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Chapter 3: Problem 50

When you are comparing two sets of data and one set is strongly skewed and the other is symmetric, which measures of the center and variation should you choose for the comparison?

Short Answer

Expert verified
For the strongly skewed data set, use the median as the measure of center and the interquartile range as the measure of variation. For the symmetric data set, use the mean as measure of center and standard deviation as the measure of variation.

Step by step solution

01

Identify Suitable Measure of Center for Skewed Data

For a skewed data set, the median is the most suitable measure of center. This is because the median, being the middle value, is not influenced by extreme values (outliers) that often cause skewness.
02

Identify Suitable Measure of Center for Symmetric Data

For a symmetric data set, the mean is typically the most appropriate measure of center. This is because, in a symmetric distribution, the mean, median and mode are all equal. Thus, the mean gives an accurate measure of the central value.
03

Identify Suitable Measure of Variation for Skewed Data

For a skewed data set, the interquartile range (IQR) is the most suitable measure of variation. The IQR, being the range of the middle 50% of the data, is resistant to outliers, which can greatly influence the variability of skewed data.
04

Identify Suitable Measure of Variation for Symmetric Data

For symmetric data, standard deviation is the best measure of variation as it considers how each data point varies from the mean. It is not resistant to outliers, but in a symmetric distribution, extreme values on both sides are balanced and do not unduly influence this measure of variation.

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Key Concepts

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

Measure of Center
When analyzing data sets, understanding the measure of center is crucial to summarize data in a single value that represents the middle point. The two most common measures are the median and the mean.

The median is the middle value in an ordered data set. It is insensitive to outliers, making it ideal for skewed data. Skewed data can have extreme values, pulling the mean away from the central tendency, but the median remains stable.

On the other hand, the mean is calculated by adding all values and dividing by the number of values. It's best suited for symmetric data where extremes don't distort it because symmetric data features balanced values, such that the mean, median, and mode converge at the same point.
Measure of Variation
The measure of variation provides insight into the spread or dispersion of the data set. For skewed data, the Interquartile Range (IQR) works best.

IQR focuses on the middle 50% of the data, ignoring extreme values that can mislead. By measuring from the first quartile (Q1) to the third quartile (Q3), it offers a robust reflection of data variability without being swayed by outliers.

For symmetric data, the standard deviation is preferred. It calculates the average distance of each data point from the mean, showing how spread out the values are. Symmetric data naturally balances out extreme values, making the standard deviation a fair representation of variation.
Skewed Data
Skewed data refers to an asymmetrical distribution where one tail is longer or fatter than the other. This asymmetry means that the majority of the data lies to one side of the graph.

In positively skewed data, the tail extends to the right, and more data points fall below the mean. In negatively skewed data, the tail extends to the left, with a concentration of data points above the mean.

Because of the presence of outliers or extremes that influence the mean, descriptive statistics for skewed data often rely on the median and IQR to minimize their impact and provide a clearer picture.
Symmetric Data
Symmetric data, on the other hand, has a balanced distribution where the left and right sides mirror each other. This type of data set has an equal distribution of values, which means the mean, median, and mode align closely.

Symmetrical data indicates consistency and uniformity. Each side of the center looks alike, creating a bell-shaped curve often known as the normal distribution. Here, measures like mean and standard deviation capture the data's central tendency and spread well, since outliers on both sides counterbalance each other, ensuring the accuracy of these statistics.

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Most popular questions from this chapter

In 2017 a pollution index was calculated for a sample of cities in the eastern states using data on air and water pollution. Assume the distribution of pollution indices is unimodal and symmetric. The mean of the distribution was \(35.9\) points with a standard deviation of \(11.6\) points. (Source: numbeo. com) see Guidance page \(142 .\) a. What percentage of eastern cities would you expect to have a pollution index between \(12.7\) and \(59.1\) points? b. What percentage of castern cities would you expect to have a pollution index between \(24.3\) and \(47.5\) points? c. The pollution index for New York, in 2017 was \(58.7\) points. Based on this distribution, was this unusually high? Explain.

In 2017 a pollution index was calculated for a sample of cities in the western states using data on air and water pollution. Assume the distribution of pollution indices is unimodal and symmetric. The mean of the distribution was \(43.0\) points with a standard deviation of \(11.3\) points. a. What percentage of western cities would you expect to have a pollution index between \(31.7\) and \(54.3\) points? b. What percentage of western cities would you expect to have a pollution index between \(20.4\) and \(65.6\) ? c. The pollution index for San Jose in 2017 was \(51.9\) points. Based on this distribution, was this unusually high? Explain.

a. In your own words, describe to someone who knows only a little statistics how to recognize when an observation is an outlier. What action(s) should be taken with an outlier? b. Which measure of the center (mean or median) is more resistant to outliers, and what does "resistant to outliers" mean?

Wechsler IQ tests have a mean of 100 and a standard deviation of \(15 .\) Which is more unusual: an IQ above 110 or an IQ below 80 ?

The mean weight gain for women during a full-term pregnancy is \(30.2\) pounds. The standard deviation of weight gain for this group is \(9.9\) pounds, and the shape of the distribution of weight gains is symmetric and unimodal. a. State the weight gain for women one standard deviation below the mean and for one standard deviation above the mean. b. Is a weight gain of 35 pounds more or less than one standard deviation from the mean?
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