The Practice of StatisticsThird Edition

Chapter 2:Describing Location in a Distribution

Copyright © 2008 by W. H. Freeman & Company

Data Analysis Toolbox

• Data – Organize & Examine

– Who, What, Why, Where, When, How, & by Whom

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• Graphs – Appropriate Graphical Display

– Box plot, histogram, etc.

• Numerical Summaries – Relevant Statistics

• Interpretation – Answer the Question!!!

Strategy for Single Variable Data

• To measure the relative standing in a

distribution.

– Plot the data.

– Look for pattern (Shape, Center, Spread)

to find deviations such as outliers.

– Calculate a numerical summary to

describe center and spread.

• The overall pattern may be so

regular that we can describe it with

a smooth curve.

• Doing so allows us to describe the

locations of individual

observations within a distribution.

947 Seventh Graders in Gary, Indiana on ITBS.

Regular distribution, tails fall off smoothly, single center peak,

no gaps or outliers.

The Density Curve

• It is a mathematical model for the

distribution.

• An idealized description of the data.

• A compact picture of the overall pattern of

the data.

• Ignores minor irregularities and outliers.

• Easier to work with curve than histogram.

Graph on left shows all the students with a score of 6 or less.

This is basically the 30th percentile. (.303)

Graph on right shows the area under the curve that is 6 or less.

The area under the curve is equal to 1.

This blue area is actually .293, so very close to .303.

So using this DENSITY CURVE is a very good estimate of the

areas given by the histogram.

This is a Density Curve That is Skewed Left.

Remember no set of real data is EXACTLY described by a

density curve.

The curve is an approximation that is easy to use and accurate

enough for practical use.

Measures of Center

We can eyeball the centers and the quartiles.

This is not true of Standard Deviation.

Please Note

• We use x-bar (mean) and s (S.D.) with our

actual observations.

• Density Curve is an IDEALIZED

description of data.

• Therefore the mean of a density curve is

denoted by μ (the Greek letter mu)

• The standard deviation of a density curve is

noted by σ (Greek letter sigma).

Assignment

• Exercises 2.7, 2.9, 2.10, 2.12, 2.13,

• Read pages 133- 136

• Watch: www.learner.org/courses/againstallodds/unitpages/unit07.html