1. Chapter 1: Exploring Data
Section 1.3
Describing Quantitative Data with Numbers
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE

2. 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

3. 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

4. Describing Quantitative Data
Measuring Center: The Mean
The most common measure of center is the ordinary arithmetic average, or mean.
Describing Quantitative Data
Definition:
To find the mean (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:
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:

5. Describing Quantitative Data
Alternate Example – McDonald’s Beef Sandwiches
Here are data for the amount of fat (in grams) for McDonald’s beef sandwiches:
Fat (g)
Hamburger
9 g
Cheeseburger
12 g
Double Cheeseburger
23 g
McDouble
19 g
Quarter Pounder®
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
Angus Mushroom & Swiss
40 g
McRib ®
Mac Snack Wrap
Describing Quantitative Data
0 9
1 2
2 34 3
3 99
4 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.
(b) The three Angus burgers are relatively new additions to the menu. How much did they increase the average when they were added?
The Angus burgers increased the mean amount of fat by 3.3 grams.

6. Describing Quantitative Data
Measuring Center: The Median
Another common measure of center is the median. In section 1.2, we learned that the median describes the midpoint of a distribution.
Describing Quantitative Data
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:
Arrange all observations from smallest to largest.
If the number of observations n is odd, the median M is the center observation in the ordered list.
If the number of observations n is even, the median M is the average of the two center observations in the ordered list.

7. Describing Quantitative Data
Measuring Center
Use the data below to calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers.
Describing Quantitative Data
Example, page 53 10 30 5 25 40 20 15 85 65 60 45
0 5
3 00 5 7
8 5
Key: 4|5 represents a New York worker who reported a 45- minute travel time to work.

8. Describing Quantitative Data
Alternate Example – McDonald’s Beef Sandwiches
Finding the median when n is odd. Here are the amounts of fat (in grams) for McDonald’s beef sandwiches, in order:
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 Chicken Sandwiches
Finding the median when n is even. Here is data for the amount of fat (in grams) for McDonald’s chicken sandwiches:
Describing Quantitative Data
0 99
2 03 8
Key: 2|3 represents 23 grams of fat in a McDonald’s chicken sandwich.
Fat
McChicken ®
16 g
Premium Grilled Chicken Classic
10 g
Premium Crispy Chicken Classic
20 g
Premium Grilled Chicken Club Sandwich
17 g
Premium Crispy Chicken Club Sandwich
28 g
Premium Grilled Chicken Ranch BLT
12 g
Premium Crispy Chicken Ranch BLT
23 g
Southern Style Crispy Chicken
Ranch Snack Wrap® (Crispy)
Ranch Snack Wrap® (Grilled)
Honey Mustard Snack Wrap® (Crispy)
Honey Mustard Snack Wrap® (Grilled)
9 g
Chipotle BBQ Snack Wrap® (Crispy)
15 g
Chipotle BBQ Snack Wrap® (Grilled)
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.

9. Describing Quantitative Data
Comparing the Mean and the Median
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.
Describing Quantitative Data
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.

10. Describing Quantitative Data
Measuring Spread: The Interquartile Range (IQR)
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.
Describing Quantitative Data
How to Calculate the Quartiles and the Interquartile Range
To calculate the quartiles:
Arrange the observations in increasing order and locate the median M.
The first quartile Q1 is the median of the observations located to the left of the median in the ordered list.
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

11. Describing Quantitative Data
Find and Interpret the IQR
Example, page 57
Describing Quantitative Data
Travel times to work for 20 randomly selected New Yorkers 10 30 5 25 40 20 15 85 65 60 45 5 10 15 20 25 30 40 45 60 65 85 5 10 15 20 25 30 40 45 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.

12. Describing Quantitative Data
Alternate Example – McDonald’s Beef Sandwiches
Alternate Example–McDonald’s Chicken Sandwiches
Here are the amounts of fat in the 15 McDonald’s beef sandwiches, in order.
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.
Describing Quantitative Data
Here are the 14 amounts of fat in order:
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.

