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Module 1: Discrete random variables

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Standard deviation as a measure of spread

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XSIQ
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Intermediate Mathematics - Standard deviation as a measure of spread

Standard deviation as a measure of spread

Two dice are rolled and the numbers showing uppermost are added. Let T be
the random variable denoting the result of one roll.

6a. Give the probability distribution of T.

6b. Find the mean of T.

6c. Find the standard deviation of T.

6d. Find the percentage of the possible values that lie within two
standard deviations of the mean.

First list all possible values for _T,_ the sum of the two numbers
showing, with all of the corresponding pairs of numbers.

Each die has the numbers 1, 2, 3, 4, 5, and 6. Therefore the value for T
might be the total 2 or the total 3 or the total 4 and so on. The total 3
might occur from throws of 1 and 2 or 2 and 1. The total of 4 might occur
from throws of 1 and 3, 2 and 2, and 3 and 1.

A list of the full range of possibilities for the varying totals would be:

2: (1, 1)

3: (1, 2), (2, 1)

4: (1, 3), (2, 2), (3, 1)

5: (1, 4), (2, 3), (3, 2), (4, 1)

6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)

7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)

8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)

9: (3, 6), (4, 5), (5, 4), (6, 3)

10: (4, 6), (5, 5), (6, 4)

11: (6, 5), (5, 6)

12: (6, 6)

There is a total of 36 possible combinations for the two numbers. The
probability distribution table could be listed as follows:

The mean might be calculated with the formula:

The smallest possible value equals 3. The largest possible value equals
11. In this case, all the values, except for 2 and 12, lie within two
standard deviations of the mean.

This proportion of the possible values represents 94% of the values. Thus
94% of the possible values lie within two standard deviations of the mean.


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