1. Section 1.6 Frequency Distributions and Histograms

2. 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

3. 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

4. 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

5. 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. ClassFrequency [ 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

6. Plot the Frequencies On the graph, draw a bar for each class The height of the bar is the frequency of that class

7. Frequency Histogram What you have made is a histogram that plots the frequency

8. Alternative methods ClassFrequencyRelative 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% 11.5% 3.8% 0 %

9. Relative Frequency Histogram Plot percentages instead of frequencies Many computer programs produce this type of histogram

10. Interpreting Histograms Histograms can be used to describe the properties of a distribution: –Location –Shape –Modality They can also identify outliers

11. 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.

12. Cumulative Distributions, con’t Add a column to the relative frequency table. Add the percents of all preceding rows for each class. ClassFreqRelative Freq. Cumulative Frequency [0,1)00 % [1,2)13.8 % [2,3)27.7 % [3,4)519.2 % [4,5)623.1 % [5,6)415.4 % [6,7)415.4 % [7,8)311.5 % [8,9)13.8 % [9,10)00 % 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 %