Name: ___________________________________ Date: ____________ Period:______

CHAPTER 1: Exploring Data CHAPTER INTRODUCTION:

Statistics is the science of _________. We will soon be mastering the art of examining and

describing sets of data. This chapter introduces us to the concept of

_______________________ ___________ __________________. We will learn how to use a

variety of graphical tools to display data as well as how to describe data numerically. By the end

of the chapter, we should understand the difference between _________________________

and _____________________________ data, how to display and describe these two types of

data, and how to move from data analysis to inference. These skills are the basis of our entire

study of Statistics!

1.1-INTRO: Data Analysis-Making Sense of Data SECTION INTRODUCTION: In this section, you will learn basic terms and definitions necessary for our study of statistics.

Some of the vocabulary will be review; although we may define it more precisely then you have

in the past. Other vocabulary will be new. You will also be introduced to the idea of moving

from data analysis to inference. One of our goals in statistics is to use data from a

_________________________ sample to make ______________ about the population from

which the sample was drawn. In order to do this, we must be able to identify the type of data

we are dealing with as our choice of statistical ______________ will depend upon this

distinction.

1.1-INTRO KEY VOCABULARY AND CONCEPTS: Individuals, Variable, Categorical Variable,

Quantitative Variable, Distribution, Inference

1.1-INTRO CONCEPT 1: Individuals and Variables (Page 2)

Sets of Data contain information about _____________________. Individuals are the objects

described by a set of data and may be people, animals, or things. The characteristics of the

individuals that are measured are _____________________. Variables can take on different

values for different individuals. It is these values and the variation in them that we will be

learning how to study in this course.

1.1-INTRO CONCEPT 2: Types of Variables (Pages 3-5)

Variables can fall into one of two categories: ____________________ or __________________.

When the characteristic we measure places individuals into one of several groups, we have a

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categorical variable. When the characteristic we measure results in numerical values for which

it makes sense to find an average, we have a quantitative variable. Don’t forget to include

__________ for quantitative variables! The distinction between categorical and quantitative

variables is very important! The _______________ we use to _______________ and

_________________ data depends on the type of variable we are studying. One of the first

things we will learn is how to display and describe the __________________________ of a

variable. Sometimes we are concerned in drawing conclusions that go beyond the data set. That

is the idea of _______________.

Recall the table about determining inference from Chapter 4:

Were individuals randomly assigned to groups? Were individuals randomly selected?

Yes No

Yes Inference about the population?

Inference about the population?

Inference about cause and effect?

Inference about cause and effect?

No Inference about the population?

Inference about the population?

Inference about cause and effect? Inference about cause and effect?

1.1-INTRO EXAMPLE 1: Mrs. Kim gathered some information on her class and organized it in a

table like the one below:

Student Gender ACT Score Favorite Subject GPA James M 34 Statistics 3.89 Jen F 35 Biology 3.75 DeAnna F 32 History 4.00 Jonathan M 28 Literature 3.00 Doug M 33 Algebra 2.89 Sharon F 30 Spanish 3.25

(a) What individuals does this data set describe?

(b) What variables were measured? Identify each as categorical or quantitative.

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(c) Describe the distribution of the ACT scores.

(d) Could we infer from this data set that students who prefer math and science perform better

on the ACT? Explain.

1.1-INTRO EXAMPLE 2:

Jake is a car buff who wants to find out more about the vehicles that students at his school

drive. He gets permission to go to the student parking lot and record some data. Later, he does

some research about each model of car on the internet. Finally, Jake makes a spreadsheet that

includes each car’s model, year, color, number of cylinders, gas mileage, weight, and whether it

comes with a navigation system.

(a) Who or what are the individuals in Jake’s study?

(b) What variables did Jake measure? Identify each as categorical or quantitative.

Important Note: It is fairly common to transform a quantitative variable into a categorical

variable! For example, a teacher may look at a student’s test score (quantitative) and transform

the score into a letter grade (categorical).

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1.1: Analyzing Categorical Data

SECTION INTRODUCTION: In this section, you will learn how to describe and analyze categorical variables. You will

practice how to display data with ______ ____________, as well as how to describe these

displays. You will also practice describing pie charts, but will not be required to construct these

by hand. This section also teaches you how to _______________ the relationship between two

categorical variables using _____________________ and ____________________ distributions.

Finally, you will be introduced to a statistical problem solving strategy, the “4-Step” Process.

