Introduction to thePractice of Statistics

Fifth Edition

Chapter 1:Looking at Data—Distributions

Copyright © 2005 by W. H. Freeman and CompanyModifications and Additions by M. Leigh Lunsford,

2005-2006

David S. Moore • George P. McCabe

Technology Requirements

• TI-83

• SPSS

• Excel Data Analysis

• Excel Macros

• Data Sets in SPSS and Excel Format on CD

• See my website for more details:

www.mathspace.com/lunsford

The Science of Learning from Data The Collection and Analysis of Data

What is Statistics??

Experimental DesignChapter 3

Descriptive Statistics(Data Exploration)

Chapters 1, 2

Inferential StatisticsChapters 5 - 8

ProbabilityChapter 4

Chapter 1 - Looking at Data

1.1 Displaying Distributions with Graphs

1.2 Describing Distributions with Numbers

1.3 Density Curves and Normal Distributions

Section 1.1

Displaying Distributions with Graphs

Data Basics

Variable Types

An Example (p. 5)

Graphs for Categorical Vars.

• Bar Graphs

• Pie Charts

Educational Level Example (page 7):– A Bar Graph by Hand– A Pie Chart by Hand

Homework: Try to do these in Excel!

Graphs for Quantitative Data

• Stemplots (Stem and Leaf Plots)– Generally for small data sets

• Histograms

• Time Plots (if applicable)

Let’s look at an example to see what types of questions one may ask and how these plots help to visualize the answers!

Example 1.7 Page 14Descriptive and Inferential Stats

1. What percent of the 60 randomly chosen fifth grade students have an IQ score of at least 120?

2. Based on this data, approximately what percent of all fifth grade students have an IQ score of at least 120?

3. What is the average IQ score of the fifth grade students in this sample?

4. Based on this data, what is the average IQ score of all fifth grade students (i.e. the population) from which the sample was drawn?

Inferential? Descriptive?2 and 4 1 and 3

Let’s Make a Stemplot!An Example (Ex. 1.7 p.14)

Data in Table 1.3 p. 14 (and on next slide)

Stem and Leaf Plot for ExampleIQ Test Scores for 60 Randomly Chosen

5th Grade StudentsGenerated Using the Descriptive Statistics Menu on Megastat

Stem and Leaf plot for iq

stem unit = 10

leaf unit = 1

Frequency Stem Leaf

3 8 1 2 9

4 9 0 4 6 7

14 10 0 1 1 1 2 2 2 3 5 6 8 9 9 9

17 11 0 0 0 2 2 3 3 4 4 4 5 6 7 7 7 8 8

11 12 2 2 3 4 4 4 5 6 7 7 8

9 13 0 1 3 4 4 6 7 9 9

2 14 2 5

60

Now Let’s Make a Histogram!

• Use the Same Data in Example 1.7 (Data in Table 1.3)

• We will start by hand….using class widths of 10 starting at 80…

• Let’s try using Megastat (Excel file on Disk)!

• Compare the Stemplot to the Histogram!

Histogram for Example

IQ Scores of Randomly Chosen Fifth Grade Students

0

5

10

15

20

25

30

80

90

100

110

120

130

140

150

IQ Score

Per

cent

  iq           cumulative

lower upper midpoint width frequency percent frequency percent

80 < 90 85 10 3 5.0 3 5.0

90 < 100 95 10 4 6.7 7 11.7

100 < 110 105 10 14 23.3 21 35.0

110 < 120 115 10 17 28.3 38 63.3

120 < 130 125 10 11 18.3 49 81.7

130 < 140 135 10 9 15.0 58 96.7

140 < 150 145 10 2 3.3 60 100.0

60 100.0

Compare this Histogram to the Stem & Leaf Plot we Generated Earlier!

Recall Our Earlier Question 1

1. What percent of the 60 randomly chosen fifth grade students have an IQ score of at least 120?

• Numerically?

• How to RepresentGraphically?

18.3%+15%+3.3%=36.6%

(11+9+2)/60=.367 or 36.7%

Grey Shaded Region corresponds to this 36.6% of data

What is Different Fromthe Histogram we Generated

In Class??

Let’s Look at the Distribution we Just Created:•Overall Pattern:

Shape (modes, tails (skewness), symmetry) Center (mean, median)Spread (range, IQR, standard deviation)

•Deviations:Outliers

Descriptors we will be interested

in for data and population

distributions.

•Overall Pattern:Shape, Center, Spread?

•Deviations:Outliers?

Example 1.9 page 18-19

Data Analysis – An Interesting Example (Example 1.10, p. 9-10)

80 Calls

•Overall Pattern:Shape, Center, Spread?

•Deviations:Outliers?

