Chapters 1 & 3
Graphical Methods for Describing Data
What is statistics?
• the science of collecting, organizing, analyzing, and drawing conclusions from data
Why should one study statistics?
1. To be informed . . .a) Extract information from tables, charts
and graphsb) Follow numerical argumentsc) Understand the basics of how data
should be gathered, summarized, and analyzed to draw statistical conclusions
Can dogs help patients with
heart failure by reducing stress
and anxiety?
When people take a vacation do they really
leave work behind?
Why should one study statistics? (continued)
2. To make informed judgments
3. To evaluate decisions that affect your life
If you choose a particular major, what are your chances of finding
a job when you graduate?
Many companies now require drug screening as a condition of
employment. With these screening tests there is a risk of a false-
positive reading. Is the risk of a false result acceptable?
What is variability?
Suppose you went into a convenience store to purchase a soft drink. Does every can on the shelf contain exactly 12 ounces?
NO – there may be a little more or less in the various cans due to the variability that is inherent in the filling process.
In fact, variability is almost universal!
It is variability that makes life interesting!!
The quality, state, or degree of being variable or changeable.
If the Shoe Fits ...
The two histograms to the right display the distribution of heights of gymnasts and the distribution of heights of female basketball players. Which is which? Why?
Heights – Figure A
Heights – Figure B
If the Shoe Fits ...
Suppose you found a pair of size 6 shoes left outside the locker room. Which team would you go to first to find the owner of the shoes? Why?
Suppose a tall woman (5 ft 11 in) tells you see is looking for her sister who is practicing with a gym. To which team would you send her? Why?
The Data Analysis Process1. Understand the nature of the
problem
2. Decide what to measure and how to measure it
3. Collect data
4. Summarize data and perform preliminary analysis
5. Perform formal analysis
6. Interpret results
It is important to have a clear direction before gathering data.
It is important to carefully define the variables to be studied and to develop appropriate methods for
determining their values.
It is important to understand how data is collected because
the type of analysis that is appropriate depends on how the
data was collected!This initial analysis provides
insight into important characteristics of the data.
It is important to select and apply the appropriate inferential
statistical methodsThis step often leads to the formulation of new research
questions.
Suppose we wanted to know the average GPA of high school graduates in the nation this year.
We could collect data from all high schools in the nation.
What term would be used to describe “all high school
graduates”?
Population
• The entire collection of individuals or objects about which information is desired
• A census is performed to gather about the entire population
What do you call it when you collect data about the entire population?
GPA Continued:Suppose we wanted to know the average GPA of high school graduates in the nation this year.
We could collect data from all high schools in the nation.
Why might we not want to use a census here?
If we didn’t perform a census, what would we do?
Sample
• A subset of the population, selected for study in some prescribed manner
What would a sample of all high school graduates across the nation look like?
High school graduates from each state (region), ethnicity, gender, etc.
GPA Continued:Suppose we wanted to know the average GPA of high school graduates in the nation this year.
We could collect data from a sample of high schools in the nation.
Once we have collected the data, what would we do with
it?
Descriptive statistics• the methods of organizing &
summarizing data
• Create a graph
If the sample of high school GPAs contained 1,000 numbers, how could the data be organized or summarized?
• State the range of GPAs• Calculate the average
GPA
GPA Continued:Suppose we wanted to know the average GPA of high school graduates in the nation this year.
We could collect data from a sample of high schools in the nation.
Could we use the data from our sample to answer this
question?
Inferential statistics• involves making generalizations
from a sample to a populationBased on the sample, if the average GPA for high school graduates was 3.0, what generalization could be made?
The average national GPA for this year’s high school graduate is approximately 3.0.Could someone claim that the average
GPA for graduates in your local school district is 3.0?No. Generalizations based on the results of a sample can only be made back to the population from which the sample came from.
Be sure to sample from the population of interest!!
Variable • any characteristic whose value
may change from one individual to another
• Suppose we wanted to know the average GPA of high school graduates in the nation this year. Define the variable of interest.The variable of interest is the GPA of high school graduates
Is this a variable . . .The number of wrecks per week at the intersection
outside school? YES
Data• The values for a variable from
individual observations
For this variable . . .The number of wrecks per week at the intersection outside . . . What could observations be?
0, 1, 2, …
Two types of variables
categorical numerical
discrete continuous
Categorical variables• Qualitative
• Identifies basic differentiating characteristics of the population
Can you name any categorical variables?
