Data analysis techniques

Copyright © 2017, 20013, and 2009, Pearson Education, Inc.
2-1
Exploring Data
with Graphs and
Numerical
Summaries
Chapter 2

2-2
Different Types of
Data
Section
2.1

2-3 Copyright © 2017, 20013, and 2009, Pearson Education, Inc.
A variable is any characteristic observed in a study.
Examples: high temperature, low temperature, cloud cover,
whether it rained, and number of centimeters of precipitation
A variable can be classified as either
 Categorical (in Categories), or
 Quantitative (Numerical)
Variable

A variable can be classified as categorical if each
observation belongs to one of a set of distinct categories.
Examples:
 Gender (Male or Female)
 Religious Affiliation (Catholic, Jewish, …)
 Type of Residence (Apartment, Condo, …)
 Belief in Life After Death (Yes or No)
Categorical Variable

A variable is called quantitative if observations on it take
numerical values that represent different magnitudes of the
variable.
Examples:
 Age
 Number of Siblings
 Annual Income
Quantitative Variable

For Quantitative variables: key features are the center and
variability (spread) of the data.
Example: What’s a typical annual amount of precipitation? Is
there much variation from year to year?
For Categorical variables: a key feature is the relative
number of observations in the various categories.
Example: What percentage of days were sunny in a given
year?
Main Features of Quantitative and Categorical
Variables

A quantitative variable is discrete if its possible values form
a set of separate numbers, such as 0,1,2,3,….
Discrete variables have a finite number of possible values.
Examples:
 Number of pets in a household
 Number of children in a family
 Number of foreign languages spoken by an individual
Discrete Quantitative Variable

A quantitative variable is continuous if its possible values
form an interval.
Continuous variables have an infinite number of possible
values.
Examples:
 Height/Weight
 Age
 Amount of time to complete an assignment
Continuous Quantitative Variable

The distribution of a variable describes how the
observations fall (are distributed) across the range of
possible values.
Graphs and frequency tables are used to look for key
features of a distribution.
Distribution of a Variable

A frequency table is a listing of possible values for a
variable, together with the number of observations and/or
relative frequencies for each value.
Frequency Table
Table 2.1 Frequency of Shark Attacks in Various Regions for 2004–2013

Proportion & Percentage (Relative
Frequencies)
The proportion of observations falling in a certain category
is the number of observations in that category divided by the
total number of observations.
proportion =
frequency of that class
sum of all frequencies
The percentage is the proportion multiplied by 100.
Proportions and percentages are also called relative
frequencies.

There were 203 reported shark attacks in Florida between
2004 and 2013. Table 2.1 classifies 689 shark attacks
reported from 2004 through 2013, so, for Florida,
 203 is the frequency.
 203/689 = 0.295 is the proportion and relative
frequency.
 29.5 is the percentage 0.295 *100= 29.5%.
Frequency, Proportion, & Percentage
Example

To show the distribution for a discrete quantitative
variable, we would similarly list the distinct values and the
frequency of each one occurring.
Discrete Quantitative Variable

For a continuous quantitative variable (or when the
number of possible outcomes is very large for a discrete
variable), we divide the numeric scale on which the variable
is measured into as set of nonoverlapping intervals and
count the number of observations that fall in each interval.
The frequency table then shows these intervals together with
the corresponding count.
Continuous Quantitative Variable

2-15
Section
2.2
Graphical
Summaries of
Data

The two primary graphical displays for summarizing a
categorical variable are the pie chart and the bar graph.
Pie Chart: A circle having a “slice of pie” for each category
Bar Graph: A graph that displays a vertical bar for each
category
Graphs for Categorical Variables

Pie Charts:
 Used for summarizing a categorical variable.
 Drawn as a circle where each category is represented
as a “slice of the pie”.
 The size of each pie slice is proportional to the
percentage of observations falling in that category.
Pie Charts

Example: Shark Attacks in the United States
Table 2.2 Unprovoked Shark Attacks in the U.S., 2004 - 2013

Figure 2.1 Pie Chart of Shark Attacks Across U.S. States. The label for each
slice of the pie gives the category and the percentage of attacks in a state. The slice that
represents the percentage of attacks reported in Hawaii is 13% of the total area of the pie.
Question Why is it beneficial to label the pie wedges with the percent?

