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Chapter 1: Exploring Data
Section 1.3
Describing Quantitative Data with Numbers
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE
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Chapter 1
Exploring Data
 Introduction:
Data Analysis: Making Sense of Data
 1.1
Analyzing Categorical Data
 1.2
Displaying Quantitative Data with Graphs
 1.3
Describing Quantitative Data with Numbers
+ Section 1.3
Describing Quantitative Data with Numbers
Learning Objectives
After this section, you should be able to…

MEASURE center with the mean and median

MEASURE spread with standard deviation and interquartile range

IDENTIFY outliers

CONSTRUCT a boxplot using the five-number summary

CALCULATE numerical summaries with technology
The most common measure of center is the ordinary
arithmetic average, or mean.
Definition:
To find the mean x (pronounced “x-bar”) of a set of observations, add
their values and divide by the number of observations. If the n
observations are x1, x2, x3, …, xn, their mean is:
sum of observations
x1  x 2  ... x n
x

n
n


In mathematics, the capital Greek letter Σis short for “add
them all up.” Therefore, the formula for the mean can be
written in more compact notation:
x

x
n
i
Describing Quantitative Data

Center: The Mean
+
 Measuring
Example – McDonald’s Beef Sandwiches
(g)
Hamburger
9g
Cheeseburger
12 g
Double Cheeseburger
23 g
McDouble
19 g
Quarter Pounder®
19 g
Quarter Pounder® with Cheese
26 g
Double Quarter Pounder® with Cheese 42 g
Big Mac®
29 g
Big N' Tasty®
24 g
Big N' Tasty® with Cheese
28 g
Angus Bacon & Cheese
39 g
Angus Deluxe
39 g
Angus Mushroom & Swiss
40 g
McRib ®
26 g
Mac Snack Wrap
19 g
0
1
1
2
2
3
3
4
9
2
999
34
6689
99
02
Key: 2|3 represents 23
grams of fat in a
McDonald’s beef
sandwich.
(a) Find the mean amount of fat total for all 15 beef sandwiches.
x
Describing Quantitative Data
Here are data for the amount of fat (in grams) for McDonald’s beef
Fat
sandwiches:
+
 Alternate
9  12  ...19
 26.3 grams
15
(b) The three Angus burgers are relatively new additions to the menu.
How much did they increase the average when they were added?
x  23.0 grams
The Angus burgers increased the mean amount of fat by 3.3
grams.
Another common measure of center is the median. In
section 1.2, we learned that the median describes the
midpoint of a distribution.
Definition:
The median M is the midpoint of a distribution, the number such that
half of the observations are smaller and the other half are larger.
To find the median of a distribution:
1)Arrange all observations from smallest to largest.
2)If the number of observations n is odd, the median M is the center
observation in the ordered list.
3)If the number of observations n is even, the median M is the average
of the two center observations in the ordered list.
Describing Quantitative Data

Center: The Median
+
 Measuring
Use the data below to calculate the mean and median of the
commuting times (in minutes) of 20 randomly selected New
York workers.
Example, page 53
10
30
5
25
40
20
10
15
30
20
15
20
85
15
65
15
60
60
40
45
10  30  5  25  ... 40  45
x
 31.25 minutes
20
0
1
2
3
4
5
6
7
8
5
005555
0005
Key: 4|5
00
represents a
005
005
5
New York
worker who
reported a 45minute travel
time to work.
20  25
M
 22.5 minutes
2
Describing Quantitative Data

Center
+
 Measuring
Alternate Example – McDonald’s Chicken Sandwiches
Finding the median when n is even. Here is data for the amount of fat (in
grams) for McDonald’s chicken sandwiches:
0 99
McChicken ®
Premium Grilled Chicken Classic
Premium Crispy Chicken Classic
Premium Grilled Chicken Club Sandwich
Premium Crispy Chicken Club Sandwich
Premium Grilled Chicken Ranch BLT
Premium Crispy Chicken Ranch BLT
Southern Style Crispy Chicken
Ranch Snack Wrap® (Crispy)
Ranch Snack Wrap® (Grilled)
Honey Mustard Snack Wrap® (Crispy)
Honey Mustard Snack Wrap® (Grilled)
Chipotle BBQ Snack Wrap® (Crispy)
Chipotle BBQ Snack Wrap® (Grilled)
Fat
16 g
10 g
20 g
17 g
28 g
12 g
23 g
17 g
17 g
10 g
16 g
9g
15 g
9g
1
1
2
2
3
002
566777
03
8
Key: 2|3 represents 23
grams of fat in a
McDonald’s chicken
sandwich.
Describing Quantitative Data
Finding the median when n is odd. Here are the amounts of fat (in grams) for
McDonald’s beef sandwiches, in order:
9 12 19 19 19 23 24 26 26 28 29 39 39 40 42
The bold 26 in the middle of the list is the median M. There are 7 values below
the median and 7 values above the median.
+
Alternate Example – McDonald’s Beef Sandwiches
Find and interpret the median.
Since there are an even number of
values (14), there is no middle value.
The center pair of values, the bold 16s
in the stemplot, have an average of 16.
Thus, M = 16. About half of the chicken
sandwiches at McDonald’s have less than 16 grams of fat and about half have more
than 16 grams of fat.