Numerical variables• quantitative
• observations or measurements take on numerical values
• makes sense to average these values
• two types - discrete & continuous
Can you name any numerical variables?
Discrete (numerical)• Isolated points along a number
line
• usually counts of items
Continuous (numerical)• Variable that can be any value in
a given interval
• usually measurements of something
Identify the following variables:
1. the color of cars in the teacher’s lot
2. the number of calculators owned by students at your school
3. the zip code of an individual
4. the amount of time it takes students to drive to school
5. the appraised value of homes in your city
Categorical
Categorical
discrete numerical
Discrete numerical
Continuous numerical
Is money a measurement or a count?
Classifying variables by the number of variables in a data
setSuppose that the PE coach records the height of each student in his class.
Univariate - data that describes a single characteristic of the population
This is an example of a univariate data
Classifying variables by the number of variables in a data
setSuppose that the PE coach records the height and weight of each student in his class.
Bivariate - data that describes two characteristics of the population
This is an example of a bivariate data
Classifying variables by the number of variables in a data
setSuppose that the PE coach records the height, weight, number of sit-ups, and number of push-ups for each student in his class.
Multivariate - data that describes more than two characteristics (beyond the scope of this course)
This is an example of a multivariate data
Graphs for categorical data
Bar Chart
When to Use Categorical data
How to construct– Draw a horizontal line; write the
categories or labels below the line at regularly spaced intervals
– Draw a vertical line; label the scale using frequency or relative frequency
– Place equal-width rectangular bars above each category label with a height determined by its frequency or relative frequency
Bar Chart (continued)
What to Look For Frequently or infrequently
occurring categories
Collect the following data and then display the data in a bar chart:
What is your favorite ice cream flavor?
Vanilla, chocolate, strawberry, or other
Double Bar Charts
When to Use Categorical data
How to construct– Constructed like bar charts, but with two (or
more) groups being compared– MUST use relative frequencies on the
vertical axis– MUST include a key to denote the different
barsWhy MUST we use relative frequencies?
Each year the Princeton Review conducts a survey of students applying to college and of parents of college applicants. In 2009, 12,715 high school students responded to the question “Ideally how far from home would you like the college you attend to be?” Also, 3007 parents of students applying to college responded to the question “how far from home would you like the college your child attends to be?” Data is displayed in the frequency table below.Frequency
Ideal Distance Students Parents
Less than 250 miles 4450 1594
250 to 500 miles 3942 902
500 to 1000 miles 2416 331
More than 1000 miles 1907 180
Create a comparative bar chart with these data.
What should you do first?
Relative Frequency
Ideal Distance Students Parents
Less than 250 miles .35 .53
250 to 500 miles .31 .30
500 to 1000 miles .19 .11
More than 1000 miles .15 .06
Found by dividing the frequency by the total number of students
Found by dividing the frequency by the total number of parents
What does this graph show about the ideal distance college should be from home?
Segmented (or Stacked) Bar Charts
When to Use Categorical data
How to construct– MUST first calculate relative frequencies– Draw a bar representing 100% of the group– Divide the bar into segments corresponding
to the relative frequencies of the categories
Relative Frequency
Ideal Distance Students Parents
Less than 250 miles .35 .53
250 to 500 miles .31 .30
500 to 1000 miles .19 .11
More than 1000 miles .15 .06
Remember the Princeton survey . . .
Create a segmented bar graph with these data.
First draw a bar that
represents 100% of the
students who
answered the survey.
Less than 250 miles
250 to 500 miles
500 to 1000 miles
More than 1000 miles
Relative Frequency
Ideal Distance Students Parents
Less than 250 miles .35 .53
250 to 500 miles .31 .30
500 to 1000 miles .19 .11
More than 1000 miles .15 .06
First draw a bar that
represents 100% of the
students who
answered the survey.
0.2
0.4
0.6
0.8
1.0
Rela
tive f
requency
Students
Next, divide the bar into segments.
Do the same thing for
parents – don’t forget a key
denoting each category
Parents
Notice that this segmented bar chart displays the same relationship
between the opinions of students and parents concerning the ideal distance
that college is from home as the double bar chart does.
Pie (Circle) ChartWhen to Use Categorical data
How to construct– Draw a circle to represent the entire data
set– Calculate the size of each “slice”:
Relative frequency × 360° – Using a protractor, mark off each slice
To describe – comment on which category had the largest
proportion or smallest proportion
Typos on a résumé do not make a very good impression when applying for a job. Senior executives were asked how many typos in a résumé would make them not consider a job candidate. The resulting data are summarized in the table below.
