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Chapter 4: Problem 16

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

Short Answer

Expert verified
The mean and standard deviation are measures of different characteristics of a data set. The mean is the average, which is a central location within the data. The standard deviation is a measure of the dispersion or variation in the data, indicating how spread out the data values are.

Step by step solution

01

Define Mean

The mean, often called the average, is calculated by adding all the values in the data set and then dividing by the number of values. It represents a central point in the dataset.
02

Define Standard Deviation

Standard deviation is a measure of how spread out numbers in a data set are. It is the square root of the variance, which is the average of the squared differences from the mean. High standard deviation means data points are generally far from the mean, and low standard deviation means data points are generally close to the mean.
03

Explain Differences

The main difference between the two lies in what they represent. The mean is a measure of central tendency, showing us where the center of our data is. The standard deviation, on the other hand, is a measure of dispersion, showing us how spread out from the mean our data is. They serve different functions in statistical analysis and are used to understand different aspects of the data.

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Most popular questions from this chapter

As a first step toward modifying his study habits, Phil keeps daily records of his study time. (a) During the first two weeks, Phil's mean study time equals 20 hours per week. If he studied 22 hours during the first week, how many hours did he study during the second week? (b) During the first four weeks, Phil's mean study time equals 21 hours. If he studied 22,18 , and 21 hours during the first, second, and third weeks, respectively, how many hours did he study during the fourth week? (c) If the information in (a) and (b) is to be used to estimate some unknown population characteristic, the notion of degrees of freedom can be introduced. How many degrees of freedom are associated with (a) and (b)? (d) Describe the mathematical restriction that causes a loss of degrees of freedom in (a) and (b).

Assume that the distribution of IQ scores for all college students has a mean of 120 , with a standard deviation of 15 . These two bits of information imply which of the following? (a) All students have an \(1 Q\) of either 105 or 135 because everybody in the distribution is either one standard deviation above or below the mean. True or false? (b) All students score between 105 and 135 because everybody is within one standard deviation on either side of the mean. True or false? (c) On the average, students deviate approximately 15 points on either side of the mean. True or false? (d) Some students deviate more than one standard deviation above or below the mean. True or false? (e) All students deviate more than one standard deviation above or below the mean. True or false? (f) Scott's IQ score of 150 deviates two standard deviations above the mean. True or false?

Days absent from school for a sample of 10 first-grade children are: 8,5,7,1,4,0,5,7,2,9 a) Before calculating the standard deviation, decide whether the definitional or computational formula would be more efficient. Why? b) Use the more efficient formula to calculate the sample standard deviation. Answers on page 425.

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 \(\$ 2,000\).

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?
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