Bar graphs are used for summarizing a categorical variable.
 Bar graphs display a vertical bar for each category.
 The height of each bar represents either counts
(“frequencies”) or percentages (“relative frequencies”) for
that category.
 It is usually easier to compare categories with a bar
graph rather than with a pie chart.
 Bar graphs are called Pareto charts when the
categories are ordered by their frequency, from the
tallest bar to the shortest bar.
Bar Graphs

Figure 2.2 Bar Graph of Shark Attacks Across U.S. States. Except for the Other
category, which is shown last, the bars are ordered from largest to smallest based on
the frequency of shark attacks. Question What is the advantage of ordering the bars
this way rather than alphabetically?

Dot Plot: shows a dot for each observation placed above its
value on a number line.
Stem-and-Leaf Plot: portrays the individual observations.
Histogram: uses bars to portray the data.
Graphs for Quantitative Variables

Dot plots are used for summarizing a quantitative variable.
To construct a dot plot
1. Draw a horizontal line.
2. Label it with the name of the variable.
3. Mark regular values of the variable on it.
4. For each observation, place a dot above its value on
the number line.
Dot Plots

Each observation is represented by a stem and a leaf.
To construct a stem-and-leaf plot
1. Sort the data in order from smallest to largest.
2. Place the stems in a column, starting with the
smallest.
3. Place a vertical line to their right.
4. On the right side of the vertical line, indicate each leaf
(final digit) that has a particular stem.
5. List the leaves in increasing order.
Stem-and-Leaf Plots

A histogram provides a versatile way to picture the
distribution of a large data set.
1. Divide the range of the data into intervals of equal width.
2. Count the number of observations in each interval,
creating a frequency table.
3. On the horizontal axis, label the values or the endpoints
of the intervals.
4. Draw a bar over each value or interval with height equal
to its frequency (or percentage), values of which are
marked on the vertical axis.
5. Label and title appropriately.
Steps for Constructing a Histogram

Example: Sodium in Cereals
Table 2.4 Frequency Table for Sodium in 20 Breakfast Cereals. The table
summarizes the sodium values using eight intervals and lists the number of
observations in each, as well as the proportions and percentages.

Figure 2.6 Histogram of Breakfast Cereal Sodium Values. The rectangular bar over
an interval has height equal to the number of observations in the interval.
Histogram for Sodium in Cereals

How do we decide which to use? Here are some guidelines:
Dot-plot and stem-and-leaf plot
 More useful for small data sets
 Data values are retained
Histogram
 More useful for large data sets
 Most compact display
 More flexibility in defining intervals
Choosing a Graph Type

Overall pattern consists of center, spread, and shape.
 Assess where a distribution is centered by finding the
median (50% of data below median and 50% of data
above).
 Assess the spread of a distribution.
 Shape of a distribution: roughly symmetric, skewed to
the right, or skewed to the left
Interpreting Histograms

Shape
A distribution is unimodal
if it has a single mound or
peak.
A distribution is bimodal
if it has two distinct
mounds.
The value that occurs the most often is called the mode.

A distribution is skewed
to the left if the left tail is
longer than the right tail.
Shape
A distribution is skewed to
the right if the right tail is
longer than the left tail.
Symmetric Distributions: if both left and
right sides of the histogram are mirror images of
each other.

Examples of Skewness

A data set collected over time is called a time series.
Time plots are used to display time series data graphically.
 Plots each observation on the vertical scale against the
time it was measured on the horizontal scale.
 Common patterns in the data over time, known as
trends, should be noted.
 To see a trend more clearly, it is beneficial to connect
the data points in their time sequence.
Time Plots

The figure shows a time plot of the incidence rate from1980 to
2012, based on numbers reported by the Center for Disease
Control and Prevention.
Time Plot Example

2-35
Section
2.3
Measuring the
Center of
Quantitative Data

The mean is the sum of the observations divided by the
number of observations.
Mean


n
x
x

Example: Center of the Cereal Sodium Data
We find the mean by adding all the observations and then
dividing this sum by the number of observations, which is 20:
0, 340, 70, 140, 200, 180, 210, 150, 100, 130,
140, 180, 190, 160, 290, 50, 220, 180, 200, 210
Mean = (0 + 340 + 70 + . . . +210)/20 = 3340/20 =167

The median is the middle value of the observations when they are
ordered from the smallest to the largest (or from the largest to
smallest).
How to Determine the Median:
Put the n observations in order of their size.
If the number of observations, n, is:
 odd, then the median is the middle observation.
 even, then the median is the average of the two middle
observations.
Median

An outlier is an observation that falls well above or well below
the overall bulk of the data.
Outlier

CO2 pollution levels in 9 largest nations measured in metric
tons per person:
The CO2 values have n = 9 observations. The ordered
values are: 0.3, 0.4, 0,8, 1.4, 1.8, 2.1, 5.9, 11.6, 16.9.
Example: CO2 Pollution

