Density Curves

Section 2.1

Strategy to explore data on a single variable

Plot the data (histogram or stemplot) CUSS Calculate numerical summary to

describe center and spread • Mean and standard deviation

• Five number summary

From Histogram to Density Curve

Sometimes the overall pattern of a histogram can be described by a smooth curve.

Density Curve

Density curve is a mathematical model for the distribution

Idealized description Scale is adjusted so area under the

curve is equal to 1

Density Curve

A density curve is a curve that• Is always on or above the horizontal axis

• Has area exactly 1 underneath it

Describes the overall pattern of a distribution

The area under the curve and above any range of values is the proportion of observations that fall in that range.

Uniform distribution If the total area under the

curve is 1, what is the height of the square?

What percent of the observations lie above 0.8?

What percent of the observations lie below 0.6?

What percent of the observations lie between 0.25 and 0.40?

Area under the curve

Is this a density curve?

21 2

11

0.50

0

0.5

1.5

Mean and Median of Density Curve

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

Median is the equal-areas point, the point that divides the area under the curve in half.

Mean and median are the same for symmetric curve.

Mean and Median

Normal Distribution

Symmetric Single-peaked Bell-shaped Can be defined by mean, , and standard

deviation, .

68-95-99.7 Rule

68% of the observations fall within 1 standard deviation.95% of the observations fall within 2 standard deviations.99.7% of the observations fall within 3 standard deviations.

Example 2.3Young Women’s Heights

The distribution of heights of young women aged 18 to 24 is approximately normal with mean = 64.5 inches and standard deviation = 2.5 inches.

How is the scale on the bottom of the next graph determined from this information?

Example 2.3Young Women’s Heights

What proportion of young women are over 67 inches tall?

Between what heights dothe middle 68% of all women fall?

What is the percentileof women with heights59.5 inches?

Section 2.2Standard Normal Calculations

By definition, what is the area under a density curve?

How can we convert a normal distribution curve to a density curve?

Standardizing

If x is an observation from a distribution that has mean and standard deviation , the standardized value of x is

xz

(This is a big deal. Memorize it!)

A standardized value is often called a z-score.

Using z-scores to compare observations

While the mean professional baseball batting average has been roughly constant over the decades, the standard deviation has dropped over time.

Using the information on the next slide, determine how far each baseball player stood above his peers.

Baseball Landmarks

Player Average Decade Mean Std dev

Ty

Cobb

.420

(1911)

1910s .266 .0371

Ted

Williams

.406

(1941)

1940s .267 .0326

George

Brett

.390

(1980)

1970s .261 .0317

Normal distribution calculations

The normal distribution probability density function is given by the formula:

Where is the mean and is the standard deviation.

Not very friendly

Standard Normal Distribution

In the standardized value, = 0 and = 1

In your book, Table A has a list of the value of p(x) for many z-scores.

If you know the area of the shaded region to the left, can you calculate the area of the unshaded region?

Finding Normal Proportions

State the problem in terms of the observed variable x. Draw a picture of the distribution and shade the area of interest under the curve.

Calculate the z-score. Draw a picture to show the area of interest under the standard normal curve.

Find the required area under the standard normal curve, using Table A.

Write your conclusion in the context of the problem.

Practice

Work through example 2.8 The distribution of heights of adult American

men is N(69,2.5). What percent of men are at least 6 feet tall? What percent of men are between 5 feet and 6

feet tall? How tall must a man be to be in the tallest 10%

of all adult men?

Using your calculator:

The normalcdf command on the TI-84+ calculator can be used to find the area under a normal distribution and above an interval.

Normalcdf(min,max,mean,sd)

Normalcdf can be found under DISTR (2nd Vars)

Use your calculator to find areas on previous slide and compare with your table results.

Normal Probability Plots

While many collections of data are normally distributed, we cannot assume that all are. You can examine histograms or box plots to see if data “looks” normal.

An easy way to check the normality is using a normal probability plot. • Put your data in one of the lists in your calculator.

Normal Probability Plot

If the data distribution is close to a normal distribution, the plotted points will lie close to a straight line.