Download Section 1.3 - Effingham County Schools

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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
Describing Quantitative Data

Center: The Mean
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 Measuring
What is the difference between

m
?
Examples: “all women in the Bay Area between the ages of 18 and
54.” “House cats raised as pets in American households.” “The light
bulbs we are manufacturing in this plant.”
Sample: Asample is a group of units selected from a larger group
(the population). By studying the sample it is hoped to draw valid
conclusions about the larger group.

Examples:“500 women in the Bay Area between the ages of 18 and
54.” “50 house cats drawn from households in San Francisco, CA.”
“10 light bulbs manufactured at this plant in the last year.”

Parameter: A parameter is a value, usually unknown (and which
therefore has to be estimated), used to represent a certain population
characteristic.

Statistic: A statistic is a quantity that is calculated from a sample of
data. It is used to give information about unknown values in the
corresponding population.
Population Parameters vs.
Sample Statistics
Population: A population is any entire collection of people, animals,
plants or things from which we may collect data. It is the entire group
we are interested in, which we wish to describe or draw conclusions
about.


x and
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
What is the difference between
x and
m
?
When we need to refer to the mean of the population,
we’ll use the symbol
m
Population Parameters vs.
Sample Statistics
x
refers to the mean of a sample. Most of the time, the
data we’ll encounter can be thought of as a sample from
some larger population.
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

A resistant measure is one that is not influenced by extreme
values in a data set.

Because the mean can not resist the influence of extreme values
(or outliers), we say that the mean is NOT a resistant measure.
Example: Class Test Scores
Calculate the mean of each data set.
71, 78, 82, 84, 85, 89, 92, 100
41, 78, 82, 84, 85, 89, 92, 100
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What is a resistant measure?
Resistant Measure

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
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 Measuring
Is the median a resistant measure?
Calculate the median of each data set.
71, 78, 82, 84, 85, 89, 92, 100
41, 78, 82, 84, 85, 89, 92, 100
Resistant Measure
Example: Class Test Scores
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

Symmetric = mean and median are about the same

Skewed Right = mean > median

Skewed Left = mean < median
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Applet:
http://bcs.whfreeman.com/tps4e/%23628644__666394__
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does the shape of a distribution affect the
relationship between the mean and the median?
Shape and Center
 How

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.
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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.
Measures of Spread:
•
•
•
Range
IQR
Standard Deviation
Describing Quantitative Data

Spread
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 Measuring
Spread: The Range
The difference between the highest value and the lowest
value in a data set

Example: Class Test Scores
Find the range of each class.
71, 78, 82, 84, 85, 89, 92, 100
41, 78, 82, 84, 85, 89, 92, 100

Is the range a resistant measure?
Describing Quantitative Data
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 Measuring
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The difference between the first and third quartiles in a data set
Before you can find the IQR, you need to find the quartiles
How to Calculate the Quartiles and the Interquartile Range
Quartiles: Points that divide the data set into 4 equal pieces
***Note: there are only 3 of them!
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)
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 Measuring
and Interpret the IQR
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 Find
Here is the data for the amount of fat (in grams) for McDonald’s
beef sandwiches. Calculate the median and the IQR.
9
12
19
19
19
Q1 = 19
23
24
26
26
M = 26
28
29
39
39
40
42
Q3= 39
IQR = Q3 – Q1
= 39 - 19
= 20
Interpretation: The range of the middle half of fat for McDonald’s
beef sandwiches is 20 grams.
Describing Quantitative Data
Example
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: McDonld’s Beef Sandwiches
In the data, we found Q1=19 grams, Q3=39 grams, and IQR=20 grams
For these data, 1.5 x IQR = 1.5(20) = 30
Q1 - 1.5 x IQR = 19 – 30 = -11
Q3+ 1.5 x IQR = 39 + 30= 69
Any beef sandwich with fat content higher than 69 g or lower than -11 g
would be an outlier. So, there are NOT any outliers in this data.
Try the example in your notes on the chicken sandwiches!
Describing Quantitative Data

Outliers
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 Identifying
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Five-Number Summary

What’s the point? 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.
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Boxplots (Box-and-Whisker Plots)
Describing Quantitative Data
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 Boxplot

Consider our beef sandwiches data. Construct a boxplot.
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Describing Quantitative Data
Example: McDonald’s Sandwiches
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Parallel Box Plots comparing the fat content in McDonald’s
beef and chicken sandwiches
Parallel Box Plots
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
Spread
Why measure spread?

We often want to know not only what the central tendency is in a
set of scores or values (i.e, the mean, the median or the mode),
we also want to know how bunched up or “spread out” the scores
are.
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 Measuring
Spread: The Standard Deviation
The most commonly used measure of spread is called the
standard deviation.

Standard deviation measures how far each observation is
from the mean.
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.
Describing Quantitative Data

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 Measuring
do you calculate standard deviation?
Consider the following data on the number of pets that 9 students own.
1) Calculate the mean.
2) Calculate each deviation.
deviation = observation – mean
deviation: 1 - 5 = -4
deviation: 8 - 5 = 3
x=5
Describing Quantitative Data
 How
Spread: The Standard Deviation
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 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= 0
Describing Quantitative Data
(xi-mean)2
xi
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 Measuring
Sum= 52
What is the variance? “average” squared deviation = 52/(9-1) = 6.5
Standard deviation = square root of variance =
6.5 = 2.55
What are the similarities and differences between range,
IQR and standard deviation?

Similarities: all are measures of variability

Differences: resistance to outliers, using all of the data

What are some properties of standard deviation?

Standard deviation is never negative.

Standard deviation is sensitive to outliers. A single outlier
can raise the standard deviation and in turn, distort the
picture of spread.

For data with approximately the same mean, the greater the
spread, the greater the standard deviation.

If all values of a data set are the same, the standard
deviation is zero (because each value is equal to the mean).
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

Standard Deviation: 34.71

Interpretation: On average, the distance between the amount
of time a student does homework and the mean amount of
time students spent doing homework is 34.71 minutes.
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Example: A random sample of 5 students was asked how
many minutes they spent doing homework the previous night.
Here are there responses (in minutes): 0, 25, 30, 60, 90.
Calculate and interpret the standard deviation.
Standard Deviation


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!
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Choosing Measures of Center and Spread
Describing Quantitative Data
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+ 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.
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Looking Ahead…
In the next Chapter…
We’ll learn how to model distributions of data…
•
Describing Location in a Distribution
•
Normal Distributions