesterol in the blood is

important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is roughly normal. For 14 year old boys the mean is μ=170 mg of cholesterol per dl of blood (mg/dl)and the standard deviation is σ=30 mg/dl. Levels above the 240 mg/dl may require medical attention. What percent of 14 year old boys have more than 240 mg/dl of cholesterol?

Example 2.8Example 2.8Step 1 : State the problem

◦P(x>240) with μ=170 mg/dl and σ=30 mg/dl.

◦Sketch the distribution: mark points of interest and shade the appropriate area.

Step 2 : Standardize x and draw a picture.◦Use z conversion formula◦Sketch standard normal curve of

distribution

Example 2.8Example 2.8Step 3 : Use Table appropriately.

◦1 – P(z<2.33)Step 4 : Write your conclusion in

context of the problem.

Example 2.9Example 2.9What percent of 14 year old boys

have blood cholesterol between 170 and 240 mg/dl?

Example 2.9Example 2.9Step 1 : State the problem

◦P(170 < x< 240◦Sketch the curve

Step 2 : Standardize the data◦Use z conversion formula◦P(0 < z < 2.33)◦Sketch the standardized curve

Step 3 : Use the table appropriately◦ Area between 0 and 2.33 = area below 2.33 – area below 0

Step 4: State your conclusion in the context of the problem.

Example 2.10 SAT Verbal Example 2.10 SAT Verbal Scores: Working BackwardsScores: Working BackwardsScores on the recent SAT Verbal

test in the recent years follow approximately the N(505,110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

Example 2.10 SAT Verbal Example 2.10 SAT Verbal ScoresScoresStep 1: State the problem and draw

a sketch.◦We want a curve that is only .10 of the

total area to the right of a z value.◦Draw Sketch◦z > something, now remember the Table

does less than’s only.Step 2: Use the Table. Look in Table A for

the entry closest to .9 since that would be the area to the left of our .1 region.

z = 1.28 is our standardized score.

Example 2.10 SAT Verbal Example 2.10 SAT Verbal ScoresScores

Assessing NormalityAssessing Normality• How can we decide if a distribution is

normal or not?◦Method 1 : Construct a frequency

histogram or stem and leaf plot. See if the graph is approximately symmetrical and bell-shaped about the mean. Histograms can reveal nonnormal features such

as outliers, pronounced skewness, or gaps and clusters.

Compare the distribution to the 68 – 95 – 99.7 rule.

Small data sets will rarely be approximately normal due to chance variation.

Assessing NormalityAssessing NormalityMethod 2: Construct a normal

probability plot. Put data in from 2.27 page 108

Put data in List 1Use 1 VarStats to compare mean and

median.Construct a dot plot using the 6th option

in your StatPlotsPlot the data and zoom in with

ZoomStatsIf the data is approximately linear then

the data is approximately normal.

Calculator Commands Calculator Commands ReviewReviewnormalcdf(xmin, xmax, μ,σ)normalcdf(zmin,zmax)shadenorm(xmin,xmax,μ,σ)Shadenorm(zmin,zmax)invnorm(p,μ,σ) gives x value that

gives this area under the curveinvnorm(p) gives z value with area

under the curve.