Time Plots – For Data Collected Over Time…

Example: Mississippi River Discharge p.19 (data p. 21)

Example – Dealing with Seasonal Variation

Extra Slides from Homework

• Problem 1.19

• Problem 1.20

• Problem 1.21

• Problem 1.31

• Problem 1.36

• Problem 1.37-1.38

Problem 1.19, page 30

Problem 1.20, page 31

Problem 1.21, page 31

Problem 1.31, page 36

Problem 1.36, page 38

Problems 1.37 – 1.39

Section 1.2

Describing Distributions with Numbers

Types of Measures

• Measures of Center:– Mean, Median, Mode

• Measures of Spread:– Range (Max-Min), Standard Deviation,

Quartiles, IQR

Means and Medians

Consider the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):

65, 65, 70, 75, 78, 80, 83, 87, 91, 94

What is the Average (or Mean) Test Score?

What is the Median Test Score?

Consider the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):

65, 65, 70, 75, 78, 80, 83, 87, 91, 94

• Draw a Stem and Leaf Plot (Shape, Center, Spread?)• Find the Mean and the Median• Let’s Use our TI-83 Calculators!

– Enter data into a list via Stat|Edit– Stat|Calc|1-Var Stats

• What happens to the Mean and Median if the lowest score was 20 instead of 65?• What happens to the Mean and Median if a low score of 20 is added to the data

set (so we would now have 11 data points?)

What can we say about the Mean versus the Median?

Quartiles: Measures of Position

A Graphical Representation of Position of Data(It really gives us an indication of how the data is spread

among its values!)

Using Measures of Position to Get Measures of Spread

And what was the range again???

5 Number Summary, IQR, Box Plot, and where Outliers would be for Test

Score Data:

65, 65, 70, 75, 78, 80, 83, 87, 91, 94

What do we notice about symmetry?

Histograms of Flower LengthsProblem 1.58

Generated via Minitab

length

Perc

ent

514845423936

48

36

24

12

0

514845423936

48

36

24

12

0

bihai red

yellow

Panel variable: variety

Histogram of Flower Length

Box Plot and 5-Number Summary for Flower Length Data

Generated via Box Plot Macro for Excel

Box Plots for Flower Lengths

30

35

40

45

50

55

Bihai Red Yellow

Flower Color

Len

gth

s (i

n m

m)

Bihai Red Yellow

Median 47.12 39.16 36.11

Q1 46.71 38.07 35.45

Min or In Fence 46.34 37.4 34.57

Max or In Fence 50.26 43.09 38.13

Q348.24

5 41.69 36.82

Outliers?

Remember this histogram from the Service Call Length Data on page 9? How do you expect the Mean and Median to compare for this data?

Mean 196.6, Median 103.5

Box Plot for Call Length Data

More on Measures of Spread

• Data Range (Max – Min)• IQR (75% Quartile minus 25% Quartile 2, range

of middle 50% of data)• Standard Deviation (Variance)

– Measures how the data deviates from the mean….hmm…how can we do this?

• Recall the Sample Test Score Data: 65, 65, 70, 75, 78, 80, 83, 87, 91, 94

Recall the Sample Mean (X bar) was 78.8…

Computing Variance and Std. Dev. by Hand and Via the TI83:

Recall the Sample Test Score Data:

65, 65, 70, 75, 78, 80, 83, 87, 91, 94

Recall the Sample Mean (X bar) was 78.8

65 70 75 80 9085 95

x

65 83

78.8

-13.8 4.2

What does the number 4.2 measure? How

about -13.8?

Consider (again!) the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):

65, 65, 70, 75, 78, 80, 83, 87, 91, 94What happens to the standard deviation and the location of the 1st and 3rd quartiles if the lowest score was 20 instead of 65?

What happens to the standard deviation and the location of the 1st and 3rd quartiles if a low score of 20 is added to the data set (so we would now have 11 data points?)

What can we say about the effect of outliers on the standard deviation and the quartiles of a data set?

Effects of Outliers on the Standard Deviation

Example 1.18:Stemplots of Annual Returns forStocks (a) and Treasury bills (b)On page 53 of text. What are the

stem and leaf units????

Consider (again!) the following sample of test scores from one of Dr. L.’s recent classes (max score = 100):

65, 65, 70, 75, 78, 80, 83, 87, 91, 94Xbar=78.8 s=10.2 (rounded)

Suppose we “curve” the grades by adding 5 points to every test score (i.e. Xnew=Xold+5). What will be new mean and standard deviation?

Suppose we “curve” the grades by multiplying every test score times 1.5 (i.e. Xnew=1.5*Xold). What will be the new mean and standard deviation?

Suppose we “curve” the grades by multiplying every test score times 1.5 and adding 5 points (i.e. Xnew=1.5*Xold+5). What will be the new mean and standard deviation?

