/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 145 The standard deviation is the mo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 145

The standard deviation is the most popular measure of variability from the mean. It uses squared deviations because the ordinary deviations sum to zero. An alternative measure is the mean absolute deviation, \(\Sigma|x-\bar{x}| / n\) a. Explain why greater variability tends to result in larger values of this measure. b. Would the MAD be more, or less, resistant than the standard deviation? Explain.

Short Answer

Expert verified
Greater variability increases MAD, as differences from the mean grow. MAD is more resistant to outliers than standard deviation.

Step by step solution

01

Understand Variability

Variability in data refers to how spread out the data points are around the mean. A dataset with high variability will have data points that are spread further from the mean, whereas a dataset with low variability has data points close to the mean.
02

Analyze the Formula for MAD

The mean absolute deviation (MAD) is calculated as the average of the absolute differences between each data point and the mean, given by the formula \( \Sigma|x-\bar{x}| / n \). The absolute value ensures that all deviations contribute positively to the total deviation, unlike squared deviations used in standard deviation.
03

Explain Larger Variability and MAD

When variability in data increases, the differences between each data point and the mean increase, or in other words, the absolute deviations \(|x-\bar{x}|\) become larger. Thus, the average of these absolute deviations, or MAD, also increases.
04

Resistance Comparison

Resistance measures how a statistical measure is affected by outliers. The mean absolute deviation (MAD) is influenced by each deviation linearly, while the standard deviation (SD) squares each deviation, making it more affected by larger deviations or outliers. Therefore, MAD is typically more resistant to outliers compared to the standard deviation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mean Absolute Deviation
The Mean Absolute Deviation (MAD) is a statistical measure used to quantify data variability, specifically the average distance between each data point and the mean of the dataset. In MAD, each deviation from the mean is considered without regard to its direction since we take the absolute value.
This results in all deviations contributing positively to the measure.
  • MAD is calculated by taking the sum of the absolute differences between each data point and the mean.
  • This sum is then divided by the number of data points, represented by the formula: \( \frac{\Sigma|x-\bar{x}|}{n} \).
MAD is particularly useful when we want to understand how much the values differ from the average in simpler terms without the complications that squaring, as done in standard deviation, brings.
Standard Deviation
The Standard Deviation (SD) is another key measure of data variability, showing how spread out the data points are from the mean. Unlike MAD, the standard deviation involves calculating squared deviations which makes the process sensitive to outliers due to the squaring process.
Here's what happens:
  • For each data point, the difference from the mean is calculated, squared, and added to the total.
  • The sum of these squared differences is divided by the number of data points for variance, and then the square root of that variance gives us the SD.
The formula for SD is: \( \sqrt{\frac{\Sigma (x-\bar{x})^2}{n}} \).
This detailed approach makes SD a valuable measure for datasets where understanding the influence of large deviations or outliers is crucial.
Data Variability
Data variability is all about understanding how varied or spread out a dataset is. A dataset with low variability will have data points clustered closely around the mean, while a dataset with high variability will spread out more widely across values.
Variability can be analyzed using:
  • Mean Absolute Deviation - simple and straightforward with linear consideration.
  • Standard Deviation - includes the effect of extreme values by squaring deviations.
By comparing these, one can get a better sense of not just the variability, but also the distribution and potential influence of outliers within the data.
Resistance to Outliers
Resistance to outliers refers to how much a measure is affected by extreme values in the dataset. Mean Absolute Deviation (MAD) and Standard Deviation (SD) differ in their resistance to these outliers.
MAD considers each deviation equally without amplification, making it more resistant to outliers.
  • Each deviation contributes linearly to the total MAD, minimizing the impact of extreme deviations.
On the other hand, SD squares each deviation, magnifying the effect of any data point far from the mean.
  • This squaring means that outliers can have a disproportionally large impact on the SD.
Overall, for datasets where robustness against outliers is important, MAD serves as a more reliable measure compared to SD.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The data values below represent the closing prices of the 20 most actively traded stocks on the NASDAQ Stock Exchange (rounded to the nearest dollar) on May \(2,2014 .\) \(\begin{array}{cccccccccc}3 & 60 & 40 & 87 & 26 & 9 & 37 & 23 & 26 & 9 \\ 4 & 78 & 4 & 7 & 26 & 7 & 52 & 8 & 52 & 13\end{array}\) a. Sketch a dot plot or construct a stem-and-leaf plot. b. Find the median, the first quartile, and the third quartile. c. Sketch a box plot. What feature of the distribution displayed in the plot in part a is not obvious in the box plot? (Hint: Are there any gaps in the data?)

Data collected over several years from college students enrolled in a business statistics class regarding their shoe size shows a roughly bell-shaped distribution, with \(\bar{x}=9.91\) and \(s=2.07\). a. Give an interval within which about \(95 \%\) of the shoe sizes fall. b. Identify the shoe size of a student which is three standard deviations above the mean in this sample. Would this be a rather unusual observation? Why?

For the following variables, indicate whether you would expect its histogram to be bell shaped, skewed to the right, or skewed to the left. Explain why. a. Number of times arrested in past year b. Time needed to complete difficult exam (maximum time is 1 hour \()\) c. Assessed value of house d. Age at death

The figure below shows the stem-and-leaf plot for the cereal sugar values from Example \(5,\) using split stems. Stem and Leaf Plot for Cereal Sugar Values with Leaf Unit \(=1000\) $$ \begin{array}{l|l} 0 & 01 \\ 0 & 33 \\ 0 & 445 \\ 0 & 67 \\ 0 & 9 \\ 1 & 011 \\ 1 & 22 \\ 1 & 445 \\ 1 & 6 \\ 1 & 8 \end{array} $$ a. What was the smallest and largest amount of sugar found in the 20 cereals? b. What sugar values are represented on the 6 th line of the plot? c. How many cereals have a sugar content less than \(5 \mathrm{~g}\) ?

For each of the following variables, would you use the median or mean for describing the center of the distribution? Why? (Think about the likely shape of the distribution.) a. Salary of employees of a university b. Time spent on a difficult exam c. Scores on a standardized test
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.