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1.3 Describing Quantitative Data with Numbers
Pages 48-73
Objectives
SWBAT:
1) Calculate measures of center (mean, median).
2) Calculate and interpret measures of spread (range,
IQR, standard deviation).
3) Choose the most appropriate measure of center and
spread in a given setting.
4) Identify outliers using the 1.5 X IQR rule.
5) Make and interpret boxplots of quantitative data.
6) Use appropriate graphs and numerical summaries to
compare distributions of quantitative variables.
• Measuring Center: The Mean
– 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
What is a resistant measure? Is the mean a resistant
measure of center?
• A resistant measure is a measure that can resist the
influence of extreme observations.
• Think about if we were going to calculate the mean
salary for students in this classroom.
• Let’s say Adam Sandler finds out he is one class short
of graduating high school, and that class happens to
be AP Statistics. He moves to Lyndhurst and transfers
into this class. What effect would his salary have on
the mean?
• What type of effect would it have on the median?
• Because the mean cannot resist the influence of
extreme observations, we say that it is not a resistant
measure of center. However, median is a resistant
measure of center.
How can you estimate the mean of a histogram or a
dotplot?
• The mean is the balancing point of the distribution, so
the mean of the deviations from the mean will add to 0
and balance there.
• Where does it look like the balancing point of this
distribution will be?
Measuring Center: The Median
Another common measure of center is the median.
The median describes the midpoint of a distribution.
The median 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 is the
center observation in the ordered list.
3. If the number of observations n is even, the median is the
average of the two center observations in the ordered list.
As previously stated, the median is a resistant measure of
center.
How does the shape of a distribution affect the
relationship between mean and median?
There is a connection between the shape of a
distribution and the relationship between the mean
and median of the distribution.
• When a distribution is symmetric, the mean and
median will be approximately the same.
• When a distribution is skewed right, the mean will
be greater than the median.
• When a distribution is skewed left, the mean will
be smaller than the median.
• This distribution of stolen bases is skewed right,
with a median of 5, as noted on the histogram.
• It does not seem plausible that the balancing
point (mean) is also 5. Because the distribution is
stretched out to the right, the mean must be
greater than 5. Think of all the extremely values
that will pull the mean up.
• Two common measures of spread (variability) are
range and IQR.
What is range? Is it a resistant measure of spread?
• The range of a distribution is the distance
between the minimum value and the maximum
value.
• Do you think it is a resistant measure of spread?
Let’s go back to our Adam Sandler example.
• Range can be a bit deceptive if there is an
unusually high or unusually low value in a
distribution. It is not a resistant measure of
spread.
What are quartiles? How do you find them?
• Quartiles are the values that divide a distribution
into four groups of roughly the same size.
• Find the quartiles:
4
6
8
12 20 22 27
How To Calculate The Quartiles And The IQR:
To calculate the quartiles:
1.Arrange the observations in increasing order and locate the median.
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.
What is the interquartile range (IQR)? Is the IQR a resistant
measure of spread?
• The interquartile range (IQR) is a single number that
measures the range of the middle half of the
distribution, ignoring the values in the lowest quarter of
the distribution and the values in the highest quarter of
the distribution
The interquartile range (IQR) is defined
as:
IQR = Q3 – Q1
• Since IQR essentially discards the lowest and highest
25% of the distribution, any outliers would have a
minimal affect on IQR. As a result, we can state that IQR
is a resistant measure of spread.
Find and Interpret the IQR
Travel times for 20 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
Median = 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.
Here are data on the
amount of fat (in grams)
in 9 different McDonald’s
fish and chicken
sandwiches. Calculate
the median and the IQR.
Median=19
What is an outlier? How do you identify them? Are there any
outliers in the chicken/fish distribution?
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.
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.
Since no values fall below the boundary of 0.75 or above the
boundary of 38.75, the distribution contains no outliers.
Are there any outliers in the
beef distribution?
No values fall below 9, so there are no small outliers.
43 falls above 41, so the Double Quarter Pounder with
Cheese is an outlier.
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.
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
Median
Q3
Maximum
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.
How To Make A Boxplot:
• A central box is drawn from the first quartile (Q1) to the
third quartile (Q3).
• A line in the box marks the median.
• Lines (called whiskers) extend from the box out to the
smallest and largest observations that are not outliers.
• Outliers are marked with a special symbol such as an
asterisk (*).
Construct a Boxplot
Consider our New York travel time data:
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
Median = 22.5
Q3= 42.5
Max=85
Recall, this is an
outlier by the
1.5 x IQR rule
• Note: When describing the shape of a distribution
using a boxplot, we cannot see peaks, so we can only
describe it using symmetric, skewed left, or skewed
right.
• To make a boxplot with the TI, see the technology
corner videos in Chapter 1 on the book website, or
page 59 of the textbook.
• In the distribution above, how far are the values
from the mean, on average?
• The concept of mean absolute deviation is similar to
standard deviation.
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.
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
Measuring Spread: The Standard Deviation
(xi-mean)2
xi
(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 =
Sum=?
This is the variance.
6.5 = 2.55
• A few words on standard deviation:
– It measures the average distance of observations
from their mean.
– The formula for standard deviation and variance
are on the formula sheet. However, the
calculations can be done right on the 84.
– For now, just know that variance is standard
deviation squared.
What are some similarities and differences
between range, IQR, and standard deviation?
• Similarity: All three measure variability.
• Differences:
– only IQR is resistant to outliers
– only standard deviation uses all the data
What are some properties of standard deviation?
• SD measures the spread about the mean and
should be used only when the mean is chosen as
the center (if median is chosen, use IQR).
• SD is always greater than or equal to 0. [A SD of 0
would mean all observations are the same value.]
• SD has the same units of measurement as the
original observations.
• SD is not resistant to outliers. A few outliers can
make SD very large.
A random sample of 5 students was asked how many
minutes they spent doing HW the previous night. Here are
their responses (in minutes): 0, 25, 30, 60, 90. Calculate
and interpret the standard deviation.
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
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!
Organizing a Statistical Problem
As you learn more about statistics, you will
be asked to solve more complex problems.
Here is a four-step process you can follow.
How to Organize a Statistical Problem: A Four-Step Process
•State: What’s the question that you’re trying to answer?
•Plan: How will you go about answering the question? What statistical
techniques does this problem call for?
•Do: Make graphs and carry out needed calculations.
•Conclude: Give your conclusion in the setting of the real-world
problem.