Since n is odd, the median is the middle value, 1.8.
The relatively high value of 16.9 falls well above the rest of the
data. It is an outlier.
The size of the outlier affects the calculation of the mean but not
the median.
Example: CO2 Pollution

The shape of a distribution influences whether the mean is
larger or smaller than the median.
 Perfectly symmetric, the mean equals the median.
 Skewed to the left, the mean is smaller than the
median.
 Skewed to the right, the mean is larger than the
median.
Comparing the Mean and Median

In a skewed distribution, the mean is farther out in the long tail
than is the median.
 For skewed distributions the median is preferred because
it better represents what is typical.
Figure 2.9 Relationship Between the Mean and Median. Question: For skewed
distributions, what causes the mean and median to differ?
Comparing the Mean and Median

A numerical summary measure is resistant if extreme
observations (outliers) have little, if any, influence on its
value.
 The Median is resistant to outliers.
 The Mean is not resistant to outliers.
Resistant Measures

Mode
 Value that occurs most often.
 Highest bar in the histogram.
 The mode is most often used with categorical data.
The Mode

2-46
Section
2.4
Measuring the
Variability of
Quantitative Data

One way to measure the variability of a distribution is to
calculate the range.
The range is the difference between the largest and
smallest values in the data set:
Range = maximum value – minimum value
The range is simple to compute and easy to understand,
but it uses only the extreme values and ignores the other
values. Therefore, it’s affected severely by outliers.
Range

The deviation of an observation x from the mean 𝑥 is
(𝑥 − 𝑥), the difference between the observation and the
sample mean.
 Each observation has a deviation from the mean.
 A deviation is positive if the value falls above the mean
and negative if the value falls below the mean.
 The sum of the deviations for all the values in a data
set is always zero.
Standard Deviation

The deviation of an observation x from the mean 𝑥 is
(𝑥 − 𝑥), the difference between the observation and the
sample mean.
 Summary measures of variability from the mean use
either the squared deviations or their absolute values.
 The average of the squared deviations is called the
variance.
 The symbol (𝑥 − 𝑥)2 is called a sum of squares.
Standard Deviation

For the cereal sodium values, the mean is 𝑥 = 167. The
observation of 210 for Honeycomb has a deviation of
210 - 167 = 43. The observation of 50 for Honey Smacks
has a deviation of 50 – 167 = -117. Figure 2.11 shows
these deviations.
Figure 2.11 Dot Plot for Cereal Sodium Data, Showing Deviations for Two
Observations. Question: When is a deviation positive and when is it negative?
Standard Deviation Example

Gives a measure of variation by summarizing the deviations
of each observation from the mean and calculating an
adjusted average of these deviations.
The larger the standard deviation, s, the greater the
variability of the data.
The Standard Deviation s of n
Observations
1
)
( 2




n
x
x
s

Standard Deviation Example
Women’s and Men’s Ideal Number of Children
Men: 0, 0, 0, 2, 4, 4, 4 Women: 0, 2, 2, 2, 2, 2, 4
Both men and women have a mean of 2 and a range of 4.
The standard deviation for men is
𝑠 =
(𝑥−𝑥)2
𝑛−1
=
24
6
= 2.0.
The standard deviation for women is 1.2.

The most basic property of the standard deviation is this:
The larger the standard deviation S, the greater the variability of the
data.
 S measures the spread of the data.
 S = 0 only when all observations have the same value, otherwise
S > 0. As the spread of the data increases, S gets larger.
 S has the same units of measurement as the original
observations. The variance = S 2 has units that are squared.
 S is not resistant. Strong skewness or a few outliers can greatly
increase S.
Properties of the Standard Deviation

Magnitude of s: The Empirical Rule
If a distribution of data is bell-shaped, then approximately:
 68% of the observations fall within 1 standard deviation of the
mean, that is between the values of 𝑥 – s and 𝑥 + s
(denoted 𝑥 ± 𝑠).
 95% of the observations fall within 2 standard deviations of
the mean (𝑥 ± 2𝑠).
 All or nearly all observations fall within 3 standard deviations
of the mean (𝑥 ± 3𝑠).

Figure 2.12 The Empirical Rule. For bell-shaped distributions, this tells us approximately
how much of the data fall within 1, 2, and 3 standard deviations of the mean. Question:
About what percentage would fall more than 2 standard deviations from the mean?
Magnitude of s: The Empirical Rule

2-56
Section
2.5
Using Measures
of Position to
Describe
Variability

The th percentile is a value such that percent of the
observations fall below or at that value.
Percentile
p p

Quartiles
Figure 2.14 The Quartiles Split the Distribution Into Four Parts. 25% is below the first
quartile (Q1), 25% is between the first quartile and the second quartile (the median, Q2), 25%
is between the second quartile and the third quartile (Q3), and 25% is above the third quartile.
Question: Why is the second quartile also the median?

