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Chapter 3: Q 3.59. (page 113)
Why you use the standard deviation as a measure of variation. What is the reference point.
When using the standard deviation as a measure of variation, the mean of the data set serves as the reference point.
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Weekly Salaries. In the following table, we repeat the salary data in Data Set II from Example 3.1.
(a) Use Definitions 3.4 and 3.6 on pages 99 and 108, respectively, to obtain the sample mean and sample standard deviation of this (ungrouped) data set.
(b) A frequency distribution for Data Set II, using single-value grouping, is presented in the first two columns of the following table. The third column of the table is for thexf- values, that is, classmark or midpoint (which here is the same as the class) times class frequency. Complete the missing entries in the table and then use the grouped-data formula to obtain the sample mean.
(c) Compare the answers that you obtained for the sample mean in parts (a) and (b). Explain why the grouped-data formula always yields the actual sample mean when the data are grouped by using single-value grouping. (Hint: What does xf represent for each class?)
(d) Construct a table similar to the one in part (b) but with columns for .Use the table and the grouped-data formula to obtain the sample standard deviation.
(e) Compare your answers for the sample standard deviation in parts (a) and (d). Explain why the grouped-data formula always yields the actual sample standard deviation when the data are grouped by using single-value grouping.
True of False: The mean, median and mode can all be used with quantitative data. Explain your answer.
What does Chebyshev's rule say about the percentage of observations that lie within one standard deviation to either side of the mean? Discuss your answer.
Outliers and trimmed means. Some data sets contain outliers, observation that fall well outside the overall pattern of the data(We discuss outliers in more detail in section 3.4) Suppose, for instance that you are interested in the ability o high school algebra student to compute square roots. You decide to give a square root exam to 10 of these students Unfortunately, one of the student had a fight with his girlfriend and cannot concentrate he gets a 0. The 10 score are displayed in increasing order in the following table. The score of 0 is an outlier.
Statisticians have a systematic method for avoiding extreme observation and outliers when they calculated means. They compute trimmed means, in which high and low observation are deleted or "trimmed off" before the mean is calculated . For instance, to compute the 10% trimmed mean of the test score data. we first delete both the bottom 10% and the top 10% of the ordered data. that is,0 and 80. Then we calculated the mean of the remaining data. Thus the10% trimmed mean of the test score data is
The following table displays a set of score for a 40 question algebra final
Part (a) Do any the score look like outliers?
Part (b) Compute the usual mean of the data.
Part (c) Compute the 5% trimmed mean of the data.
Part (d) Compute the 10% trimmed mean of the data.
Part (e) Compare the means you obtained in parts (b) (d) which of the three provides the best measure of center for the data?
What condition on a data set is required to apply the Empirical rule?
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