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Chapter 4: Problem 6

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.

Short Answer

Expert verified
a) The computational formula would be more efficient for this data set. b) The sample standard deviation is approximately \( \sqrt{4.18} \), which is around 2.05 when rounded to two decimal places.

Step by step solution

01

Decide Which Formula to Use

In this case, as we have a small data set with only 10 values, both the definitional and computational formulas can be employed with similar ease. However, the computational formula tends to be more efficient as it requires fewer calculations generally. Hence, computational formula would be more efficient in this case.
02

Calculating the Sum of Scores and Their Squares

Before you can use the computational formula for standard deviation, first compute the sum of the scores and the sum of the squared scores. \[ \sum X = 8 + 5 + 7 + 1 + 4 + 0 + 5 + 7 + 2 + 9 = 48 \] and \[ \sum X虏 = 8虏 + 5虏 + 7虏 + 1虏 + 4虏 + 0虏 + 5虏 + 7虏 + 2虏 + 9虏 = 268 \]
03

Insert the Values into the Formula

Insert these values and the sample size (N=10) into the computational formula for standard deviation: \[ s=\sqrt{\frac{1}{N-1}[\sum X虏 - (\frac{(\sum X)虏}{N}]} = \sqrt{\frac{1}{10 - 1}[268 - (\frac{(48)^2}{10})]} \]
04

Compute for Final Answer

Perform the specified operations to get the final answer: \[ s= \sqrt{\frac{1}{9}(268 - 230.4)} = \sqrt{\frac{1}{9}(37.6)} = \sqrt{4.18} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation Formula
Understanding the standard deviation formula is crucial when aiming to measure the amount of variation or dispersion of a set of values. It provides insight into how spread out the numbers in your data set are in relation to the mean (average). The formula is expressed as:

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Most popular questions from this chapter

Using the definition formula for the sum of squares, calculate the sample standard deviation for the following four scores: 1,3,4,4

Add 10 to only the smallest score in Question 4.4(1,3,4,4) to produce another new distribution (11,3,4,4) . Would you expect the value of \(s\) to be the same for both the original and new distributions? Explain your answer, and then calculate s for the new distribution.

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

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?

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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