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Chapter 12: Problem 54

Use the data \(0,1,3,4,4,6\). Add 2 to each of the numbers. How does this affect the mean? How does this affect the standard deviation?

Short Answer

Expert verified
When each number in the data set is increased by 2, the mean of the set also increases by 2. However, the operation does not affect the standard deviation as it is a measure of dispersion or how spread out numbers are, which remains constant in this case as the increase is uniform.

Step by step solution

01

Calculate original mean

Add all the numbers together, and divide by how many numbers there are. Let's use this formula for the mean: \[Mean = \frac{\sum X}{N}\], where \(\sum X\) is the sum of all numbers and \(N\) is the count of numbers. So, for our data set \([0,1,3,4,4,6]\), Mean = \((0+1+3+4+4+6) / 6\).
02

Calculate new mean

Now, add 2 to each number in the data set, and calculate the new mean. The new data set is \([2,3,5,6,6,8]\). Mean' = \((2+3+5+6+6+8) / 6\).
03

Calculate original Standard Deviation

The Standard Deviation is a measure of how spread out numbers are. Calculate standard deviation using formula: \[SD = \sqrt{\frac{\sum (X-mean)^2}{N}}\].Here \(X\) represents each value in the data set, \(mean\) is the mean we calculated in step 1, and \(N\) is the total count of numbers. So, SD = \(\sqrt{(0-mean)^2+(1-mean)^2+(3-mean)^2+(4-mean)^2+(4-mean)^2+(6-mean)^2 / 6}\).
04

Calculate New Standard Deviation

Measure the spread of the new data set (2 added to each number). Hence, the new set is \([2,3,5,6,6,8]\) and the new mean is the one calculated in step 2. So, SD' = \(\sqrt{(2-mean')^2+(3-mean')^2+(5-mean')^2+(6-mean')^2+(6-mean')^2+(8-mean')^2 / 6}\).

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