3.3 Density Curves and Normal Distributions

• Density Curves • Measuring Center and Spread for Density Curves• Normal Distributions• The 68-95-99.7 Rule• Standardizing Observations• Using the Standard Normal Table • Inverse Normal Calculations• Normal Quantile Plots

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We now have a kit of graphical and numerical tools for describing distributions. We also have a strategy for exploring data on a single quantitative variable. Now, we’ll add one more step to the strategy.

Exploring Quantitative Data

1. Always plot your data: Make a graph.2. Look for the overall pattern (shape, center, and spread) and

for striking departures such as outliers.3. Calculate a numerical summary to briefly describe center

and spread.4. Sometimes, the overall pattern of a large number of

observations is so regular that we can describe it by a smooth curve.

1. Always plot your data: Make a graph.2. Look for the overall pattern (shape, center, and spread) and

for striking departures such as outliers.3. Calculate a numerical summary to briefly describe center

and spread.4. Sometimes, the overall pattern of a large number of

observations is so regular that we can describe it by a smooth curve.

Exploring Quantitative DataExploring Quantitative Data

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Density curvesA density curve is a mathematical model of a distribution…

The total area under the curve, by definition, is equal to 1, or 100%.

The area under the curve for a range of values is the proportion of all observations for that range.

Area under Density Curve ~ Relative Frequency of Histogram

Histogram of a sample with the smoothed, density curve

describing theoretically the population.

rel. freq of left histogram=287/947=.303

area = .293 under rt. curve

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A density curve is a curve that:

Is always on or above the horizontal axis Has an area of exactly 1 underneath it

A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values on the horizontal axis is the proportion of all observations that fall in that range.

A density curve is a curve that:

Is always on or above the horizontal axis Has an area of exactly 1 underneath it

A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values on the horizontal axis is the proportion of all observations that fall in that range.

Density Curves

Density curves come in many

shapes. Some are well known

mathematically and others aren’t

– but they all lie above the

horizontal axis and have total

area = 1.

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Our measures of center and spread apply to density curves as well as to actual sets of observations.

The median of a density curve is the equal-areas point―the point that divides the area under the curve in half.

The mean of a density curve is the balance point, at which the curve would balance if made of solid material.

The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

The median of a density curve is the equal-areas point―the point that divides the area under the curve in half.

The mean of a density curve is the balance point, at which the curve would balance if made of solid material.

The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

Distinguishing the Median and Mean of a Density CurveDistinguishing the Median and Mean of a Density Curve

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Density Curves

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The mean and standard deviation computed from actual observations (data) are denoted by and s, respectively, and are called the sample mean and sample standard deviation.

The mean and standard deviation of the idealized distribution represented by the density curve are denoted by µ (“mu”) and (“sigma”), respectively, and are sometimes called the population mean and population standard deviation.

x

Density Curves

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One particularly important class of density curves are the Normal curves, which describe Normal distributions.

All Normal curves are symmetric, single-peaked, and bell-shaped.

A specific Normal curve is described by giving its mean µ and standard deviation σ.

Normal Distributions

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A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ.

The mean of a Normal distribution is the center of the symmetric Normal curve.

The standard deviation is the distance from the center to the change-of-curvature points on either side, the points of inflection of the density.

We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).

A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ.

The mean of a Normal distribution is the center of the symmetric Normal curve.

The standard deviation is the distance from the center to the change-of-curvature points on either side, the points of inflection of the density.

We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).

Normal Distributions

Normal distributions

e = 2.71828… The base of the natural logarithm

π = pi = 3.14159…

Normal – or Gaussian – distributions are a family of symmetrical, bell-

shaped density curves defined by a mean (mu) and a standard

deviation (sigma) : N().

xx

A family of density curves

Here, means are different

( = 10, 15, and 20) while

standard deviations are the

same ( = 3).

Here, means are the same ( = 15)

while standard deviations are

different ( = 2, 4, and 6).

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The 68-95-99.7 Rule

In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.

The 68-95-99.7 Rule

In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.

The 68-95-99.7 Rule

Here’s a N(64.5”, 2.5”)distribution of heights ofcollege-aged females.

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The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th-grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).

Sketch the Normal density curve for this distribution.

What percent of ITBS vocabulary scores are less than 3.74?

What percent of the scores are between 5.29 and 9.94?

The 68-95-99.7 Rule

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All Normal distributions are the same if we measure in units of size σ from the mean µ as center.

If a variable x has a distribution with mean µ and standard deviation σ, then the standardized value of x, or its z-score, is

Note z= the number of s.d.’s away from mu that x is…(sorry about the grammer!)

