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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

The heights of five starting players on a basketball team have a mean of 76 inches, a median of 78 inches, and a range of 11 inches. a. If the tallest of these five players is replaced by a substitute who is 2 inches taller, find the new mean, median, and range. b. If the tallest player is replaced by a substitute who is 4 inches shorter, which of the new values (mean, median, range) could you determine, and what would their new values be?

The mean age of six persons is 46 years. The ages of five of these six persons are \(57,39,44,51\), and 37 years, respectively. Find the age of the sixth person.

The following data give the 2010 gross domestic product (in billions of dollars) for all 50 states. The data are entered in alphabetical order by state (Source: Bureau of Economic Analysis). \(\begin{array}{rrrrrrrrrr}173 & 49 & 254 & 103 & 1901 & 258 & 237 & 62 & 748 & 403 \\ 67 & 55 & 652 & 276 & 143 & 127 & 163 & 219 & 52 & 295 \\ 379 & 384 & 270 & 97 & 244 & 36 & 90 & 126 & 60 & 487 \\ 80 & 1160 & 425 & 35 & 478 & 148 & 174 & 570 & 49 & 164 \\\ 40 & 255 & 1207 & 115 & 26 & 424 & 340 & 65 & 248 & 39\end{array}\) a. Calculate the mean and median for these data. Are these values of the mean and the median sample statistics or population parameters? Explain. b. Do these data have a mode? Explain.

The following data give the numbers of text messages sent by a high school student on 40 randomly selected days during 2012: \(\begin{array}{llllllllll}32 & 33 & 33 & 34 & 35 & 36 & 37 & 37 & 37 & 37 \\\ 38 & 39 & 40 & 41 & 41 & 42 & 42 & 42 & 43 & 44 \\ 44 & 45 & 45 & 45 & 47 & 47 & 47 & 47 & 47 & 48 \\ 48 & 49 & 50 & 50 & 51 & 52 & 53 & 54 & 59 & 61\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 49 fall in relation to these quartiles? b. Determine the approximate value of the 91 st percentile. Give a brief interpretation of this percentile. c. For what percentage of the days was the number of text messages sent 40 or higher? Answer by finding the percentile rank of 40 .

Is it possible for a (quantitative) data set to have no mean, no median, or no mode? Give an example of a data set for which this summary measure does not exist.
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