SUMMARY: Finding Quartiles
 Arrange the data in order.
 Consider the median. This is the second quartile, Q2.
 Consider the lower half of the observations (excluding
the median itself if n is odd). The median of these
observations is the first quartile, Q1.
 Consider the upper half of the observations (excluding
the median itself if n is odd). Their median is the third
quartile, Q3.

Consider the sodium values for the 20 breakfast cereals. What are the
quartiles for the 20 cereal sodium values? From Table 2.3, the sodium values, in
ascending order, are:
The median of the 20 values is the average of the 10th and 11th
observations, 180 and 180, which is Q2 = 180 mg.
The first quartile Q1 is the median of the 10 smallest observations (in the
top row), which is the average of 130 and 140, Q1 = 135 mg.
The third quartile Q3 is the median of the 10 largest observations (in the
bottom row), which is the average of 200 and 210, Q3 = 205 mg.
Example: Cereal Sodium Data

The interquartile range is the distance between the third
quartile and first quartile:
IQR = Q3  Q1
IQR gives the spread of middle 50% of the data.
The Interquartile Range (IQR)
In Words: If the interquartile range of U.S. music teacher salaries equals $16,000, this
means that for the middle 50% of the distribution stretches over a distance of $16,000.

Examining the data for unusual observations, such as
outliers, is important in any statistical analysis. Is there a
formula for flagging an observation as potentially being an
outlier?
The 1.5 x IQR Criterion for Identifying Potential Outliers
An observation is a potential outlier if it falls a distance of
more than 1.5 x IQR below the first quartile or a distance of
more than 1.5 x IQR above the third quartile.
Detecting Potential Outliers

The five-number summary is the basis of a graphical display
called the box plot, and consists of
 Minimum value
 First Quartile
 Median
 Third Quartile
 Maximum value
Five-Number Summary of Positions

 A box goes from Q1 to Q3.
 A line is drawn inside the box at the median.
 A line goes from the lower end of the box to the
smallest observation that is not a potential outlier and
from the upper end of the box to the largest
observation that is not a potential outlier.
 The potential outliers are shown separately.
SUMMARY: Constructing a Box Plot

Figure 2.15 shows a box plot for the sodium values. Labels
are also given for the five-number summary of positions.
Example: Box Plot for Cereal Sodium
Data
Figure 2.15 Box Plot and Five-Number Summary for 20 Breakfast Cereal Sodium Values.
The central box contains the middle 50% of the data. The line in the box marks the median.
Whiskers extend from the box to the smallest and largest observations, which are not identified
as potential outliers. Potential outliers are marked separately. Question: Why is the left whisker
drawn down only to 50 rather than to 0?

Side-by-Side Box Plots Help to Compare
Groups
A box plot does not portray certain features of a distribution, such as
distinct mounds and possible gaps, as clearly as does a histogram.
Box plots are useful for identifying potential outliers.
Figure 2.16 Box Plots of Male and Female College Student Heights. The box plots use the
same scale for height. Question: What are approximate values of the quartiles for the two
groups?

The z-score also identifies position and potential outliers.
The z-score for an observation is the number of standard
deviations that it falls from the mean. A positive z-score indicates
the observation is above the mean. A negative z-score indicates
the observation is below the mean. For sample data, the z -score
is calculated as:
An observation from a bell-shaped distribution is a potential outlier
if its z-score < -3 or > +3 (3 standard deviation criterion).
Z-Score
observation mean
.
standard deviation
z



2-68
Section
2.6
Recognizing and
Avoiding
Misuses of
Graphical
Summaries

 Label both axes and provide proper headings.
 To better compare relative size, the vertical axis should start
at 0.
 Be cautious in using anything other than bars, lines, or points.
 It can be difficult to portray more than one group on a single
graph when the variable values differ greatly. Consider
instead using separate graphs, or plotting relative sizes such
as ratios or percentages.
Summary: Guidelines for Constructing
Effective Graphs

Example: Beware of Poor Graphs
Figure 2.18 An Example of a Poor Graph. Question: What’s misleading about the
way the data are presented?

Figure 2.19 A Better Graph for the Data in Figure 2.18. Question: What trends do
you see in the enrollments from 2004 to 2012?
Example: Beware of Poor Graphs

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