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Chapter 4: Problem 15

In what sense is the variance (a) a type of mean? (b) not a readily understood measure of variability? (c) a stepping stone to the standard deviation?

Short Answer

Expert verified
(a) Variance is a type of mean since it is the average of the squared differences from the mean; (b) It is not a commonly understood measure of variability because it is presented in squared units, which can be unintuitive for interpretation; (c) Variance is a stepping stone to standard deviation because the standard deviation is the square root of the variance.

Step by step solution

01

Understanding Variance as a Type of Mean

Variance is a type of mean because it represents the average of the squared differences from the Mean. It is calculated by taking the differences between each number in the data set and the mean, then squaring the result (the squared differences), and finally calculating the average of those squared differences. This is why we can consider variance as a type of mean.
02

Intepreting Variance as a Measure of Variability

While variance provides a measure of data dispersion, it is not easily understood because it is expressed in squared units. For example, if our dataset measures height in centimeters, variance will be in square centimeters. This makes it less intuitive for interpretation.
03

Role of Variance in Computing Standard Deviation

Variance serves as a stepping stone to the standard deviation because standard deviation is simply the square root of the variance. In doing so, standard deviation enables a more interpretable measure of variability since it is in the original units of measure. Therefore, considering variance as a stepping stone doesn't seem inappropriate at all.

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Most popular questions from this chapter

Employees of Corporation A earn annual salaries described by a mean of \(\$ 90,000\) and a standard deviation of \(\$ 10,000\). (a) The majority of all salaries fall between what two values? (b) A small minority of all salaries are less than what value? (c) A small minority of all salaries are more than what value? (d) Answer parts (a), (b), and (c) for Corporation B's employees, who earn annual salaries described by a mean of \(\$ 90,000\) and a standard deviation of \(\$ 2,000\).

(a) While in office, a former governor of California proposed that all state employees receive the same pay raise of \(\$ 70\) per month. What effect, if any, would this raise have had on the mean and the standard deviation for the distribution of monthly wages in existence before the proposed raise? Hint: Imagine the effect of adding \(\$ 70\) to the monthly wages of each state employee on the mean and on the standard deviation (or on a more easily visualized measure of variability, such as the range). (b) Other California officials suggested that all state employees receive a pay raise of 5 percent. What effect, if any, would this raise have had on the mean and the standard deviation for the distribution of monthly wages in existence before the proposed raise? Hint: Imagine the effect of multiplying the monthly wages of each state employee by 5 percent on the mean and on the standard deviation or on the range.

Specify an important difference between the standard deviation and the mean.

Determine the values of the range and the IQR for the following sets of data. (a) Retirement ages: 60,63,45,63,65,70,55,63,60,65,63 (b) Residence changes: 1,3,4,1,0,2,5,8,0,2,3,4,7,11,0,2,3,4

Using the computation formula for the sum of squares, calculate the population standard deviation for the scores in (a) and the sample standard deviation for the scores in (b). (a) 1,3,7,2,0,4,7,3 (b) 10,8,5,0,1,1,7,9,2
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