CHAPTER 2 Modeling Distributions of Data -...
Transcript of CHAPTER 2 Modeling Distributions of Data -...
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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