1.1 KEY VOCABULARY AND CONCEPTS: Frequency table, Relative frequency table, Pie chart, Bar

graph, Two-way table, Marginal distribution, Conditional distribution, Association, 4-Step Process

1.1-CONCEPT 1: Displaying Categorical Data (Pages 8-11)

A frequency table (or relative frequency table) displays the counts (or percents) of individual

that take on each value of a variable. Tables are sometimes difficult to read and don’t always

highlight important features of the distribution. __________________ displays of data are

much easier to read and often reveal interesting patterns and departures from patterns in the

distribution of data. We can use _______ _____________ or ________ _______________ to

display the distribution of _________________________ variables. Pie charts are not on the AP

Topic Outline, but we mention them here briefly because they are so common in the media.

Statisticians prefer ________ _____________ to pie charts because they are easier to read and

compare. When constructing graphical displays, we must be careful not to distort the

quantities. Beware of pictographs and watch the scales when displaying or reading graphs!

Important Features of Bar Graphs:

1. Bars should be __________________________________________________________

And must have _________________________________________________________

2. The horizontal axis should include the _______________________________________

_______________________________________________________________________

3. The vertical axis shows ____________________________________________________

and has a scale that _______________________________________________________.

If your smallest count is large, you may use a “zig-zag” to represent the space frequency

Between zero and your variable’s frequencies.

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1.1 EXAMPLE 1: A local business owner was interested in knowing the coffee-shop preferences

of her town’s residents. According to her survey of 250 residents, 75 preferred “Starbucks”, 50

preferred “Coffee Bean”, 38 chose “Morning Joe’s”, 50 said “One Mean Bean”, and 37 brewed

their own coffee. Construct a bar graph display.

1.1-CONCEPT 2: Two-Way Tables (Pages 12-21)

Bar graphs and pie charts are helpful when analyzing a _________________ categorical

variable. However, often we want to explore the relationship between ______ categorical

variables. To do this, we organize our data in a ______-______ ___________ with a row

variable and a column variable. The _______________ of the individuals in each intersecting

category make up the entries in the table.

When exploring a two-way table, you should start by describing each variable separately. This

can be done by describing the _________________________ distribution of the row or column

variable. The marginal distributions are the totals for the row variable only, or the column

variable only. The “Total” columns at the right or on the bottom of the table (the margins)

contain just that, the totals. If these are missing, your first step should be to write them in

yourself!

_______________ are often more informative than counts, especially when we are combining

groups of different sizes. We can display the marginal distribution in percents by dividing each

row total by the table total and converting it to a percent.

Conditional distributions of a variable describe the values of that variable among individuals

who have a specific value of ________________ variable. There is a separate _______________

distribution for each value of the other variable.

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1.1 EXAMPLE 2: A sample of 200 children from the United Kingdom aged 9-17 was selected

from the CensusAtSchool website. The gender of each student was recorded along with which

superpower they would most like to have: invisibility, super strength, telepathy, ability to fly, or

the ability to freeze time. Here are the results.

Superpower Female Male Total Percent

Invisibility 17 13 30

Super Strength 3 17 20

Telepathy 39 5 44

Ability to Fly 36 18 54

Freeze Time 20 32 52

Total 115 85 200

(a) What are the row and column variables?

(b) Use the data in the two-way table to calculate the marginal distribution (in percents) of

preferred super power. Fill in the table above.

(c) Make a graph to display the marginal distribution.

(d) Calculate the conditional distribution of preferred super power among female students.

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STRATEGY: The 4-Step Process

As we study statistics, we will encounter increasingly complex problems. In our book, we will

organize our thinking around four steps:

1. State: What is the question we are trying to answer?

2. Plan: What statistical technique does this problem call for?

3. Do: Make graphs and carry out the necessary calculations.

4. Conclude: Give a practical conclusion in the context of the situation.

When you encounter a statistical problem, think through these four steps. Always try to

______________ what it is you are trying to answer with the data. Think about what technique

would be most appropriate to answer that question. Carry out that technique, ______________

as much work as possible! Finally, state your conclusion in the ___________________ of the

problem. Never, never, never leave a simple numeric answer in a statistics course!

1.1 EXAMPLE 3: A survey of 1,000 randomly chosen residents of a Minnesota town asked

“Where do you prefer to purchase your daily coffee?” The two-way table below shows the

responses sorted by gender:

Preference Male Female Total National Chain 95 65 160

One Mean Bean 15 85 100

The Ugly Mug 145 25 170

Goodbye Blue Monday 170 90 260

Home-brewed 100 160 260

Don’t drink coffee 10 40 50

Total 535 465 1000

Based on the data, can we conclude that there is an association between gender and coffee

preferences? Use appropriate graphical and numerical evidence to support you conclusion.

