Probability And Statistics Worksheets Page 33
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429-7
Standard
Deviation
■
Goals
Find the variance, standard deviation and z-scores for
a set of data.
■
Use standard deviation to interpret data.
Applications
Sports, Test-Taking, Education
Work in groups of 4–5 students.
1. Collect a set of data about students in class, such as heights, arm lengths,
head circumference, lengths of thumbs and so on.
2. Study the data. Look for new ways to describe the data. Instead of focusing on
central tendencies, study how the data are spread out, or dispersed.
3. Consider these questions: How much do individual values in your data differ
from the greatest value? The least value? The mean, median or mode of the
values?
4. Share your results with your classmates.
B
U
UILD
NDERSTANDING
Statistics that show how data is spread out are called measures of dispersion. For
example, you know that the range of a set of data is the difference between the
largest and smallest item.
Variance is another measure of dispersion. The variance of a set of numbers is the
mean of the squared differences between each number in the set and the mean
of all numbers in the set. For the set of numbers x
, x
, . . . x
, with a mean of m,
1
2
n
use this formula.
2
2
2
(x
m)
(x
m)
. . .
(x
m)
1
2
n
v
n
E x a m p l e 1
SPORTS
During a basketball tournament, the five starters for the Bulldogs made
the following number of 3-pointers: Bowen, 3; White, 4; Fillmore, 5; Graham 6;
and Bonilla, 7. Find the variance for the set of numbers.
Solution
1. Divide the sum of scores by 5 to find the mean, m.
2
number
(
)
(m
5)
2
3
3
5
( 2)
4
2
4
4
5
( 1)
1
2. Find the difference between each number and the
2
5
5
5
0
0
mean. Then find the square of each difference.
2
6
6
5
1
1
3. Find the mean of all the squares in Step 2.
2
7
7
5
2
4
4
1
0
1
4
10
10
5
2
The variance is 2.
The standard deviation, s, of a set of numbers is the square root of the variance.
412
Chapter 9 Probability and Statistics
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