Effects of Linear Transformations on the MeanAnd Standard Deviation

Box Plots for Problems 1.62-1.64

Section 1.3

Density Curves and Normal Distributions

Basic Ideas• One way to think of a density curve is as a smooth

approximation to the irregular bars of a histogram.• It is an idealization that pictures the overall pattern of the

data but ignores minor irregularities.• Oftentimes we will use density curves to describe the

distribution of a single quantitative continuous variable for a population (sometimes our curves will be based on a histogram generated via a sample from the population).– Heights of American Women

– SAT Scores

• The bell-shaped normal curve will be our focus!

Shape?Center?Spread?

Density Curve

Page 64

Sample Size =105

Shape?Center?Spread?

Density Curve

Page 65

Sample Size=72 Guinea pigs

1. What proportion (or percent) of seventh graders from Gary,Indiana scored below 6?

2. What is the probability (i.e. how likely is it?)that a randomly chosenseventh grader from Gary, Indiana will have a test score less than 6?

Two Different butRelated Questions!

Example 1.22Page 66

Sample Size = 947

Relative “area under the curve”

VERSUSRelative “proportion of

data” in histogrambars.

Page 67 of text

Shape?Center?Spread?

The classic “bell shaped” Density curve.

A “skewed” density curve.Median separates area under curve into two equal areas

(i.e. each has area ½)

What is the geometric interpretationof the mean?

The mean as “center of mass” or “balance point” of the density curve

The normal density curve!Shape? Center? Spread?

Area Under Curve?

How does the magnitude of the standard deviation affect a density curve?

How does the standard deviation affect the shape of the normal density curve?

Assume Same Scale onHorizontal and Vertical

(not drawn) Axes.

The distribution of heights of young women (X) aged 18 to 24 is approximately normal with mean mu=64.5 inches and standard deviation sigma=2.5 inches (i.e. X~N(64.5,2.5)). Lets draw the density curve for X and observe the empirical rule!

(aka the “Empirical Rule”)

Example 1.23, page 72How many standard deviations from the mean height is the height of a woman who is 68 inches? Who is 58 inches?

The Standard Normal Distribution

(mu=0 and sigma=1)

Horizontal axis in units of z-score!

Notation:Z~N(0,1)

Let’s find some proportions (probabilities) using normal distributions!

Example 1.25 (page 75)Example 1.26 (page 76)(slides follow)

Let’s draw the distributions by hand

first!

Example 1.25, page 75

TI-83 Calculator Command: Distr|normalcdfSyntax: normalcdf(left, right, mu, sigma) = area under curve from left to right

mu defaults to 0, sigma defaults to 1Infinity is 1E99 (use the EE key), Minus Infinity is -1E99

Example 1.26, page 76

Let’s find the same probabilities using z-scores!

On the TI-83: normalcdf(720,820,1026,209)

The Inverse Problem:Given a normal density proportion or

probability, find the corresponding z-score!

What is the z-score such that 90% of the data has a z-score less than that z-score?

(1) Draw picture!(2) Understand what you are solving for!(3) Solve approximately! (we will also use the invNorm

key on the next slide)

Now try working Example 1.30 page 79!(slide follows)

TI-83: Use Distr|invNorm

Syntax:invNorm(area,mu,sigma) gives value of x with area to left of x under normal curve with mean mu and standard deviation sigma.

invNorm(0.9,505,110)=?invNorm(0.9)=?

Page 79

How can we use our TI-83s to solve this??

How can we tell if our data is “approximately normal?”

Box plots and histograms should show essentially symmetric, unimodal data. Normal Quantile plots are also used!

Histogram and Normal Quantile Plot for Breaking Strengths (in pounds) of Semiconductor Wires

(Pages 19 and 81 of text)

Histogram and Normal Quantile Plot for Survival Time of Guinea Pigs (in days) in a Medical Experiment

(Pages 38 (data table), 65 and 82 of text)

Using Excel to Generate Plots

• Example Problem 1.30 page 35– Generate Histogram via Megastat– Get Numerical Summary of Data via Megastat

or Data Analysis Addin– Generate Normal Quantile Plot via Macro (plot

on next slide)

Normal Quantile plot for Problem 1.30 page 35

Extra Slides from Homework

• Problem 1.80• Problem 1.82• Problem 1.119• Problem 1.120• Problem 1.121• Problem 1.222• Problem 1.129• Problem 1.135

Problem 1.80 page 84

Problem 1.83 page 85

Problem 1.119 page 90

Problem 1.120 page 90

Problem 1.121 page 92

Problem 1.122 page 92

Problem 1.129 page 94

Problem 1.135 page 95-96