13. Describing Quantitative Data
Identifying Outliers
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.
Describing Quantitative Data
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.
For these data, 1.5 x IQR = 1.5(27.5) = 41.25
Q x IQR = 15 – =
Q x IQR = = 83.75
Any travel time shorter than minutes or longer than minutes is considered an outlier.
0 5
3 00 5 7
8 5

14. Describing Quantitative Data
Alternate Example–McDonald’s Chicken Sandwiches
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:
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 Q IQR = (7) = Thus, the value 28 is an outlier.
Describing Quantitative Data

15. The 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.
Describing Quantitative Data
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

16. Describing Quantitative Data
Boxplots (Box-and-Whisker Plots)
The five-number summary divides the distribution roughly into quarters. This leads to a new way to display quantitative data, the boxplot.
Describing Quantitative Data
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.

17. Describing Quantitative Data
Construct a Boxplot
Consider our NY travel times data. Construct a boxplot.
Example
Describing Quantitative Data 10 30 5 25 40 20 15 85 65 60 45 5 10 15 20 25 30 40 45 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

18. Describing Quantitative Data
Alternate Example – The Previous Home Run
Here are the number of home runs that Hank Aaron hit in each of his 23 seasons:
Here is the data in order, with the 5 number summary identified:
The boundaries for outliers are 26 – 1.5(44 – 26) = -1 and (44 – 26) = 71, so there are no outliers.
Here is a boxplot of the data:
Describing Quantitative Data

19. Describing Quantitative Data
Measuring Spread: The Standard Deviation
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.
Describing Quantitative Data
Calculate the mean.
Calculate each deviation.
deviation = observation – mean
deviation: = -4
deviation: = 3
= 5

20. Describing Quantitative Data
Measuring Spread: The Standard Deviation
Describing Quantitative Data xi
(xi-mean)
(xi-mean)2 1
1 - 5 = -4
(-4)2 = 16 3
3 - 5 = -2
(-2)2 = 4 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
Sum=?
3) Square each deviation.
4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance.
5) Calculate the square root of the variance…this is the standard deviation.
“average” squared deviation = 52/(9-1) = This is the variance.
Standard deviation = square root of variance =

21. Describing Quantitative Data
Measuring Spread: The Standard Deviation
Describing Quantitative Data
Definition:
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.

22. Describing Quantitative Data
Alternate Example – Foot lengths
Here are the foot lengths (in centimeters) for a random sample of 7 fourteen-year-olds from the United Kingdom.
25, 22, 20, 25, 24, 24, 28
The mean foot length is 24 cm. Here are the deviations from the mean (xi – ) and the squared deviations from the mean (xi – )2:
Describing Quantitative Data x
xi –
(xi – )2 25
25 – 24 = 1
(1)2 = 1 22
22 – 24 = -2
(-2)2 = 4 20
20 – 24 = -4
(-4)2 = 16 24
24 – 24 = 0
(0)2 = 0 28
28 – 24 = 4
(4)2 = 16
Sum = 0
Sum = 38

23. Describing Quantitative Data
Choosing Measures of Center and Spread
We now have a choice between two descriptions for center and spread
Mean and Standard Deviation
Median and Interquartile Range
Describing Quantitative Data
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!

24. Describing Quantitative Data
Alternate Example – Who has more contacts-males or females?
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:
Female:
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.
Describing Quantitative Data

25. Describing Quantitative Data
Alternate Example – Who has more contacts-males or females?
Do: Here are the numerical summaries and boxplots for each distribution: 
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. N sx
min Q1 M Q3
max
Male 20
122.15
73.83 27
64.5
109.5
167.5
290
Female 14
126.71
65.75 22 83
129
180
218
Describing Quantitative Data

26. 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.

27. 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.

28. Looking Ahead… In the next Chapter…
We’ll learn how to model distributions of data…
Describing Location in a Distribution
Normal Distributions
In the next Chapter…