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Chapter 2: Problem 119

According to the \(95 \%\) rule, the largest value in a sample from a distribution which is approximately symmetric and bell-shaped should be between 2 and 3 standard deviations above the mean, while the smallest value should be between 2 and 3 standard deviations below the mean. Thus the range should be roughly 4 to 6 times the standard deviation. As a rough rule of thumb, we can get a quick estimate of the standard deviation for a bell-shaped distribution by dividing the range by \(5 .\) Check how well this quick estimate works in the following situations. (a) Pulse rates from the StudentSurvey dataset discussed in Example 2.17 on page \(77 .\) The five number summary of pulse rates is \((35,62,70,\) 78,130) and the standard deviation is \(s=12.2\) bpm. Find the rough estimate using all the data, and then excluding the two outliers at 120 and \(130,\) which leaves the maximum at 96 . (b) Number of hours a week spent exercising from the StudentSurvey dataset discussed in Example 2.21 on page 81 . The five number summary of this dataset is (0,5,8,12,40) and the standard deviation is \(s=5.741\) hours. (c) Longevity of mammals from the MammalLongevity dataset discussed in Example 2.22 on page 82 . The five number summary of the longevity values is (1,8,12,16,40) and the standard deviation is \(s=7.24\) years.

Short Answer

Expert verified
Rough standard deviation estimates: For Pulse Rates (using all data): 19 bpm, (excluding outliers): 12.2 bpm. For Exercise hours: 8 hours. For Mammal longevity: 7.8 years.

Step by step solution

01

Calculate rough estimate for Pulse Rates

First, consider the pulse rates from the StudentSurvey dataset. The five number summary given is (35,62,70,78,130). The range of this dataset is \(130 - 35 = 95\). Now, to get a quick estimation of the standard deviation, divide the range by 5: \(95 / 5 = 19\) bpm. This is the rough estimate using all the data. For the second part, excluding the outliers 120 and 130 and taking the maximum as 96, the range would be \(96 - 35 = 61\). So, the rough standard deviation estimate would be \(61 / 5 = 12.2\) bpm.
02

Calculate rough estimate for exercise hours

In this step, consider the number of hours a week spent exercising from the StudentSurvey dataset. The five number summary of this dataset is (0,5,8,12,40). The range here is \(40 - 0 = 40\). Dividing the range by 5 gives us the rough estimate of standard deviation which is \(40 / 5 = 8\) hours.
03

Calculate rough estimate for mammal longevity

Lastly, evaluate the MammalLongevity dataset. The five number summary of the longevity values is (1,8,12,16,40). The range here is \(40 - 1 = 39\). Divide this range by 5 to get the rough estimate of the standard deviation, which is \(39 / 5 = 7.8\) years.

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Key Concepts

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

Standard Deviation
Standard deviation is a measure of how spread out the numbers are in a data set. Think of it as an average distance from the mean (average) value. For example, in a group of students, if we measure the heights and calculate the mean height, standard deviation tells us, on average, how far away each student's height is from the mean height.

When we have a set of numbers, like pulse rates, exercise hours, or lifespan of mammals, standard deviation gives us a precise numerical value that helps quantify their variability. It's calculated using a specific formula, but for a quick estimate, the range rule of thumb comes in handy鈥攅specially when dealing with symmetrical, bell-shaped distributions. In real-life situations, you鈥檒l often find that data won鈥檛 be perfect, and outliers can skew your standard deviation, revealing the potential for extreme values in a dataset.
Normal Distribution
A normal distribution, often depicted as a bell-shaped curve, represents a distribution where most of the data points are clustered around the mean and less are found as you move away from the center. It's symmetric, meaning the left and right sides are mirror images of each other.

Characteristics such as pulse rates, hours spent exercising, or longevity of mammals often follow this pattern when enough random samples are collected. The amazing thing about normal distributions is that they enable us to predict probabilities and make inferences about the entire population based on sample data. This is why knowing if a dataset closely follows a normal distribution is essential in statistics.
Range Rule of Thumb
The range rule of thumb is like a quick math trick to estimate the standard deviation without doing any complex calculations. As we saw in the exercise solutions, you simply take the range of your data (the difference between the largest and smallest values) and divide it by 5 for a dataset that follows a normal distribution.

Why 5, though?

It鈥檚 based on the empirical rule that in a normal distribution, approximately 95% of the data falls within 2 standard deviations of the mean鈥攖hus the range is often around 4 standard deviations. Dividing by a number a bit larger than 4 (like 5) accounts for slight asymmetry and provides a buffer, making the rule applicable to a broader range of bell-shaped distributions.
Data Variability
Data variability, or dispersion, tells us how much the data points in a set differ from each other and from the mean value. It's not just one number鈥攄ifferent statistics measure it, including range, variance, and standard deviation. High variability means that the data points are spread out over a wider range of values. Low variability means that the data points are more clustered closely to the mean.

Understanding variability is crucial because it affects the conclusions we can draw from data. For instance, two sets of mammal longevity could have the same mean life span, but one could have high variability with some mammals living much longer or shorter than others, while the other set could have low variability with most mammals living close to the average age.

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