Chapter 1: Exploring Data
AP Stats, 2010-2011
Questionnaire
“Please take a few minutes to answer the
following questions. I am collecting data for
my doctoral dissertation, which is on
characteristics of American private school
students. After you complete the
questionnaire, please return it in the enclosed
SASE. Thank you for your participation.”
Definitions, pp. 4-6

Individuals and variables


Categorical and quantitative


On questionnaire, what are the individuals and
what are the variables?
Which variables are categorical and which are
quantitative?
Distribution
Practice

From Questionnaire:

#11 (dot plot)

#12 (pie chart)

#13 (bar chart)
A really nice bar graph (I made this myself, so
just nod approvingly):
Percent of Total
Education Level in U.S. (adults age 25+)
50
40
33.1
25.4
30
20
25.6
15.9
10
0
No high
school
degree
High school 1-3 years of 4+ years of
only
college
college
Years of Schooling
Bar Graph, Figure 1.1 (p. 9)
Pie Chart, Figure 1.1 (p. 9)
Practice

Exercises 1.1, 1.2, 1.4, p. 7.
Dot Plot, Figure 1.3 (p. 11)
Interpreting the Dotplot

Shape, center, spread


Look for overall patterns and striking deviations
from that pattern.
Outliers


Individual observation(s) that falls outside the
overall pattern on a graph of a distribution.
In the next section, we will learn a mathematical
rule of thumb for deeming an observation an
outlier. For now, we’ll just talk in general terms.
Stemplots


Sometimes called “stem and leaf plots.”
Useful when there are a lot of data points, or
the range of values is large.
Dotplot?

What would a dotplot look like for these
data?
A stemplot (Figure 1.4, p. 14)
A stemplot (Figure 1.4, p. 14)—Split stems
How to create a stemplot

Example 1.5, p. 13

Rules of thumb:

Choosing the number of stems:




No magic number, but a minimum of 5 is good. Too few stems will
result in a skyscraper effect, too many make a pancake graph.
10 is a good starting point.
For data points with decimals, round the data so that the final digit
after rounding is suitable as a leaf.
Let’s try one:

Exercise 1.8, p. 17
Stemplot for 1.8 (StatCrunch)
Stem and Leaf
Variable: MPG
2 : 113444444
2 : 5556678888888888999
3 : 0002
Another Exercise

1.9, p. 17
Homework

Reading: Section 1.1 through p. 30.

Exercises: 1.10 and 1.11 (pp. 17-18)
Histograms


The most common way to display the
distribution of a quantitative variable.
How to make a histogram:



Example 1.6, p. 19
Read the interpretation of this graph, p.
20.
Choose between 6 and 15 classes (bars on
your graph)
Example 1.6, p. 19
Figure 1.7, p. 20
Notice y-axis: number
of values in a particular
class.
Notice the x-axis:
It is the variable of interest.
Practice Problem

1.14, p. 23
How to make a histogram with your calculator

Technology Toolbox, p. 21

Just enter raw data in L1,
then construct a histogram.


Read this carefully tonight.
Another way:

Summarize the data and put in, say, L2. Put the midpoint of the
class the data are in in L1.
Exercise 1.14, p. 23

Now, make a stemplot for these data.

Which do you prefer?
Homework

Reading: Through p. 34


Pay careful attention to Example 1.8, pp. 28-30:
how to create an ogive.
Exercise:

1.12 (p. 22)
Right Skew?
Practice Problems

Exercises:

1.13 (p. 23)

1.17 (p. 27)
Percentile


Would you rather score at the 70th or
95th percentile on the SATs?
If you scored at the 95th percentile,
what does that mean?
Ogive


Probably my favorite word to say in statistics.
Let’s practice saying it …
Used when we would like to see the relative
standing of an individual observation.

Does a histogram give us this?
Example 1.8, pp. 28-30

Look at the table on p. 29.


The two columns on the far left could be used to create a
histogram.
The fourth and fifth columns are of particular importance when we
want to construct an ogive.


Look over these briefly to see that you know where these data come
from.
Steps 2 and 3:

3: Plot a point corresponding to the relative cumulative frequency
in each class interval at the left endpoint of the next class interval.
Example 1.8, pp. 28-30
Practice problem

Exercise 1.19, p. 31


Create a frequency table.
Then, create cumulative frequency and
relative cumulative frequency columns.
Exercise 1.19, p. 31 (Data on p. 23)
Class
Freq.
3-5.99
6-8.99
9-11.99
12-14.99
15-17.99
18-20.99
Rel. Cum.
Cum. Freq. Freq. (%)
Exercise 1.19, p. 31
Class
Freq.
3-5.99
1
6-8.99
1
9-11.99
12
12-14.99
31
15-17.99
4
18-20.99
1
Rel. Cum.
Cum. Freq. Freq. (%)
Exercise 1.19, p. 31
Class
Freq.
3-5.99
1
6-8.99
1
9-11.99
12
12-14.99
31
15-17.99
4
18-20.99
1
Rel. Cum.
Cum. Freq. Freq. (%)
1
2
2
4
14
28
45
90
49
98
50
100
Relative Cum. Frequency
Ogive for Exercise 1.19
100
80
60
40
20
0
3
6
9
12
15
% of people 65 or older
18
21
Time plots

Used to plot the value of a variable vs.
the time in was measured.