Number of Typos
Frequency
Relative Frequency
1 60 .40
2 54 .36
3 21 .14
4 or more 10 .07
Don’t know 5 .03
Create a pie chart for these data.
Number of Typos
Frequency
Relative Frequency
1 60 .40
2 54 .36
3 21 .14
4 or more 10 .07
Don’t know 5 .03
First draw a circle to
represent the entire data set.
Next, calculate the size of the
slice for “1 typo”
.40×360º =144º
Draw that slice.
Repeat for each slice.
Here is the completed pie chart created
using Minitab.
What does this pie chart tell us about the number of typos occurring in
résumés before the applicant would not be considered for a job?
Graphs for numerical data
Dotplot
When to Use Small numerical data sets
How to construct– Draw a horizontal line and mark it with an
appropriate numerical scale– Locate each value in the data set along the
scale and represent it by a dot. If there are two are more observations with the same value, stack the dots vertically
Dotplot (continued)
What to Look For – The representative or typical value– The extent to which the data values spread out– The nature of the distribution along the number line– The presence of unusual values
Collect the following data and then display the data in a dotplot:
How many body piercings do you have?
How to describe a numerical, univariate
graph
What strikes you as the most distinctive difference among the distributions of exam scores in
classes A, B, & C ?
1. Center
• discuss where the middle of the data falls
• three measures of central tendency–mean, median, & mode
The mean and/or median is typically reported rather than the
mode.
What strikes you as the most distinctive difference among the
distributions of scores in classes D,
E, & F?
2. Spread
• discuss how spread out the data is
• refers to the variability in the data
• Measure of spread are–Range, standard deviation, IQR
Remember,Range = maximum value – minimum
value
Standard deviation & IQR will be discussed in Chapter 4
What strikes you as the most distinctive difference among the distributions of exam scores in
classes G, H, & I ?
3. Shape
• refers to the overall shape of the distribution
The following slides will discuss these shapes.
Symmetrical
• refers to data in which both sides are (more or less) the same when the graph is folded vertically down the middle
• bell-shaped is a special type–has a center mound with two sloping tails
1. Collect data by rolling two dice and recording the sum of the two dice. Repeat three times.
2. Plot your sums on the dotplot on the board.
3. What shape does this distribution have?
Uniform
• refers to data in which every class has equal or approximately equal frequency
1. Collect data by rolling a single die and recording the number rolled. Repeat five times.
2. Plot your numbers on the dotplot on the board.
3. What shape does this distribution have?
To help remember the name for this shape,
picture soldier standing in
straight lines. What are they
wearing?
Skewed
• refers to data in which one side (tail) is longer than the other side
• the direction of skewness is on the side of the longer tail
1. Collect data finding the age of five coins in circulation (current year minus year of coin) and record
2. Plot the ages on the dotplot on the board.
3. What shape does this distribution have?
The directions are right skewed or left skewed.
Name a variable with a distribution that is skewed left.
Bimodal (multi-modal)
• refers to the number of peaks in the shape of the distribution
• Bimodal would have two peaks• Multi-modal would have more
than two peaksBimodal distributions can occur when the data set consist of observations
from two different kinds of individuals or objects.
Suppose collect data on the time it takes to drive from San Luis Obispo, California to Monterey, California. Some people may take the inland route (approximately 2.5 hours) while others may take the coastal route (between 3.5 and 4 hours).
What shape would this distribution have?
What would a distribution be called if it had ONLY one peak? Unimodal
3. Shape
• refers to the overall shape of the distribution
• symmetrical, uniform, skewed, or bimodal
What strikes you as the most distinctive difference among the
distributions of exam scores in class J ?
4. Unusual occurrences
• Outlier - value that lies away from the rest of the data
• Gaps
• Clusters
5. In context• You must write your answer in
reference to the context in the problem, using correct statistical vocabulary and using complete sentences!
Dotplot (continued)
What to Look For – The representative or typical value– The extent to which the data values spread out– The nature of the distribution along the number line– The presence of unusual values
Collect the following data and then display the data in a dotplot:
How many body piercings do you have?
Describe the distribution of the number of body
piercings the class has.
