### 6.2   Picturing the Distribution

Given a set of  n  values  x1, x2, ..., xn  from a random sample, how can we graphically picture their 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
example:
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!
Q:  how many categories should be used?
A:  depends; textbook gives some guidelines

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)
example:
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

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