Follow the 4-Step Process.

State:

Plan:

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Do:

Conclude:

1.2: Displaying Quantitative Data with Graphs SECTION INTRODUCTION: I this section, you will learn how to display quantitative data. We are also interested in

describing the distribution of the data by focusing on the SHAPE, OUTLIERS, CENTER, and

SPREAD (_________). We will use dotplots, stemplots, and histograms to help reveal the

distributions features. These basic plots will become handy throughout the course, so be sure

to master them in this section!

1.2 KEY VOCABULARY AND CONCEPTS: Dotplot, Shape, Center, Spread, Outliers, Symmertic,

Skewed to the Right, Skewed to the Left, Unimodal, Bimodal, Stemplot, Histogram

1.2-CONCEPT 1: Describe the SOCS! (Pages 27-30)

The reason we construct graphs of quantitative data is so that we can get a better

understanding of it. Constructing a plot helps us examine the data and identify its

__________________ ________________. When examining the distribution of a quantitative

variable, start by looking for the overall pattern and then describe any major

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_________________________ from it. Note its _________________. Is it symmetric? Is it

skewed? How many peaks does it have? What is its __________________? That is, roughly

what value would split the distribution in half? How variable are the data? Are the values

bunched up around the center, or are they ______________ out? Finally, are there

________________? Do any values fall far away from the rest of the distribution? The answers

to each of these questions are very important when describing a set of data. Be sure to address

all of them as you explore data sets through this course and don’t forget your SOCS

(________________, ___________________, _______________, and _________________)!

1.2 EXAMPLE 1: Describe the following distribution.

More on the 4 ways you must describe distributions:

1. Center: 2. Shape:

Symmetric

Skewed left/negatively or skewed right/positively:

Uniform Unimodal

Bimodal Multimodal

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3. Spread: 4. Unusual Values:

-Outliers: -Clusters: -Gaps:

1.2-CONCEPT 2: Dotplots and Stemplots (Pages 27-34)

Dotplots and stemplots (or stem and leaf plots) are some of the easiest graphs to construct,

especially if you have a __________ set of data. These plots are helpful for describing

distributions because they keep the data intact. That is, you can determine the individual data

values directly from the plot. When comparing two sets of data, we can construct __________-

____-__________ stemplots that share the same scale. Remember, when constructing plots,

ALWAYS label your axes and provide a key!

Important Features of Dot Plots:

1. Has only a ______________________________ axis.

2. Dots are ________________________________________________________________

3. It is very important to _____________________________________________________

1.2 EXAMPLE 2: A recent study by the Environmental Protection Agency (EPA) measured the

gas mileage (miles per gallon) for 30 models of cars. The results are below:

EPA-Measured MPG for 30 Cars

36.3 32.7 40.5 36.2 38.5 36.3 41.0 37.0 37.1 39.9

41.0 37.3 36.5 37.9 39.0 36.8 31.8 37.2 40.3 36.9

36.7 33.6 34.2 35.1 39.7 39.3 35.8 34.5 39.5 36.9

Construct a dotplot to display this distribution and describe the distribution.

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Important Features of Stem Plots:

1. Can be from a single set of data or ___________________________________________

and are constructed _______________________________________________________

2. The stem is _____________________________________________________________

and can be ______________________________________________________________

3. A leaf is _________________________________________________________________

and must be _____________________________________________________________

4. The plot must ___________________________________________________________

1.2 EXAMPLE 3: Recall the previous examples data set:

EPA-Measured MPG for 30 Cars

36.3 32.7 40.5 36.2 38.5 36.3 41.0 37.0 37.1 39.9

41.0 37.3 36.5 37.9 39.0 36.8 31.8 37.2 40.3 36.9

36.7 33.6 34.2 35.1 39.7 39.3 35.8 34.5 39.5 36.9

Construct a stemplot to display the data.

1.2-CONCEPT 3: Histograms (Pages 35-41)

When dealing with ________________ sets of data, dotplots and stemplots can be a bit

cumbersome and time-consuming to construct. In these cases, it may be easier to construct a

________________________. Instead of plotting each value in the data set, a histogram

displays the __________________ of the values that fall within equal-width ___________ (or

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ranges). Be sure not to confuse histograms with ______ ____________. Even though they look

similar, they have some major differences and describe different types of data!

