Section 2.1: Frequency Distributions, Histograms and Related Topics

Organizing Data Data when collected in original form is called “raw data”. For example

Frequency Distribution We organize the raw data by using a frequency distribution. The frequency is the number of values in a specific class of data. A frequency distribution is the organizing of raw data in table form, using classes and frequencies.

Frequency Distribution Cont. For the raw data set, a frequency distribution is shown as follow: We use a frequency distribution to construct a histogram. Class limits Tally Frequency 1-3 ///// 10 4-6 //// 14 7-9 10-12 / 6 13-15 5 16-18

Histograms Histograms are bar graphs in which The bars have the same width and always touch (the edges of the bars are on class boundaries which are described below). The width of a bar represents a quantity. The height of each bar indicates frequency.

To construct a histogram from raw data: Decide on the number of classes (5 to 15 is customary). Find a convenient class width. Organize the data into a frequency table. Find the class midpoints and the class boundaries. Sketch the histogram. 7

To find Class Width First compute: Largest value - smallest value Desired number of classes Increase the value computed to the next highest whole number. The lower class limit of a class is the lowest data that can fit into the class, the upper class limit is the highest data value that can fit into the class. The class width is the difference between lower class limits of adjacent classes.

Class Width Raw Data: 10.2 18.7 22.3 20.0 6.3 17.8 17.1 5.0 2.4 7.9 0.3 2.5 8.5 12.5 21.4 16.5 0.4 5.2 4.1 14.3 19.5 22.5 0.0 24.7 11.4 Use 5 classes. 24.7 – 0.0 5 = 4.94 Round class width up to 5. 9

Frequency Table Determine class width. Create the classes. May use smallest data value as lower limit of first class and add width to get lower limit of next class. Tally data into classes. Compute midpoints for each class. Determine class boundaries. 10

Tallying the Data tally frequency 0.0 - 4.9 |||| | 6 5.0 - 9.9 |||| 5 0.0 - 4.9 |||| | 6 5.0 - 9.9 |||| 5 10.0 - 14.9 |||| 4 15.0 - 19.9 |||| 5 20.0 - 24.9 |||| 5 11

Computing Class Width difference between the lower class limit of one class and the lower class limit of the next class

Finding Class Widths f class widths 0.0 - 4.9 6 5 5.0 - 9.9 6 5 0.0 - 4.9 6 5 5.0 - 9.9 6 5 10.0 - 14.9 4 5 15.0 - 19.9 5 5 20.0 - 24.9 5 5 14

Computing Class Midpoints lower class limit + upper class limit 2

Finding Class Midpoints f class midpoints 0.0 - 4.9 6 2.45 5.0 - 9.9 5 7.45 10.0 - 14.9 4 12.45 15.0 - 19.9 5 17.45 20.0 - 24.9 5 22.45 18

Class Boundaries Class boundaries cannot belong to any class. Upper limit of one class + lower limit of next class 2 Class boundaries cannot belong to any class. Class boundaries between adjacent classes are the midpoint between the upper limit of the first class, and the lower limit of the higher class. Differences between upper and lower boundaries of a given class is the class width.

Finding Class Boundaries f class boundaries 0.0 - 4.9 6 0.05 - 4.95 5.0 - 9.9 5 4.95 - 9.95 10.0 - 14.9 4 9.95 - 14.95 15.0 - 19.9 5 14.95 - 19.95 20.0 - 24.9 5 19.95 - 24.95 25

Constructing the Histogram f f 0.0 - 4.9 6 5.0 - 9.9 5 10.0 - 14.9 4 15.0 - 19.9 5 20.0 - 24.9 5 | | | | | | 6 5 4 3 2 1 - -0.05 4.95 9.95 14.95 19.95 24.95 26

Relative Frequencies The relative frequency of a class is f/n where f is the frequency of the class, and n is the total of all frequencies. Relative frequency tables are like frequency tables except the relative frequency is given. Relative frequency histograms are like frequency histograms except the height of the bars represent relative frequencies.

Relative Frequency Histogram f relative frequency 0.0 - 4.9 6 0.24 5.0 - 9.9 5 0.20 10.0 - 14.9 4 0.16 15.0 - 19.9 5 0.20 20.0 - 24.9 5 0.20 | | | | | | .24 .20 .16 .12 .08 .04 - -0.05 4.95 9.95 14.95 19.95 24.95 Relative frequency f/n 29

Common Shapes of Histograms When folded vertically, both sides are (more or less) the same. Symmetrical f 30

Common Shapes of Histograms Also Symmetrical f 31

Common Shapes of Histograms Uniform f 32

Common Shapes of Histograms Skewed Histograms Skewed left Skewed right 34

Common Shapes of Histograms Bimodal f The two largest rectangles are approximately equal in height and are separated by at least one class. 35

Frequency Polygon A frequency polygon or line graph emphasizes the continuous rise or fall of the frequencies. Dots are placed over the midpoints of each class. Dots are joined by line segments. Zero frequency classes are included at each end.

Constructing the Frequency Polygon 2 - 4 6 5 - 7 5 8 - 10 4 11 - 13 5 f 6 5 4 3 2 1 - | | | | | | 0 3 6 9 12 15 38

Cumulative Frequencies & Ogives The cumulative frequency of a class is the frequency of the class plus the frequencies for all previous classes. An ogive is a cumulative frequency polygon.

Cumulative Frequency Table Greater than 1.5 20 Greater than 4.5 14 Greater than 7.5 9 Greater than 10.5 5 Greater than 13.5 0 f 2 - 4 6 5 - 7 5 8 - 10 4 11 - 13 5 20

Constructing the Ogive f Greater than 1.5 20 Greater than 4.5 14 Greater than 7.5 9 Greater than 10.5 5 Greater than 13.5 0 20 15 10 5 - Cumulative frequency | | | | | | 1.5 4.5 7.5 10.5 13.5 pounds