The mean and median measure center in different ways, and
both are useful.

Don’t confuse the “average” value of a variable (the mean) with its
“typical” value, which we might describe by the median.
Comparing the Mean and the Median
The mean and median of a roughly symmetric distribution are
close together.
If the distribution is exactly symmetric, the mean and median
are exactly the same.
In a skewed distribution, the mean is usually farther out in the
long tail than is the median.
+
Comparing the Mean and the Median
Describing Quantitative Data


A measure of center alone can be misleading.
A useful numerical description of a distribution requires both a
measure of center and a measure of spread.
How to Calculate the Quartiles and the Interquartile Range
To calculate the quartiles:
1)Arrange the observations in increasing order and locate the
median M.
2)The first quartile Q1 is the median of the observations
located to the left of the median in the ordered list.
3)The third quartile Q3 is the median of the observations
located to the right of the median in the ordered list.
The interquartile range (IQR) is defined as:
IQR = Q3 – Q1
Describing Quantitative Data

Spread: The Interquartile Range (IQR)
+
 Measuring
and Interpret the IQR
+
 Find
Travel times to work for 20 randomly selected New Yorkers
10
30
5
25
40
20
10
15
30
20
15
20
85
15
65
15
60
60
40
45
5
10
10
15
15
15
15
20
20
20
25
30
30
40
40
45
60
60
65
85
Q1 = 15
M = 22.5
Q3= 42.5
IQR = Q3 – Q1
= 42.5 – 15
= 27.5 minutes
Interpretation: The range of the middle half of travel times for the
New Yorkers in the sample is 27.5 minutes.
Describing Quantitative Data
Example, page 57
+
Alternate Example – McDonald’s Beef Sandwiches
9
12 19 19 19 23 24 26 26 28 29 39 39 40 42
Since there are an odd number of observations, the median is the middle one, the
bold 26 in the list. The first quartile is the median of the 7 observations to the left of
the median, which is the highlighted 19. The third quartile is the median of the 7
observations to the right of the median, which is the highlighted 39. Thus, the
interquartile range is IQR = 39 – 19 = 20 grams of fat.
Alternate Example–McDonald’s Chicken Sandwiches
Here are the 14 amounts of fat in order:
9 9 10 10 12 15 16 16 17 17 17 20 23 28
Since there is no middle value, the median falls between the two middle values
(the two 16s). The first quartile is the median of the 7 values to the left of the
median (Q1 = 10) and the third quartile is the median of the 7 values to the right
of the median (Q3 = 17). Thus, IQR = 17 – 10 = 7 grams of fat. The range of the
middle 50% of amounts of fat for McDonald’s chicken sandwiches is 7 grams.
Describing Quantitative Data
Here are the amounts of fat in the 15 McDonald’s beef sandwiches, in order.
In addition to serving as a measure of spread, the
interquartile range (IQR) is used as part of a rule of thumb
for identifying outliers.
Definition:
The 1.5 x IQR Rule for Outliers
Call an observation an outlier if it falls more than 1.5 x IQR above the
third quartile or below the first quartile.
Example, page 57
In the New York travel time data, we found Q1=15
minutes, Q3=42.5 minutes, and IQR=27.5 minutes.
0
1
2
For these data, 1.5 x IQR = 1.5(27.5) = 41.25
3
Q1 - 1.5 x IQR = 15 – 41.25 = -26.25
4
Q3+ 1.5 x IQR = 42.5 + 41.25 = 83.75
5
Any travel time shorter than -26.25 minutes or longer than 6
7
83.75 minutes is considered an outlier.
8
5
005555
0005
00
005
005
5
Describing Quantitative Data