Important Features of a Histogram: Note: Similar to bar graphs EXCEPT:

1. There are no _____________________________________________________________

2. The horizontal axis has _____________________________________________________

3. The vertical axis shows _____________________________________________________

________________________________________________________________________

1.2 EXAMPLE 4: The EPA expands its study of gas mileage to include a total of 50 models of

cars. The results are below:

EPA-Measured MPG for 30 Cars

36.3 32.7 40.5 36.2 38.5 36.3 41.0 37.0 37.1 39.9

41.0 37.3 36.5 37.9 39.0 36.8 31.8 37.2 40.3 36.9

36.7 33.6 34.2 35.1 39.7 39.3 35.8 34.5 39.5 36.9

36.9 41.2 37.6 36.0 35.5 32.5 37.3 40.7 36.7 32.9

42.1 37.5 40.0 35.6 38.8 38.4 39.0 36.7 34.8 38.1

(a) Construct a histogram for these data by hand. Describe the distribution.

Histogram:

CALCULATOR NOTES: Sorting Data

But first…let’s use our calculator to SORT our data! That will make this so much easier

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Description:

(b) Now, use your calculator to construct a histogram. Did you get the same graph? _______ If not, how

can you make your calculator match the histogram you constructed in part (a)?

1.3: Describing Quantitative Data with Numbers SECTION INTRODUCTION: In this section, you will learn how to use ___________________ ______________________ to

describe the ______________ and _______________ of a distribution of quantitative data. You

will also learn how to identify ________________________ in a distribution. Being able to

accurately describe a distribution and calculate and interpret numerical summaries forms the

foundation for our statistical study. By the end of this section, you will want to make sure you

are comfortable selecting appropriate numerical summaries, calculating them (by hand and by

using technology), and interpreting them for a set of quantitative data.

1.3 KEY VOCABULARY AND CONCEPTS: Mean, Median, Range, Quartiles, Interquartile Range

(IQR), Five-Number Summary, Boxplot, Standard Deviation, Variance, Resistant, 1.5 x IQR Rule

CALCULATOR NOTES: Constructing a Histogram

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1.3-CONCEPT 1: Exploratory Data Analysis (Pages 50)

Statistical tools and ideas can help you examine data in order to describe their main features.

This examination is called exploratory data analysis. To organize our exploration we want to:

Examine each variable first by itself, then move on to study relationships among the

variables.

Always always always plot your data, ____________!

Begin with a graph or graphs. Make sure to construct and interpret appropriate graphs

based on the data.

Add numeric summaries. For quantitative data, don’t forget your ___________!

Lastly, check for CONTEXT. Make sure your description always includes the context and

units of your problem.

1.3-CONCEPT 2: Exploratory Data Analysis (Pages 50-55)

The __________ and ________________ measure center in different ways. While the mean, or

__________________, is the most common measure of center, it is not always the most

appropriate. Extreme values can “pull” the mean towards them. The median, or ____________

value, is ____________________ to extreme values and is sometimes a better measure of

center. In ______________________ distributions, the mean and median will be approximately

equal. Always consider the __________________ of the distribution when deciding which

measure of center to use to describe your data!

1.3 EXAMPLE 1: Finding the median with odd and even amounts of data values.

(a) 5, 10, 10, 10, 20, 20, 25, 30, 55 (b) 5, 10, 10, 10, 20, 20, 25, 30

Mean and Median Relationships:

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1.3 EXAMPLE 2: Consider the following stemplot of the lengths of time (in seconds) it took

students to complete logic puzzles. Use it to answer the following questions.

1 5 8 2 2 3 2 6 7 7 7 7 8 8 8 8 9 9 9 3 2 3 4 4 3 6 8 4 0 1 1 2 2 3 4 6 5 0 0

(a) Based only on the plot, how does the mean compare to the median? How do you know?

(b) Calculate and interpret the mean. Calculate and interpret the median.

(c) Which measure of center would be the more appropriate summary of the center of this

distribution? Why?

(d) Use technology to find the mean and median, and then verify they are the same as the

results you found.

CALCULATOR NOTES: Using you calculator to find measures of center

Key: 1 5 = 15 seconds

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1.3-CONCEPT 3: Measures of Spread and Boxplots (Pages 55-55)

Like measures of center, we have several different ways to measure __________________. The

easiest way to describe the spread of a distribution is to calculate the _______________, found

by: maximum-minimum. However, extreme values can cause this measure of spread to be

much greater than the spread of the majority of values.

