6.2   Picturing the Distribution

Histogram: bar graph for data from random sample

• divide values into categories (ranges), usually of equal width
• determine the number of values which lie in each category (frequencies) or the percentage of the total which lie in each category (relative frequencies)
• histogram is bar graph of frequencies or relative frequencies

Heights of students; suppose a random sample of 25 students is selected from the student population at large, and their heights are recorded, giving following values:
70, 68, 65, 69, 77, 62, 70, 70, 61, 72, 64, 62, 69, 72, 73, 69, 63, 72, 69, 71, 70, 64, 68, 75, 61

Create a histogram with category widths of 2 inches:

• the number of values in each category, and the percentages these represent, are given in the following table:
 
 

Categories	number	fraction
61-62 inches	4	.16
63-64	3	.12
65-66	1	.04
67-68	2	.08
69-70	8	.32
71-72	4	.16
73-74	1	.04
75-76	1	.04
77-78	1	.04

 
• the percentages in this table can be represented as a bar graph, giving the histogram:


Create a histogram with category widths of 5 inches:
• the number of values in each category, and the percentages these represent, are given in the following table:
 

Categories	number	fraction
59-63	5	.20
64-68	5	.20
69-73	13	.52
74-78	2	.08

 
• representing the table as a bar graph, get the histogram:


Note that the appearance of the histogram depends on the number of categories used!

Notes:

• histogram gives an approximation to density function f(x) for the entire population
• can use to estimate a few probabilities (those involving our categories)

Use the first histogram above to estimate the probability that a student selected at random will have a height of 65 to 68 inches.

Just use the percentages from the bars in the histogram for 65-66 and 67-68 inches; get
P(65 <= X <= 68)   will equal approximately   .04 + .08  =  .12