CHAPTER 2 Modeling Distributions of Data · The Practice of Statistics, 5th Edition 3 Exploring Quantitative Data 1. Always plot your data: make a graph, usually a dotplot, stemplot,

The Practice of Statistics, 5th Edition

Starnes, Tabor, Yates, Moore

Bedford Freeman Worth Publishers

CHAPTER 2 Modeling Distributions of Data 2.2 Density Curves and

Normal Distributions

Learning Objectives

After this section, you should be able to:

The Practice of Statistics, 5th Edition 2

Density Curves: define and describe the mean and median location

Normal Distribution: estimate areas/ proportion of values/

probability. Use 68-95-99.7 Rule

The Standard normal distribution: find the proportion of z-values in

a specified interval, or a z-score from a percentile in the standard

Normal distribution.

Density Curves and Normal Distributions


Exploring Quantitative Data

1. Always plot your data: make a graph, usually a dotplot, stemplot, or histogram.

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.

Exploring Quantitative Data

4. Sometimes the overall pattern of a large number of

observations is so regular that we can describe it by a

smooth curve.


Density Curves

A density curve is a curve that

• is always on or above the horizontal axis, and

• has area exactly 1 underneath it.

A density curve describes the overall pattern of a distribution.

The area under the curve and above any interval of values on

the horizontal axis is the proportion, percentile or probability

of all observations that fall in that interval.

The overall pattern of this histogram of the scores

of all 947 seventh-grade students in Gary, Indiana,

on the vocabulary part of the Iowa Test of Basic

Skills (ITBS) can be described by a smooth curve

drawn through the tops of the bars.

Example

• Approximated area – George Box (famous statistician “Essentially all models are

wrong but some are useful)


Batting averages

• The histogram below shows the distribution of batting average

(proportion of hits) for the 432 Major League Baseball players with at

least 100 plate appearances in a recent season. The smooth curve

shows the overall shape of the distribution.


Describing Density Curves: measure of center

• median -> equal-areas point, the point that divides the area under the curve in half.

• mean -> balance point (µ (the Greek letter mu))

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 Curve



Normal Distributions

• Described by Normal curves (symmetric, single-peaked, and bell-

shaped)

• Described by mean µ (right at the center) and its standard

deviation σ. (from center up to the inflection point)

• Notation: N(µ,σ).


Why are Normal Distributions important?

• Normal distributions are good descriptions for some distributions of real

data.(scores on tests, repeated measurements on volleyball diameter.

Characteristics of biological population)

• Normal distributions are good approximations of the results of many

kinds of chance outcomes (number of heads in many tosses with fair

coin).

• Many statistical inference procedures are based on Normal distributions.


Discovery Applet Activity

• Follow instructions on page 110

• http://bcs.whfreeman.com/tps5e/default.asp#923932__929331__

• Summarize: For any normal density curve, the area under the curve

within one, two or three standard deviations of the mean is about

____% ____% ____%.

• Page: 110 read, + example page 11


The 68-95-99.7 Rule Although there are many Normal curves, they all have properties in

common.

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.


Steps to Draw a Normal Distribution

1. Draw a number line with 7 evenly spaced stick marks (mean, 𝜇 ± 𝜎,

𝜇 ± 2𝜎, 𝜇 ± 3𝜎)

2. Draw a dot above the mean value where the peak will be

3. Above the mark where we have one standard deviation, 60% as

high draw a dot

4. Above the two standard deviation mark, draw dots about 15% as

high draw a dot

5. Above the three standard deviation mark draw two dots right above

the horizontal axis


Practice: Batting Average

• In the previous alternate example about batting averages for Major

League Baseball players, the mean of the 432 batting averages was

0.261 with a standard deviation of 0.034. Suppose that the

distribution is exactly Normal with = 0.261 and = 0.034.

• Problem:

• (a) Sketch a Normal density curve for this distribution of batting

averages. Label the points that are 1, 2, and 3 standard deviations

from the mean.

• (b) What percent of the batting averages are above 0.329? Show

your work.

• (c) What percent of the batting averages are between 0.193 and

0.295? Show your work.



The Standard Normal Distribution All Normal distributions are the same if we measure in units of size σ

from the mean µ as center.

The standard Normal distribution is the Normal distribution with mean 0 and

standard deviation 1.

If a variable x has any Normal distribution N(µ,σ) with mean µ and standard

deviation σ, then the standardized variable

has the standard Normal distribution, N(0,1).

z =x - m

s


The Standard Normal 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.

Z .00 .01 .02

0.7 .7580 .7611 .7642

0.8 .7881 .7910 .7939

0.9 .8159 .8186 .8212

P(z < 0.81) = .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:





Page 114 problem

• Practice (you’ll be asked to find L , R tail area and in between z-

scores)

Finding areas under the standard Normal curve

Problem: Find the proportion of observations from the standard Normal

distribution that are between −0.58 and 1.79.


Homework

• Calculator Activity (in the AP exam choose whichever method is

easiest: table A or calculator)

• Page 128 # 33 to 51

Documents

CHAPTER 2 Modeling Distributions of Data · The Practice of Statistics, 5th Edition 3 Exploring Quantitative Data 1. Always plot your data: make a graph, usually a dotplot, stemplot,