Outliers
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 Identifying
Problem: Determine whether the Premium Crispy Chicken Club
Sandwich with 28 grams of fat is an outlier.
Solution:
Here are the 14 amounts of fat in order:
9 9 10 10 12 15 16 16 17 17 17 20 23 28
Earlier we found that Q1 = 10, M = 16 and Q3 = 17.
Outliers are any values below Q1 – 1.5IQR = 10 – 1.5(7) = -0.5
and any values above Q3 + 1.5IQR = 17 + 1.5(7) = 27.5. Thus,
the value 28 is an outlier.
Describing Quantitative Data

+
Alternate Example–McDonald’s Chicken Sandwiches
+
Five-Number Summary

The minimum and maximum values alone tell us little about
the distribution as a whole. Likewise, the median and quartiles
tell us little about the tails of a distribution.

To get a quick summary of both center and spread, combine
all five numbers.
Definition:
The five-number summary of a distribution consists of the
smallest observation, the first quartile, the median, the third
quartile, and the largest observation, written in order from
smallest to largest.
Minimum
Q1
M
Q3
Maximum
Describing Quantitative Data
 The

The five-number summary divides the distribution roughly into
quarters. This leads to a new way to display quantitative data,
the boxplot.
How to Make a Boxplot
•Draw and label a number line that includes the
range of the distribution.
•Draw a central box from Q1 to Q3.
•Note the median M inside the box.
•Extend lines (whiskers) from the box out to the
minimum and maximum values that are not outliers.
+
Boxplots (Box-and-Whisker Plots)
Describing Quantitative Data

a Boxplot
+
 Construct
Consider our NY travel times data. Construct a boxplot.

10
30
5
25
40
20
10
15
30
20
15
20
85
15
65
15
60
60
40
45
5
10
10
15
15
15
15
20
20
20
25
30
30
40
40
45
60
60
65
85
Min=5
Q1 = 15
M = 22.5
Q3= 42.5
Max=85
Recall, this is
an outlier by the
1.5 x IQR rule
Describing Quantitative Data
Example
Example – The Previous Home Run
+
 Alternate
13 27 26 44 30 39 40 34 45 44 24 32 44 39 29 44 38 47
34 40 20 12 10
Here is the data in order, with the 5 number summary identified:
10 12 13 20 24 26 27 29 30 32 34 34 38 39 39 40 40 44 44 44 44 45 47
The boundaries for outliers are 26 – 1.5(44 – 26) = -1 and 44 + 1.5(44
– 26) = 71, so there are no outliers.
Collection 1
Box Plot
Here is a boxplot of the data:
10 15 20 25 30 35 40 45 50
HomeRuns
Describing Quantitative Data
Here are the number of home runs that Hank Aaron hit in each of his
23 seasons:

The most common measure of spread looks at how far each
observation is from the mean. This measure is called the
standard deviation. Let’s explore it!
Consider the following data on the number of pets owned by
a group of 9 children.
1) Calculate the mean.
2) Calculate each deviation.
deviation = observation – mean
deviation: 1 - 5 = -4
deviation: 8 - 5 = 3
x =5
Describing Quantitative Data

Spread: The Standard Deviation
+
 Measuring
Spread: The Standard Deviation
(xi-mean)
1
1 - 5 = -4
(-4)2 = 16
3
3 - 5 = -2
(-2)2 = 4
3) Square each deviation.
4
4 - 5 = -1
(-1)2 = 1
4) Find the “average” squared
deviation. Calculate the sum of
the squared deviations divided
by (n-1)…this is called the
variance.
4
4 - 5 = -1
(-1)2 = 1
4
4 - 5 = -1
(-1)2 = 1
5
5-5=0
(0)2 = 0
7
7-5=2
(2)2 = 4
8
8-5=3
(3)2 = 9
9
9-5=4
(4)2 = 16
5) Calculate the square root of the
variance…this is the standard
deviation.
Sum=?
“average” squared deviation = 52/(9-1) = 6.5
Standard deviation = square root of variance =
Describing Quantitative Data
(xi-mean)2
xi
+
 Measuring
Sum=?
This is the variance.
6.5  2.55
Spread: The Standard Deviation
The standard deviation sx measures the average distance of the
observations from their mean. It is calculated by finding an average of
the squared distances and then taking the square root. This average
squared distance is called the variance.
(x1  x ) 2  (x 2  x ) 2  ... (x n  x ) 2
1
variance = s 

