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Chapter 13: Problem 4

What summary statistics are best used to report the "typical" value of a data set when the distribution is strongly skewed?

Short Answer

Expert verified
The best summary statistics to report the 'typical' value of a strongly skewed data set are the median and interquartile range. The median is less influenced by extreme values than the mean and the interquartile range provides an understanding of the statistical dispersion.

Step by step solution

01

Understand the concept of Skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. In other words, skewness identifies the direction and relative size of a distribution's tail. When a distribution is asymmetric, or 'skewed', it means that it has more observations on one side of the graph than the other.
02

Understand Summary Statistics in context of Skewness

Summary statistics, like mean, median, and mode, are calculated to give a 'typical' value for a data set. However, when a data set is skewed, using the mean can be misleading because it is influenced by the extreme values in the data set. The median and the mode are less affected by these extreme values.
03

Select the Best-suited Summary Statistics

Since the median and the mode are less affected by extreme values, these summary statistics are more reliable for skewed distributions. Especially, the median is often preferable because it is the value that separates the highest half from the lowest half in a data set, providing a 'typical' value that isn't overly influenced by the skewness. However, it's also valuable to consider the interquartile range, which gives an impression of the statistical dispersion.

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Most popular questions from this chapter

Suppose a college career center was interested in the starting salaries of recent graduates in Communications Studies and Sociology. The center randomly samples 15 recent graduates from each of these fields and records the starting salary for the graduates. The center wants to determine whether there is a difference in the starting salaries for graduates in these majors. Which test(s) should be used in each of these situations? a. Assume the starting salary for both majors is approximately Normally distributed. b. Assume that one of the salary distributions is strongly right-skewed.

You have recorded the time slept on a Tuesday and the time slept on a Sunday for a random sample of 15 students. You want to investigate whether students tend to sleep more on weekends than on weekdays. Which test(s) can you use? Answer for each circumstance. a. Assume the distribution of sleep hours for both Tuesday and Sunday are approximately Normal. b. Assume the distribution of sleep hours for both Tuesday and Sunday are not approximately Normal and assume that the distribution of differences in sleep hours for each student is not Normal. c. Assume the distributions of sleep hours for both Tuesday and Sunday are not Normal but assume the distribution of the differences in sleep hours is approximately Normal.

Suppose you want to determine whether meditation can cause a decrease in pulse rate. You randomly select 15 students, teach them a meditation technique, and then measure their pulse rates before and after meditation. Which test(s) should you choose for each situation? a. Assume that your analysis shows that the differences in pulse rates are Normally distributed. b. Assume that the distributions of differences in pulse rates are strongly skewed.

A doctor says he can predict the height (in inches) of a child between 2 and 9 years old from the child's age (in years) by using the equation Predicted Height \(=31.78+2.45\) Age This tells us the deterministic part of the regression model. What factors might contribute to the random component? In other words, why might a child's height not fall exactly on this line?

A professor tells his class that he knows their second exam score without their having to take the test. He tells them that the second exam score can be predicted from the first with this equation: Predicted second exam score \(=5+0.75\) (first exam score) This tells us that the deterministic part of the regression model that predicts second exam score on the basis of first exam score is a straight line. What factors might contribute to the random component? In other words, why might a student's score not fall exactly on this line?
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