Frequency Distributions, Histograms, and Related Topics Section 2.5

Objectives Organize raw data using a frequency table. Construct histograms and relative-frequency histograms Recognize basic distribution shapes: uniform, symmetric, skewed, and bimodal. Interpret graphs in the context of the data setting.

Frequency Tables… A frequency table partitions data into classes or intervals and shows how many data values are in each class. The classes or intervals are constructed so that each data value falls into exactly one class. In general we should try to use between 5 and 15 classes to accurately explain the data.

Frequency table… 1) Decide how many classes (5 to 15) 2) Find Width of the classes (round to higher whole number) 3) Determine data range for each class (lower limit of one class to lower limit of next class) 4) Tally/Frequency

Example–Frequency table… A task force to encourage car pooling did a study of one-way commuting distances of workers in the downtown Dallas area. A random sample of 60 of these workers was taken. The commuting distances of the workers in the sample are given below. Make a frequency table for these data. One-Way Commuting Distances (in Miles) for 60 Workers in Downtown Dallas

Example–Frequency table… One-Way Commuting Distances (in Miles) for 60 Workers in Downtown Dallas Step 1; Decide how many classes (between 5 and 15) Smallest #:1 Largest #:47 Go with 6 classes this time

Example–Frequency table… Largest value: 47 Smallest value: 1 Go with 6 classes this time

Example–Frequency table… Step 3; Determine data range for each class Class width is the difference between the lower class limit of one class and lower class limit of the next class. Start at lower class limit (1) add class width (8). Lower class limit 2nd Class = 1+8=9 Lower class limit 3rd Class = 9+8=17 Lower class limit 4th Class = 17+8=25 Lower class limit 5th Class = 25+8=33 Lower class limit 6th Class = 33+8=41

Example–Frequency table… Class Limits Tally Frequency 1-8 |||| |||| |||| 14 9-16 |||| |||| |||| |||| | 21 |||| |||| | 11 17-24 |||| | 25-32 6 |||| 33-40 4 41-48 |||| 4

The center of each class is called the midpoint (or class mark) The center of each class is called the midpoint (or class mark). The midpoint is often used as a representative value of the entire class.

Frequency Tables… The relative frequency of a class is the proportion of all data values that fall into that class.

Example–Relative Freq. table… Class Limits Frequency Relative Freq. 1-8 14 9-16 21 17-24 11 25-32 6 33-40 4 41-48

Histograms and Relative-Frequency Histograms… Histograms and relative-frequency histograms provide effective visual displays of data organized into frequency tables. In these graphs, we use bars to represent each class, where the width of the bar is the class width. For histograms, the height of the bar is the class frequency, whereas for relative-frequency histograms, the height of the bar is the relative frequency of that class.

Histograms and Relative-Frequency Histograms…

Frequency Histograms… Class Limits Tally Freq. 1-8 |||| |||| |||| 14 9-16 |||| |||| |||| |||| | 21 17-24 |||| |||| | 11 25-32 |||| | 6 33-40 |||| 4 41-48 24 18 12 6 .5 8.5 16.5 24.5 32.5 40.5 48.5

Relative Frequency Histograms… Class Limits Freq. Rel. Freq. 1-8 14 .23 9-16 21 .35 17-24 11 .18 25-32 6 .10 33-40 4 .07 41-48 .40 .30 .20 .10 .5 8.5 16.5 24.5 32.5 40.5 48.5

Distribution Shapes…

2.4 Frequency Distributions, Histograms, and Related Topics Summarize Notes Read section Homework