(x i  x ) 2

n 1
n 1
2
x
1
2
standard deviation = sx 
(x

x
)

i
n 1
Describing Quantitative Data
Definition:
+
 Measuring
x
Example – Foot lengths
25, 22, 20, 25, 24, 24, 28
The mean foot length is 24 cm. Here are the deviations from the mean
(xi – x ) and the squared deviations from the mean (xi – x )2:
x
25
22
20
25
24
24
28
xi –
x
25 – 24 = 1
22 – 24 = -2
20 – 24 = -4
25 – 24 = 1
24 – 24 = 0
24 – 24 = 0
28 – 24 = 4
Sum = 0
( x i – x )2
(1)2 = 1
(-2)2 = 4
(-4)2 = 16
(1)2 = 1
(0)2 = 0
(0)2 = 0
(4)2 = 16
Sum = 38
The " average" squared deviation is
38
6.33, so the standard deviation
7 -1
6.33  2.52 cm. This 2.52 cm is
roughly th e average distance each
foot length is from the mean.
Describing Quantitative Data
Here are the foot lengths (in centimeters) for a random sample of 7
fourteen-year-olds from the United Kingdom.
+
 Alternate

We now have a choice between two descriptions for center
and spread

Mean and Standard Deviation

Median and Interquartile Range
Choosing Measures of Center and Spread
•The median and IQR are usually better than the mean and
standard deviation for describing a skewed distribution or a
distribution with outliers.
•Use mean and standard deviation only for reasonably
symmetric distributions that don’t have outliers.
•NOTE: Numerical summaries do not fully describe the
shape of a distribution. ALWAYS PLOT YOUR DATA!
+
Choosing Measures of Center and Spread
Describing Quantitative Data


The following data show the number of contacts that a sample of
high school students had in their cell phones. Feel free to use the
data below, but it will probably be more interesting for your students
if you collect and use their number of contacts.

Male: 124 41 29 27 44 87 85 260 290 31 168 169 167 214
135 114 105 103 96 144

Female: 30 83 116 22 173 155 134 180 124 33 213 218 183 110

State: Do the data give convincing evidence that one gender has
more contacts than the other?

Plan: We will compute numerical summaries, graph parallel
boxplots, and compare shape, center, spread and unusual values.
+
Alternate Example – Who has more contacts-males or females?
Describing Quantitative Data

Male
Female
20
14
122.15
126.71
73.83
65.75
27
22
64.5
83
109.5
129
167.5
180
290
218
Shape: The female distribution is approximately symmetric but the male
distribution is slightly skewed to the right.
Center: The median number of contacts for the females is slightly higher
than the median number of contacts for the males.
Spread: The distribution of contacts for males is more spread out than
the distribution of females since both the IQR and range is larger.
Outliers: Neither of the distributions have any outliers.
 Conclude: In the samples, females have more contacts than males,
since both the mean and median values are slightly larger. However,
the differences are small so this is not convincing evidence that one
gender has more contacts than the other.
+
Describing Quantitative Data
Alternate Example – Who has more contacts-males or females?
 Do: Here are the numerical summaries and boxplots for each
distribution:
N
sx
min
Q1
M
Q3
max
x
+ Section 1.3
Describing Quantitative Data with Numbers
Summary
In this section, we learned that…

A numerical summary of a distribution should report at least its
center and spread.

The mean and median describe the center of a distribution in
different ways. The mean is the average and the median is the
midpoint of the values.

When you use the median to indicate the center of a distribution,
describe its spread using the quartiles.

The interquartile range (IQR) is the range of the middle 50% of the
observations: IQR = Q3 – Q1.
+ Section 1.3
Describing Quantitative Data with Numbers
Summary
In this section, we learned that…

An extreme observation is an outlier if it is smaller than
Q1–(1.5xIQR) or larger than Q3+(1.5xIQR) .

The five-number summary (min, Q1, M, Q3, max) provides a
quick overall description of distribution and can be pictured using a
boxplot.

The variance and its square root, the standard deviation are
common measures of spread about the mean as center.

The mean and standard deviation are good descriptions for
symmetric distributions without outliers. The median and IQR are a
better description for skewed distributions.
+
Looking Ahead…
In the next Chapter…
We’ll learn how to model distributions of data…
•
Describing Location in a Distribution
•
Normal Distributions