Numerical Graphs Continued
Stem-and-Leaf DisplaysWhen to Use Univariate numerical data
How to construct– Select one or more of the leading digits for
the stem– List the possible stem values in a vertical
column– Record the leaf for each observation beside
each corresponding stem value– Indicate the units for stems and leaves in a
key or legend
To describe – comment on the center, spread, and shape of
the distribution and if there are any unusual features
Each number is split into two parts:
Stem – consists of the first digit(s)Leaf - consists of the final digit(s)
Use for small to moderate sized
data sets. Doesn’t work well for large data sets.
Be sure to list every stem
from the smallest to the largest value
If you have a long lists of leaves behind a few
stems, you can split stems in order to
spread out the distribution.
Can also create comparative stem-and-leaf displays
Remember the data set collected in Chapter 1 – how many piercings do you have? Would a stem-and-leaf display be a
good graph for this distribution? Why or why not?
The following data are price per ounce for various brands of different brands of dandruff shampoo at a local grocery store.
0.32 0.21 0.29 0.54 0.17 0.28 0.36 0.23
Create a stem-and-leaf display with this data? Stem Leaf
1
2
3
4
5
What would an appropriate stem be?
List the stems vertically
For the observation of “0.32”, write the
2 behind the “3” stem.
2
Continue recording each leaf with the
corresponding stem 1 9
4
7
8
6
3
Describe this distribution.
The median price per ounce for dandruff
shampoo is $0.285, with a range of $0.37. The
distribution is positively skewed with an outlier at
$0.54.
The Census Bureau projects the median age in 2030 for the 50 states and Washington D.C. A stem-and-leaf display is shown below.
Notice that you really cannot see a distinctive
shape for this distribution due to the long list of
leaves
We can split the stems in order to better see the
shape of the distribution.
Notice that now you can see the
shape of this distribution.
We use L for lower leaf values (0-4) and H for higher leaf values (5-
9).
The following is data on the percentage of primary-school-aged children who are enrolled in school for 19 countries in Northern Africa and for 23 countries in Central African.
Northern Africa54.6 34.3 48.9 77.8 59.6 88.5 97.4 92.5 83.9 98.891.6 97.8 96.1 92.2 94.9 98.6 86.6 96.9 88.9
Central Africa58.3 34.6 35.5 45.4 38.6 63.8 53.9 61.9 69.9 43.085.0 63.4 58.4 61.9 40.9 73.9 34.8 74.4 97.461.0 66.7 79.6
Create a comparative stem-and-leaf display. What is an appropriate
stem?
Let’s truncate the leaves to the unit place.
“4.6” becomes “4”
Be sure to use comparative language when describing
these distributions!
The median percentage of primary-school-aged children enrolled in school is larger for countries in Northern Africa than in Central Africa, but the ranges are the same. The
distribution for countries in Northern Africa is strongly negatively skewed, but the
distribution for countries in Central Africa is approximately symmetrical.
HistogramsWhen to Use Univariate numerical
data
How to construct Discrete data―Draw a horizontal scale and mark it with the
possible values for the variable―Draw a vertical scale and mark it with frequency
or relative frequency―Above each possible value, draw a rectangle
centered at that value with a height corresponding to its frequency or relative frequency
To describe – comment on the center, spread, and shape of the
distribution and if there are any unusual features
Constructed differently for
discrete versus continuous data
For comparative histograms – use two separate graphs with the same scale on the horizontal axis
Queen honey bees mate shortly after they become adults. During a mating flight, the queen usually takes several partners, collecting sperm that she will store and use throughout the rest of her life. A study on honey bees provided the following data on the number of partners for 30 queen bees.
12 2 4 6 6 7 8 7 8 11 8 3 5 6 7 10 1 9 7 6 9 7 5 4 7 4 6 7 8 10
Create a histogram for the number of partners of the queen bees.
First draw a horizontal
axis, scaled
with the possible values of
the variable of interest.
Next draw a vertical
axis, scaled
with frequency or relative frequency.
Suppose we use relative frequency instead of frequency on the
vertical axis.
Draw a rectangle
above each value with a
height correspondin
g to the frequency.
What do you notice about the shapes of these two histograms?
0 1 2 3 4 5 6 7 8 9 10 11 120
1
2
3
4
5
6
7
HistogramsWhen to Use Univariate numerical
data
How to construct Continuous data―Mark the boundaries of the class intervals on the
horizontal axis―Draw a vertical scale and mark it with frequency
or relative frequency―Draw a rectangle directly above each class
interval with a height corresponding to its frequency or relative frequency
To describe – comment on the center, spread, and shape of the
distribution and if there are any unusual features
This is the type of histogram that most students are familiar with.