If a variable x has a distribution with mean µ and standard deviation σ, then the standardized value of x, or its z-score, is

Note z= the number of s.d.’s away from mu that x is…(sorry about the grammer!)

μx

z-

The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. That is, the standard Normal distribution is N(0,1) – it is represented by Z and we write Z ~ N(0,1)

The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. That is, the standard Normal distribution is N(0,1) – it is represented by Z and we write Z ~ N(0,1)

Standardizing Observations

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Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.

The Standard Normal Table

Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.

The Standard Normal Table

Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.

The Standard Normal Table

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Z 0.00 0.01 0.02

0.7 0.7580 0.7611 0.7642

0.8 0.7881 0.7910 0.7939

0.9 0.8159 0.8186 0.8212

P(z < 0.81) = 0.7910

Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81. We can use Table A:

The Standard Normal Table

Tips on using Table A

To calculate the area between 2 z- values, first get the area under N(0,1) to the

left for each z-value from Table A.

area between z1 and z2 =

area left of z1 – area left of z2

A common mistake made by

students is to subtract both z

values - it is the areas that are

subtracted, not the z-scores!

Then subtract the smaller

area from the larger area.

area right of z = 1 - area left of z

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Express the problem in terms of the observed variable X.

Draw a picture of the distribution of X and shade the area of interest under the curve.

Perform calculations. Standardize X to restate the problem in terms of a standard

Normal variable Z. Use Table A and the fact that the total area under the curve is

1 to find the required area under the standard Normal curve.

Write your conclusion in the context of the problem.

Express the problem in terms of the observed variable X.

Draw a picture of the distribution of X and shade the area of interest under the curve.

Perform calculations. Standardize X to restate the problem in terms of a standard

Normal variable Z. Use Table A and the fact that the total area under the curve is

1 to find the required area under the standard Normal curve.

Write your conclusion in the context of the problem.

How to Solve Problems Involving Normal DistributionsHow to Solve Problems Involving Normal Distributions

Normal Calculations

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According to the Health and Nutrition Examination Study of 1976–1980, the heights X (in inches) of adult men aged 18–24 are N(70, 2.8).How tall must a man be (? below) to be in the lower 10% for men

aged 18–24?

0.10

? 70

N(70, 2.8)

Inverse Normal Calculations

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z 0.07 0.08 0.09

–1.3 0.0853 0.0838 0.0823

–1.2 .1020 0.1003 0.0985

–1.1 0.1210 0.1190 0.1170

How tall must a man be in the lower 10% for men aged 18–24?

0.10 ? 70

N(70, 2.8)

Look up the closest probability (closest to 0.10) in the table.

Find the corresponding z the standardized score.

The value you seek is that many standard deviations from the mean.

Z = –1.28

Inverse Normal Calculations

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How tall must a man be in the lower 10% for men aged 18–24?

0.10

? 70

N(70, 2.8)

We need to “unstandardize” the z-score to find the observed value (x):

Z = –1.28

zx

x z x = 70 + z(2.8) = 70 + [(1.28 ) (2.8)] = 70 + (–3.58) = 66.42

A man would have to be approximately 66.42 inches tall or less to place in the lower 10% of all men in the population.

Normal Calculations

One way to assess if a distribution is indeed approximately Normal is

to plot the data on a Normal quantile plot.

The data points are ordered from smallest to largest and their

percentile ranks are converted to z-scores with Table A. These z-

scores are then plotted against the data to create a Normal quantile

plot.

If the distribution is indeed Normal, the plot will show a straight

line, indicating a good match between the data and a Normal

distribution – in JMP the points fall within the dotted lines.

Systematic deviations from a straight line indicate a non-Normal

distribution. Outliers appear as points that are far away from the

overall pattern of the plot – some points fall outside the dotted

lines in JMP.22

Normal Quantile Plots

Normal quantile plots are complex to do by hand, but they are easy to do in JMP – under the red triangle, choose Normal Quantile Plot – but notice the difference when compared to the above plots…

Curved pattern: The data are not

Normally distributed. Instead, it shows

a right skew: A few individuals have

particularly long survival times.

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Normal Quantile Plots

Good fit to a straight line: the distribution

of rainwater pH values is close to

normal. The intercept of the line ~ mean

of the data and the slope of the line ~

s.d. of the data

HW: •Finish reading section 3.3•Work over all the examples!•I’ve put up some videos on computing Normal Probabilities as an online assignment… due 9/19 & 9/22 at 9:00am•Work on #3.82-3.93, 3.95, 3.100-3.102,3.104, 3.106-3.109, 3.110-3.128. Do as manyof these problems in order to really understand what’s going on here! •Quiz in class on Monday 9/22•Test #1 on October 1, covering Chapts. 1-4