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Chapter 3: Problem 39

When is the value of the standard deviation for a data set zero? Give one example. Calculate the standard deviation for the example and show that its value is zero.

Short Answer

Expert verified
The standard deviation for a data set is zero when all the values in that set are the same because there is no spread in the data. For example, in the data set {5, 5, 5, 5, 5}, every step of calculating the standard deviation results in the number 0, demonstrating that the standard deviation is indeed 0.

Step by step solution

01

Define the Data Set

Create a data set where all values are equal. For example, the data set can be: \{5, 5, 5, 5, 5\}. All values are the same (5). There are 5 values (n = 5).
02

Calculate the Mean

The mean (or average) is calculated as the sum of all values divided by the number of values. In this case, the sum of all values is 25 (5+5+5+5+5) and there are 5 values, so the mean is \( \frac{25}{5} = 5 \).
03

Calculate the Difference between Each Data Point and the Mean

The next step is to subtract the mean from each data point. This gives us: \{5-5, 5-5, 5-5, 5-5, 5-5\} = \{0, 0, 0, 0, 0\}.
04

Square Each of These Differences

Each of these differences is then squared: \{0^2, 0^2, 0^2, 0^2, 0^2\} = \{0, 0, 0, 0, 0\}.
05

Calculate the Mean of the Squared Differences

The next step is to calculate the mean of the squared differences. This is also known as the variance. The variance is \( \frac{0+0+0+0+0}{5} = 0 \).
06

Calculate the Square Root of the Variance

Finally, to find the standard deviation, take the square root of the variance. The square root of 0 is 0. Therefore, the standard deviation of this data set, where all numbers are equal, is 0.

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Most popular questions from this chapter

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