Frequency Distributions and Histograms Statistics: Section 2.2

Histograms Look like bar graphs but also have the following criteria: The bars have the same width and always touch The width of a bar represents a quantitative value, such as age, rather than a category The height of each bar indicates frequency They give information about a range of individuals not just a single individual.

Histograms Your first objective is to figure out how many bars (or classes) you want. Usually 5 to 15 groups are used. Next, find a class width. (range)/number of classes Always increase to the next whole number, even if you got a whole number as an answer.

Class Limits The lower class limit is the lowest value in a particular class, as the upper class limit is the highest value in a particular class. The class width is the difference between the lower class limit of one class with the lower class limit of the next class.

Midpoint (class mark) The center of the class [(lower class limit) + (upper class limit)]/2

Frequency Table A frequency table lists the following: The limits of each class The frequency with which the data fell into a class The class midpoint

Class Boundaries We don’t want a space between the bars, so we “meet halfway” between the difference of the “lower-upper limit” and the “higher-lower limit”.

Example: Commuting Distance in Dallas 13 47 10 3 16 20 17 40 4 2 7 25 8 21 19 15 14 6 12 45 1 11 18 23 46 9 34 41 28 36 24 27 29 26 37 31

Example: Step 1 – Class Width I want there to be six classes – bars eventually. [(largest value)-(smallest value))]/Number of classes If I want 6 classes (47-1)/6 = 7.7 -> 8 So my class width is 8.

Example – Step 2: Class Limits Determine the lower limits Smallest value is 1 and my class width is 8. So my lower class limits are 1, 9, 17, 25, 33, 41 Determine the upper limits The second class begins at 9 so my upper limit for my first class must be 8. 8, 16, 24, 32, 40, 48

Example – Step 3: Find Midpoints The center of the class [(lower class limit) + (upper class limit)]/2 (1+8)/2 = 9/2 = 4.5 (9+16)/2 = 25/2 = 12.5 (17+24)/2 = 41/2 = 20.5 28.5 36.5 44.5

Example – Step 4: Find the Class Boundaries Extend your class limits by ½ both ways. Class 1: 0.5 – 8.5 Class 2: 8.5 – 16.5 Class 3: 16.5 – 24.5 Class 4: 24.5 – 32.5 Class 5: 32.5 – 40.5 Class 6: 40.5 – 48.5

Example – Step 5: Create a tally Count how many people fall into each class and create a bar graph from that.

Relative-frequency Histograms Relative frequency = Class frequency / total of all frequencies Percentages The graphs should look the same except the vertical scales will be different