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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 distribution of the lengths of fish in a certain lake is not known, but it is definitely not bell shaped. It is estimated that the mean length is 6 inches with a standard deviation of 2 inches. a. At least what proportion of fish in the lake are between 3 inches and 9 inches long? b. What is the smallest interval that will contain the lengths of at least \(84 \%\) of the fish? c. Find an interval so that fewer than \(36 \%\) of the fish have lengths outside this interval.

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?

Explain how the value of the median is determined for a data set that contains an odd number of observations and for a data set that contains an even number of observations.

Explain the concept of the percentile rank for an observation of a data set.

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 .
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