Can detect seasonal variation, for instance
(See Figure 1.15, p. 32)
Used effectively in designed experiments.
Practice problem: Exercise 1.21, p. 33

Use your calculator—the line graph
function.
Figure 1.15 p. 32 (Example 1.9)
Time plot for Exercise 1.21
Homework

Reading: pp. 37-47

Exercise:


1.29, p. 36
1.1 Quiz on Friday

Probably 20-30 minutes
Section 1.1 Review Problems:
Displaying Distributions with Graphs

Exercises, pp. 34-36:

1.23, 1.24, 1.27, 1.28, 1.30
1.2 Describing Distributions with Numbers

Measuring center


Measuring spread





Mean, median
With quartiles: Inter-quartile range
Standard deviation (and variance)
Range
Statistical summaries
Boxplots
Measuring the Center of a Distribution

Mean


Numerical average
Median


Middle value in a data set, if an odd number of
values, or the average of the middle two values, if
an even number of values
Splits the distribution exactly in half
Practice

Exercise 1.14, p. 23

Create a histogram using your calculator


Discuss shape, center, and spread of the
distribution.
Calculate 1-variable statistics using your
calculator.

Discuss difference between the mean and
median.
Resistant Measure


The mean cannot resist the influence of
extreme observations and/or skew. The
median can, however.

Mean: not resistant

Median: Resistant
We generally prefer to use the median when
dealing with skewed distributions.
Powerful Numerical Summary


The 5-number summary …

min

Q1

Median (Q2)

Q3

Max
… Plus mean and standard deviation
The Boxplot

Let’s use the data from Exercise 1.14, p. 23 to
create a boxplot:

Now, use your calculator to create a modified
boxplot (box on page 46).

Outliers
Comparing Distributions


Draw side-by-side boxplots to compare
distributions.
Exercise 1.36, p. 47
Exercise 1.36, p. 47 (Statcrunch)
Column
n
Mean
Variance
Std. Dev.
Median
Range
Min
Women
18
141.05556
698.8791
26.436321
138.5
99
101
Men
20
121.25
1079.25
32.85194
114.5
117
70
Max
Q1
Q3
200
126
154
187
98
143
Measuring Spread

Calculating the central tendency of a
distribution is only half the story. We also
need to consider the spread.
Quartiles and IQR

The first quartile, Q1, is the median of the
first half of the data.


The third quartile, Q3, is the median of the
second half of the data.


25th percentile
75th percentile
Inter-Quartile Range:

IQR=Q3 - Q1
Outliers


Before, we said something is an outlier if it
looked to be one.
Now, we can use the IQR to create a
mathematical rule:
Low End:
xi  Q1  1.5 * IQR ?
High End:
xi  Q3  1.5 * IQR ?
Another Measure of Spread:
Standard Deviation


Probably the most important and mathematically useful
measure of spread.
Used along with mean.


Like the mean, it is not a resistant measure.
Calculating it (see p. 49): In your notes, write this out in
words.
n
s
2
(
x

x
)
 i
i 1
n 1
Variance

s2
Properties of Standard Deviation (p. 51)

Measures spread about the mean

Is greater than or equal to zero

Strong skewness or a few outliers can make s
very large.
More on Comparing Distributions



Example 1.17, p. 57
Also include numerical
summaries
Back-to-back stemplot:
HW


Reading through end of chapter
Exercises:


1.40, p. 52
1.47, p. 59
Effects of a linear transformation

Includes adding and/or multiplying by a constant.


Example 1.15, p. 53




Xnew=a+bxold
1. Enter salary data in L1. Compute summary stats.
2. Add a constant amount of 0.5 ($500,000) to each salary,
and put these new data in L2. Compute summary stats.
3. Now suppose each player receives a 10% raise (multiply
L1 by 1.10). Put these data in L3, then compute summary
stats.
Compare summary stats. What happened?
Effects of a linear transformation

Summary, p. 55:

Multiply by b:


Center (mean and median) and quartiles and
spread (s, IQR) multiply by b.
Add a:

Center (mean and median) and quartiles: add a

Spread (s, IQR): no change
Preparing for your chapter 1 test




Section 1.2 summary, p. 61
Chapter review, pp. 64-66
Exercise 1.55, p. 63
Exercises, pp. 66-72:





1.63
1.64
1.65
1.67
1.68