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Chapter 12: Problem 42

Describe what the standard deviation reveals about a data set.

Short Answer

Expert verified
The standard deviation reveals the degree of variation or dispersion of a set of values in a data set. A low standard deviation suggests that the values are close to the mean, while a high standard deviation suggests a wider spread around the mean. In simple terms, it tells how much the data varies from the average.

Step by step solution

01

Understanding Standard Deviation

Standard deviation is a statistical measure that captures the difference between the actual data points and their mean (average). It's a means of showing 'dispersion' in statistics. If the data points are closer to the mean, the standard deviation is smaller, and if the data points are spread out over a broader range of values, the standard deviation is larger.
02

Function of Standard Deviation

Standard deviation aids in the analysis of whether a specific data point is standard or unusual when compared to the overall data set. If a data point is closer to the mean, it's more typical in the context of the data, whereas if a data point is farther away from the mean (hence larger standard deviation), it's deemed more unusual.
03

Distribution of values

The standard deviation tells about the spread of numbers in a set. More significantly, if the values in a data set adhere to a bell-shaped curve (normal distribution), around 68% of the data will fall within one standard deviation of the mean, around 95% will fall within two standard deviations, and around 99.7% will fall within three standard deviations.

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Most popular questions from this chapter

The scores on a test are normally distributed with a mean of 100 and a standard deviation of 20.Find the score that is \(2 \frac{1}{2}\) standard deviations below the mean.

Find the standard deviation for each group of data items. Round answers to two decimal places. \(3,3,3,4,5,5,5\)

Intelligence quotients (IQs) on the Stanford-Binet intelligence test are normally distributed with a mean of 100 and a standard deviation of 16. Use the 68-95-99.7 Rule to find the percentage of people with IQs below \(52 .\)

A set of data items is normally distributed with a mean of 60 and a standard deviation of 8.Convert each data item to a z-score. 30

Determine whether each statement makes sense or does not make sense, and explain your reasoning. If a set of test scores has a large range but a small standard deviation, describe what this means about students' performance on the test.
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