Section 1.6

Frequency Distributions and Histograms

Constructing Histograms

Say we have the following dataset:

5 4 3 4 6 5 2 3 2 7 4 6 3

4 6 4 5 3 7 6 1 4 7 5 8 3

And we want to construct a histogram

The Classes

• Define the interval width for the horizontal axis• All intervals must be the same width

– The range of our data is small, so define the bars to be width = 1

• The range includes the left endpoint but not the right endpoint.– So the first interval is the range 0 <= X < 1

The Classes

• There is no rule governing how wide to make the intervals.

• However, too many intervals will spread out your data over many bars so that your histogram looks flat, and too few intervals will condense your data into 1 or 2 bars. Neither of these is a useful plot.

• The more data you have, the more intervals you can have

Frequency Tables

• Make a table to define each class and the number of elements in that class (frequency)

Note:

A square bracket means the endpoint is included.

A parenthesis means the endpoint is NOT included.

Class Frequency

[ 0 , 1 )

[ 1, 2 )

[ 2 , 3 )

[ 3 , 4 )

[ 4 , 5 )

[ 5 , 6 )

[ 6 , 7 )

[ 7 , 8 )

[ 8 , 9 )

[ 9, 10 )

0

1

5

4

4

1

2

3

0

6

Plot the Frequencies

• On the graph, draw a bar for each class• The height of the bar is the frequency of that

class

Frequency Histogram• What you have made is a histogram that

plots the frequency

Alternative methods

Class Frequency Relative Frequency

[ 0 , 1 ) 0

[ 1, 2 ) 1

[ 2 , 3 ) 2

[ 3 , 4 ) 5

[ 4 , 5 ) 6

[ 5 , 6 ) 4

[ 6 , 7 ) 4

[ 7 , 8 ) 3

[ 8 , 9 ) 1

[ 9, 10 ) 0

• Sometimes it is more useful to draw relative frequency histograms.

• Divide the frequency of each class by the sample size to obtain relative frequency.

• Convert to a percent.

0 %

3.8%

7.7%

19.2%

23.1%

15.4%

15.4%

11.5%

3.8%

0 %

Relative Frequency Histogram• Plot percentages instead of frequencies

• Many computer programs produce this type of histogram

Interpreting Histograms

• Histograms can be used to describe the properties of a distribution: – Location– Shape– Modality

• They can also identify outliers

Cumulative Distributions

• Another way to express our data is in a cumulative distribution.

• Instead of percentages for each class, we use percentages for <= (less than or equal to) each class.

Cumulative Distributions, con’t

• Add a column to the relative frequency table.

• Add the percents of all preceding rows for each class.

Class Freq Relative Freq.

Cumulative Frequency

[0,1) 0 0 %

[1,2) 1 3.8 %

[2,3) 2 7.7 %

[3,4) 5 19.2 %

[4,5) 6 23.1 %

[5,6) 4 15.4 %

[6,7) 4 15.4 %

[7,8) 3 11.5 %

[8,9) 1 3.8 %

[9,10) 0 0 %

0 %

0 + 3.8 = 3.8 %

3.8 + 7.7 = 11.5 %

11.5 + 19.2 = 30.7 %

30.7 + 23.1 = 53.8 %

53.8 + 15.4 = 69.2 %

69.2 + 15.4 = 84.6 %

84.6 + 11.5 = 96.1 %

96.1 + 3.8 = 99.9 %

99.9 ~ 100 %