MATH& 146

Lesson 10

Section 1.6

Graphing Numerical Data

1

Graphs of Numerical Data

One major reason for constructing a graph of

numerical data is to display its distribution, or the

pattern of variability displayed by the data of a

variable.

Three popular methods for displaying distributions

of numerical data are the dotplot, the histogram,

and the box plot.

2

Dotplots

The dotplot displays the data of a sample by

representing each data value with a dot positioned

along a scale, either horizontally or vertically.

The frequency of the dotplot is represented along the

other scale.

3

Example 1

Create a dotplot of the following exam scores.

4

Histograms

For much of the work you do in this course, you

will use a histogram to display the data. One

advantage of a histogram is that it can readily

display large data sets.

5

Histograms

Unlike dotplots, histograms use ranges of values

instead of individual values. These ranges of values

are represented by bars (called classes), with the

heights equal to the frequency of each class.

6

Constructing histograms

The basic steps to construct a histogram are as

follows:

1) Find the minimum and maximum values of the

data.

2) Create classes by slicing data into intervals of

equal width (choose "nice" numbers).

3) Make a table (called a frequency table) to count

the number of values in each class.

4) Make a bar for each class, using the heights to

determine the height of each bar.

7

Example 2

The following are the scores on a measure of

sensitivity to smell taken by 13 chefs attending a

national conference:

96, 83, 59, 64, 73, 74, 80, 68, 87, 67, 64, 92, 76

Make a histogram of the data.

8

Shape of a Distribution

When describing the shape of a distribution (the

outline of a histogram), you should answer the

following three questions:

1) Does it have a single, central peak or several

separated peaks, or none at all?

2) Is it symmetric or is it skewed one way or the

other?

3) Do any unusual features (e.g. outliers) stick out?

9

Peaks

1) Does the distribution have a single, central

peak or several separated peaks, or none at

all?

A distribution with one main peak is dubbed

unimodal.

10

Peaks

A distribution with two peaks is bimodal.

A distribution with no peaks and shaped more or

less like a rectangle is called uniform.

11

Bimodal Distributions

Bimodal distributions usually occur when the data

of two separate groups are combined.

Diastolic Blood Pressure12

Uniform Distributions

A distribution that doesn't appear to have any

mode and in which all the bars are approximately

the same height (in the "real world," the bars will

never be exactly the same) is called uniform:

Proportion of Wins 13

Symmetry

2) Is the distribution symmetric?

Essentially, a distribution is symmetric if you can fold

the distribution along a vertical line through the middle

and have the edges match pretty closely.

14

Skewness

The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail.

15

skewed left skewed right

Skewness

Symmetric graphs are ideal for inferential statistics, though skewed graphs can also work, provided the sample size is large enough. Generally, the more skewed the graph, the larger the sample size is needed to be.

16

skewed left skewed right

Outliers

3) Do any unusual features stick out?

Sometimes it's the unusual features that tell us something interesting or exciting about the data.

You should always mention any stragglers, or outliers, that stand off away from the body of the distribution.

17

Outliers

Often, not always, outliers are due to mistakes

(such as writing 5,000 instead of 50). Other

outliers may indicate that something unusual is

happening. If you see an outlier, proceed carefully.

18

Example 3

What can be said about the following histogram?

19

Example 4

What can be said about the following histogram?

20

Example 5

What can be said about the following histogram?

21

Example 6

What can be said about the following histogram?

22

Box Plots

Box plots, or box-and-whisker plots, give a

graphical image of the concentration of the data.

The box plot is constructed from five values, called

the five-number summary:

23

The Five-Number Summary

The five-number summary includes:

• The minimum

• The lower quartile, Q1

• The median

• The upper quartile, Q3

• The maximum

These numbers divide the data into four more or less

equal pieces.

24

Min Q1 MaxMed Q3

25% 25%25%25%

The Interquartile Range

The middle fifty percent of all data is represented by

the box. The length of this box is the Interquartile

Range. In other words:

The length of the entire boxplot is the Range.

3 1IQR Q Q

25

Range Max Min

Range

IQR

Construct the Box Plot

To construct a box plot, use a number line and mark

each of the five numbers: minimum, first quartile,

median, third quartile, and maximum (use a dotted tick

mark for the median). Draw a top and bottom around

the middle three numbers to make a box, and then

draw lines connecting the box to the minimum and

maximum.

26

Example 7

Construct a box plot and find the range and

interquartile range.

91, 96, 84, 100, 92, 23, 84

27

Comparing Groups

Boxplots are ideal when it comes to comparing two or more groups or categories.

28

Outliers

Box Plots can be used to show extreme values by using dots or asterisks (• or *) to represent potential outliers.

Any potential outlier should be examined carefully in your data analysis.

29

Example 8

The boxplots below show the number of millionaires by

state per 1000 households, as reported by

Netscape.com in 2006.

30

Example 8 continued

a) List the regions from lowest to highest in terms of

the median rate of millionaires in that region.

b) Which region has the smallest interquartile range?

c) Which region has potential outliers?

31

Example 9

The following box plot shows the U.S. population for

1990.

a) Are there fewer or more children (age 17 and

under) than senior citizens (age 65 and over)?

b) 12.6% are age 65 and over. Approximately what

percent of the population are of working age adults

(above age 17 to age 65)?

32

Example 10

Match each histogram with

its boxplot.

33

X

Y

Z