A study examined the length of hours spent watching TV per day for a sample of children age 1 and for a sample of children age 3. Below are comparative histograms.
Children Age 1 Children Age 3
Notice the common scale on the horizontal axis
Write a few sentences comparing the distributions.
The median number of hours spent watching TV per day was greater for the 1-
year-olds than for the 3-year-olds. The distribution for the 3-year-olds was more
strongly skewed right than the distribution for the 1-year-olds, but the two distributions had similar ranges.
Cumulative Relative Frequency Plot
When to use- used to answer questions about percentiles.
How to construct- Mark the boundaries of the intervals on the horizontal axis- Draw a vertical scale and mark it with relative frequency- Plot the point corresponding to the upper end of each interval with its cumulative relative frequency, including the beginning point- Connect the points.
Percentiles are a value with a given percent of
observations at or below that value.
The National Climatic Center has been collecting weather data for many years. The annual rainfall amounts for Albuquerque, New Mexico from 1950 to 2008 were used to create the frequency distribution below.
Annual Rainfall(in inches)
Relative frequency
Cumulative relative frequency
4 to <5 0.052
5 to <6 0.103
6 to <7 0.086
7 to <8 0.103
8 to <9 0.172
9 to <10 0.069
10 to < 11 0.207
11 to <12 0.103
12 to <13 0.052
13 to <14 0.052
Find the cumulative relative frequency for
each interval
0.052
0.155
0.241
+
+
Continue this pattern to
complete the table
The National Climatic Center has been collecting weather data for many years. The annual rainfall amounts for Albuquerque, New Mexico from 1950 to 2008 were used to create the frequency distribution below.
Annual Rainfall(in inches)
Relative frequency
Cumulative relative frequency
4 to <5 0.052 0.052
5 to <6 0.103 0.155
6 to <7 0.086 0.241
7 to <8 0.103 0.344
8 to <9 0.172 0.516
9 to <10 0.069 0.585
10 to < 11 0.207 0.792
11 to <12 0.103 0.895
12 to <13 0.052 0.947
13 to <14 0.052 0.999
In the context of this problem, explain
the meaning of this value.
In the context of this problem, explain
the meaning of this value.
Why isn’t this value one (1)?
To create a cumulative relative frequency plot, graph a point for the upper value
of the interval and the cumulative relative frequency
Plot a point for each interval. Plot a starting point at (4,0).
Connect the points.
2 4 6 8 10 12 14
0.2
0.4
0.6
0.8
1.0
Rainfall
Cum
ula
tive r
ela
tive f
requency
What proportion of years had rainfall amounts that were
9.5 inches or less?
Approximately 0.55
2 4 6 8 10 12 14
0.2
0.4
0.6
0.8
1.0
Rainfall
Cum
ula
tive r
ela
tive f
requency
Approximately 30% of the years had annual rainfall less than what amount?
Approximately 7.5 inches
2 4 6 8 10 12 14
0.2
0.4
0.6
0.8
1.0
Rainfall
Cum
ula
tive r
ela
tive f
requency
Which interval of rainfall amounts
had a larger proportion of
years –9 to 10 inches or 10 to 11 inches?
Explain The interval 10 to 11 inches, because its slope is steeper, indicating a larger proportion occurred.
Displaying Bivariate Numerical Data
ScatterplotsWhen to Use Bivariate numerical data
How to construct - Draw a horizontal scale and mark it with
appropriate values of the independent variable
- Draw a vertical scale and mark it appropriate values of the dependent variable
- Plot each point corresponding to the observations
To describe - comment the relationship between the variables
Scatterplots are discussed in much greater depth in
Chapter 5.
Time Series PlotsWhen to Use
- measurements collected over time at regular intervalsHow to construct
- Draw a horizontal scale and mark it with appropriate values of time
- Draw a vertical scale and mark it appropriate values of the observed variable
- Plot each point corresponding to the observations and connect
To describe - comment on any trends or patterns over time
Can be considered bivariate data where the y-variable is the
variable measured and the x-variable is time
The accompanying time-series plot of movie box office totals (in millions of dollars) over 18 weeks in the summer for 2001 and 2002 appeared in USA Today (September 3, 2002).
Describe any trends or patterns that you see.