A measure of spread that is resistant to the effect of ____________________ is the

_________________________________ range or the IQR. To find this value, arrange the

observations from smallest to largest and determine the median. Then find the median of the

lower half of the data. This is the _________ ___________________ (Q1). The median of the

upper half of the data is called the __________ ___________________(Q3).The distance

between the first and third quartiles is the Interquartile Range (IQR). Not only does the IQR

provide a measure of spread, it also provides us with a way to identify outliers. According to the

1.5 x IQR rule, any value that falls more than 1.5xIQR above the third quartile or below the first

quartile is considered an outlier.

The minimum, maximum, median, and quartiles make up the “five-number summary”. This set

of numbers describes the center and spread of a set of quantitative data and leads to a useful

display-the boxplot (or box and whisker plot).

Another measure of spread that we will use to describe data is the ________________

________________________. The standard deviation measures roughly the average distance of

the observations from their mean. The calculation can be quite time-consuming to do by hand,

so we’ll rely on technology to provide the standard deviation for us. However, be sure to

understand how it is calculated! As the formula and process is often tested on the multiple

choice portion of the AP exam.

Standard Deviation: 2( )

1

ix xs

n

Variance:

2 2 22 1 2( ) ( ) ( )

1

nx x x x x xs

n

Where n = ___________________________________________________________________

ix = ___________________________________________________________________

x = ___________________________________________________________________

Notation Note: The sample mean is represented as ______ and the sample standard deviation is

represented as ______. The population mean is represented as ______ and the population

standard deviation is represented as _____. To find the population standard deviation we would

divide by _____. Rarely will we have the population data. Keep this notation straight, using

improper notation will guarantee you deductions in points on exams!

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1.3 EXAMPLE 3: The length (in pages) of Mr. Stan’s favorite books are noted below:

242 346 314 330 340 322 284 342 368 170 344 318 318 374 332

(a) Determine the 5 number summary for this data set and the IQR.

(b) Check for outliers.

(c) Use these data to construct a boxplot.

(d) Describe the distribution.

(e) Calculate and interpret the mean and standard deviation for these data.

Page | 18

1.3 EXAMPLE 4: The following data show the number of contacts that a sample of high school

students had in their cell phones. Do the data give convincing evidence that one gender has

more contacts than the other?

Males: 124 41 29 27 44 87 85 260 290 31 168 169 167 214 135 114 105 103 96 144 Females: 30 83 116 22 173 155 134 124 33 180 213 218 183 110

To help organize your response, first complete the table below:

n ̅ sx Min Q1 Med Q3 Max

Male

Female

Parallel Boxplots: use ONE SCALE for both plots!

Description: Describe and COMPARE both SOCS!

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Properties of Standard Deviation:

1. s measures spread about the ____________

2. s = 0 only when ___________________________________________________________

3. s, like x , is not _____________________. So a few outliers can make a big difference in these values!

When describing data numerically, you must always make sure to note a measure of center and

a measure of spread along with the shape and any outliers. If you choose the median as your

measure of center, you should use the _________ to describe the spread. If you choose the

mean to describe the center, use the ___________________ _____________________ to

measure the spread.

CHAPTER SUMMARY:

In this chapter, we learned that statistics is the art and science of data. When working with

data, it is important to know whether the variables are _______________________ or

__________________________ as this will determine the most appropriate displays for the

distribution. For categorical data, the display will help us describe the distribution. For

quantitative data, the display will help us describe the _____________ of the distribution and

suggest the most appropriate numeric measures of ________________ and _______________.

Always begin with the _____________ of the distribution, then move to a _________________

description. Which graph, numerical summary, etc. you choose will depend on the context of

the situation and the type of data you are dealing with. When exploring quantitative data, we

want to be sure to interpret the _______________, _______________________,

_________________ and __________________. Look for an overall pattern to describe your

data and not any striking departures from that pattern.

A recurring theme in our class will be to focus on understanding, not just the mechanics. While

it may be easy to “plug the data” into your calculator and generate plots and numerical

summaries, simply generating graphs and values is not the point of statistics. Rather, you

should focus your studies on being able to explain ________ a graph or values is constructed

and ________ you would choose a certain display or numerical summary. Get in this habit

early…your calculator is a powerful tool, but cannot replace your thinking and communication

skills!