Concept Byte: Standard Deviation

Use With Lesson 12-3

EXTENSION

You have learned about one measure of dispersion, range. Another measure of dispersion is standard deviation. Standard deviation is a measure of how the values in a data set vary, or deviate, from the mean.

Statisticians use several special symbols in the formula for standard deviation.

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Example

Find the mean and standard deviation of the data set 12.6, 15.1, 11.2, 17.9, and 18.2. Use a table to help organize your work.

• Step 1 Find the mean: x ¯ = 12.6 + 15.1 + 11.2 + 17.9 + 18.2 5 = 15 . x bar , equals . fraction 12.6 , plus , 15.1 , plus , 11.2 , plus , 17.9 , plus , 18.2 , over 5 end fraction . equals 15 . .

• Step 2 Find the difference between each data value and the mean: x − x ¯ . x minus , x bar . .

• Step 3 Square each difference: ( x − x ¯ ) 2 . open . x minus , x bar . close squared . .

• Step 4 Find the average (mean) of these squares: ∑ ( x − x ¯ ) 2 n . fraction sum . open . x minus , x bar . close squared , over n end fraction . .

  5.76 + 0.01 + 14.44 + 8.41 + 10.24 5 = 7.772 fraction 5.76 , plus , 0.01 , plus , 14.44 , plus , 8.41 , plus , 10.24 , over 5 end fraction . equals , 7.772

• Step 5 Take the square root to find the standard deviation: ∑ ( x − x ¯ ) 2 n = 7.772 ≈ 2.79 . square root of fraction sum . open . x minus , x bar . close squared , over n end fraction end root . equals , square root of 7.772 , almost equal to , 2.79 . .

The mean is 15 and the standard deviation is about 2.79.

A small standard deviation (compared to the data values) means that the data are clustered tightly around the mean. As the data become more widely distributed, the standard deviation increases.

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