The Practice of Statistics, 5th Edition

Starnes, Tabor, Yates, Moore

Bedford Freeman Worth Publishers

CHAPTER 2 Modeling Distributions of Data 2.1 Describing Location in a Distribution

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BIG PICTURE

• Chapter 2 covers contents from Section I and Section III of AP Statistics

• The content is section 2.1 (z-scores, percentiles, cumulative frequency

plots and the effect of changing units on summary measures) comes

from Section I Exploring Data: describing patterns and departures from

patterns

• The content is section 2.2 (Density curves and normal distributions)

comes from section III: Anticipating patterns: Exploring Random

phenomena using probability and simulations

• In Chapter 1 we learned about the shape, center, and spread of a

distribution of quantitative data. Now we can ask additional questions:

• Where is a particular observation located within a distribution

(above average, or below average and if so by how much)

• What happens to the summary statistics if I convert the

measurements from inches to meters?

• Is the overall pattern of the distribution so regular that we can

model it using a smooth curve?

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Learning Objectives

After this section, you should be able to:

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Learned about the measure of center:

Percentile (shown by cumulative relative frequency graphs)

Z-scores (standardizing score): (x-mean)/standard deviation

Effect of adding, subtracting, multiplying by, or dividing by a

constant on the shape, center, and spread of a distribution of data.

Describing Location in a Distribution

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Estimate the Width of your classroom • Dot plot

• Find the 5 numbers (can use calculator), sx IQR, mean

• Describe SOCS

– Shape (skewed, symmertric, bimodal, unimodal)?

– Outliers1.5 *IQR Rule

– Center: chose which measure to use (median or mean)

– Spread:

• Chose which measure to use (sx or IQR)

• Talk about:

– Q1 (approximately the 25th percentile)

– Q3 the 75th percentile

• Reveal the classroom width

• How close were the students (graph error): error = guess – 13

• What happened to the Shape? Center? Spread?

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Transforming Data

Transforming converts the original observations from the original units

of measurements to another scale. Transformations can affect the

shape, center, and spread of a distribution.

Adding the same number a to (subtracting a from) each observation:

• adds a to (subtracts a from) measures of center and location

(mean, median, quartiles, percentiles), but

• Does not change the shape of the distribution or measures of

spread (range, IQR, standard deviation).

Effect of Adding (or Subtracting) a Constant

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Where do you stand Activity

• Transform the heights from

– centimeter into inches: multiply by 0.3937

• Plot the two dot plots

• What happened to the Shape? Center? Spread?

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Transforming Data

Transforming converts the original observations from the original units

of measurements to another scale. Transformations can affect the

shape, center, and spread of a distribution.

Multiplying (or dividing) each observation by the same number b:

• multiplies (divides) measures of center and location (mean,

median, quartiles, percentiles) by b

• multiplies (divides) measures of spread (range, IQR, standard

deviation) by |b|, but

• does not change the shape of the distribution

Effect of Multiplying (or Dividing) by a Constant

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Example: Taxi Cabs

Taxi cabs

• In 2010, taxi cabs in New York City charged an initial fee of $2.50

plus $2 per mile. In equation form, fare = 2.50 + 2(miles). At the end

of a month, a businessman collects all his taxi cab receipts and

calculates some numerical summaries. The mean fare he paid was

$15.45, with a standard deviation of $10.20.

• Problem: What are the mean and standard deviation of the lengths

of his cab rides in miles?

TRY: Question 19 page102

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Example of putting it all together

• Pg 95 Too Cool At the Cabin

Section Summary

In this section, we learned how to…

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FIND and INTERPRET the percentile of an individual value within a

distribution of data.

ESTIMATE percentiles and individual values using a cumulative

relative frequency graph.

FIND and INTERPRET the standardized score (z-score) of an

individual value within a distribution of data.

DESCRIBE the effect of adding, subtracting, multiplying by, or dividing

by a constant on the shape, center, and spread of a distribution of

data.

Describing Location in a Distribution

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Homework

Page 102-103 #19-32 Everything!

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WARM-UP

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Guess The Width of your Classroom Activity

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Chapter 2.1 – Effects of Transformations

Transforming converts the original observations from the

original units of measurements to another scale.

Transformations can affect the shape, center, and spread

of a distribution.

Effect of adding/subtracting by a constant?

Effect of multiplying/dividing by a constant?

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Too Cool at the Cabin Solution

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Example: Taxi Cabs

Taxi cabs

• In 2010, taxi cabs in New York City charged an initial fee of $2.50

plus $2 per mile. In equation form, fare = 2.50 + 2(miles). At the end

of a month, a businessman collects all his taxi cab receipts and

calculates some numerical summaries. The mean fare he paid was

$15.45, with a standard deviation of $10.20.

• Problem: What are the mean and standard deviation of the lengths

of his cab rides in miles?

TRY: Question 19 page102

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Homework and announcements

• Page 102 # 19-32 (ALL)

• Extra Credit Project

– due Monday October 6th

– Replaces one of the PAST POP Quiz grade ONLY

• Expected Chapter 1& 2 Cumulative Quiz

– October 17

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