MAT 1000
Mathematics in Today's World
Last Time
1. Collecting data requires making
measurements.
2. Measurements should be valid.
3. We want to minimize bias and
variability, as much as possible.
Today
1. Three keys for summarizing a
collection of data
2. The distribution of a data set
3. Two ways to visualize a distribution
Summarizing data
The best summary of a large collection of
data tells us about three things
• Shape
• Center
• Spread
Today we focus on the “shape” of a
collection of data
Visualization
A graph is a visual presentation of a
collection of data.
Graphing is an excellent way to reveal the
shape of a collection of data.
Visualization
There are many different types of graph,
each with advantages and disadvantages.
We will look at two types of graph
• Histograms
• Stemplots
Organizing data
Before we can visualize the data, it may be
necessary to organize it.
One way is to count how often particular values
occur in our data set.
For example: how many students in this class
are psychology majors?
Organizing data
The number of times a value occurs is called
the value’s frequency.
Number of psychology majors = frequency of
psychology majors.
The proportion of times a value occurs is called
the relative frequency of that value.
Percent of psychology majors = relative
frequency of psychology majors.
Organizing data
The variable “a student’s major” is not numeric.
For non-numeric variables we can always find
frequencies or relative frequencies.
What about numeric variables?
Organizing data
We can find the frequency or relative frequency
for numeric variables, but often there’s a
better option:
Organize by grouped frequencies.
This means we put the data into classes,
lumping together numbers which are close.
Organizing data
However we choose to organize the data—by
count, proportion, or in classes—we produce
a list of different values and how often they
occur.
Distribution: a list of different data values and
how often each value occurs.
A distribution shows the “shape” of the data.
This shape is best presented visually.
Example
Consider the set:
3, 11, 12, 19, 22, 23, 24, 25, 27, 29, 35, 36, 37, 38, 45, 49
(the ages of a population consisting of 16
people)
Example (continued)
Knowing the frequency (how many 1s, how
many 2s, how many 3s, etc.) would be
useless—no number occurs more than once.
Instead, let’s look at grouped frequencies.
Data Range Frequency
0-9
1
10-19
3
20-29
6
30-39
4
40-49
2
Example (continued)
3, 11, 12, 19, 22, 23, 24, 25, 27, 29, 35, 36, 37, 38, 45, 49
Example (continued)
Now we can make a chart of the frequency
distribution of the data
The following is called a frequency histogram:
Histograms
Bars for each class.
Height of the bar is the number of data in the
class.
Note that the bars touch each other.
Only leave a blank space for empty classes.
The shape of a distribution
Important features to identify:
Number of peaks
Symmetric or asymmetric
Asymmetric: skewed to the left, the right, or
neither
Outliers: values that stand out from the
overall shape.
Clusters
•
•
•
•
•
Symmetric Distributions
Bell-Shaped
Symmetric Distributions
Mound-Shaped
Symmetric Distributions
Uniform
Asymmetric Distributions
Skewed to the Left
Asymmetric Distributions
Skewed to the Right
The shape of a distribution
Earlier example
Symmetric with one peak and no outliers or clusters
The shape of a distribution
8
7
6
Frequency
5
4
3
2
1
0
0-9
10-19
20-29
30-39
40-49
Home Runs
50-59
60-69
70-79
Asymmetric with one peak, skewed to the left, no clusters,
one outlier in the 70-79 class.
The shape of a distribution
Asymmetric with one peak, skewed to the right, and no
outliers or clusters
The shape of a distribution
Asymmetric with multiple peaks, not skewed, no outliers,
two clusters
The disadvantage of histograms
In a histogram the original data points are lost.
8
7
Frequency
6
5
4
3
2
1
0
0-9
10-19
20-29
30-39
40-49
Home Runs
50-59
60-69
70-79
We can see that there is one data value in the 70-79
range, but there is no way to determine the value.
Stemplots
Here is a sample of a stemplot
The numbers on the left are the “stems.” The other
numbers are the “leaves.”
Stemplots
The leaf is the rightmost digit of the data value.
The stem is the rest of the data value.
For example, the 0 in the last row means that the
number 60 is in this data set.
Notice there are no leaves on the 1 stem, but we
still include it in the stemplot.
How to make a stemplot
1. Each observation gets separated into a stem
(all but the rightmost digit) and a leaf (the final
digit).
2. The stems get put in a vertical column with the
smallest at the top. A vertical line is then drawn.
3. Each leaf is then written in the row to the right
of its stem, in increasing order out of the stem.
4. Make sure to line up the leaves in columns.
Example
The following data is a list of the annual home run totals of
the baseball player Barry Bonds over his entire 22 year
career, sorted from smallest to largest.
5 16 19 24 25 25 26 28 33 33 34 34 37 37 40 42 45 45 46
46 49 73
0
5
1
6 9
2
4 5 5 6 8
3
3 3 4 4 7 7
4
0 2 5 5 6 6 9
5
6
7
3
Example
The following data is a list of the annual home run totals of
the baseball player Barry Bonds over his entire 22 year
career, sorted from smallest to largest.
5 16 19 24 25 25 26 28 33 33 34 34 37 37 40 42 45 45 46
46 49 73
Comparing histograms and
stemplots
Let’s compare our stemplot to a histogram of the same
data.
0
5
8
6 9
2
4 5 5 6 8
3
3 3 4 4 7 7
4
0 2 5 5 6 6 9
7
6
Frequency
1
5
4
3
2
5
1
0
6
7
0-9
3
10-19
20-29
30-39
40-49
Home Runs
50-59
60-69
70-79
Comparing histograms and
stemplots
Stemplots are like histograms that are “tipped
over.”
Stemplots gives all of the same information about
the shape of the distribution.
In addition, stemplots show all of the data values,
which histograms do not.
But, we can’t use stemplots for large data sets.
How to make a stemplot
Sometimes you may need to round the data to
improve a stemplot.
Example
8.623 8.735 9.529 9.873 10.023
After rounding to the nearest tenth, these are
8.6
8.7
9.5
9.9
10.0