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Chapter 2: Problem 50

Shape of home prices? According to the National Association of Home Builders, the median selling price of new homes in the United States in January 2007 was \(\$ 239,800\). Which of the following is the most plausible value for the standard deviation: $$ -\$ 15,000, \$ 1000, \$ 60,000, \text { or } \$ 1,000,000 ? \text { Why? } $$ Explain what's unrealistic about each of the other values.

Short Answer

Expert verified
The most plausible standard deviation is \(\$60,000\); the others are unrealistic because they are either negative, too small, or excessively large.

Step by step solution

01

Understanding the problem

We need to estimate a plausible standard deviation for home prices given that the median selling price is \(\\(239,800\). The values given are \(-\\)15,000\), \(\\(1,000\), \(\\)60,000\), and \(\$1,000,000\).
02

Analyzing negative standard deviation

The standard deviation cannot be negative. Therefore, \(-\$15,000\) is not realistic as the standard deviation represents the average amount by which the values deviate from the mean, and it must always be non-negative.
03

Evaluating small standard deviation

A standard deviation of \(\\(1,000\) is too small given the median home price of \(\\)239,800\). This would imply that most home prices are very close to the median, which is unlikely in large datasets like nationwide home prices.
04

Evaluating large standard deviation

A standard deviation of \(\$1,000,000\) is extremely large and would imply that home prices vary widely and some prices may be negative or exceedingly high, which is unrealistic for the national market.
05

Choosing the plausible standard deviation

A standard deviation of \(\$60,000\) is reasonable considering the diverse range of home prices in the U.S. This value accounts for variation due to factors like location, size, and amenities, without being excessively large or small.

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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 key concept in statistics that measures the amount of variation or spread in a set of data. It tells us how much the values in a data set differ from the mean (average) value. In simple terms, it's a way to quantify how diverse or similar the data points are.

For housing prices, understanding standard deviation helps to get a sense of how much the prices of different houses might differ from the average home price. Unlike the mean, which can be misleading due to extremely high or low values, the standard deviation gives a more nuanced view of data distribution. Let's summarize why:
  • The standard deviation is always a non-negative value. A negative standard deviation is not possible, as it represents a distance or spread, which cannot be less than zero.
  • An extremely low standard deviation indicates that the prices are very close to each other, which is unlikely for something like housing prices that have various influencing factors such as location and home features.
  • Conversely, a very high standard deviation would suggest wildly varying prices, potentially reaching unrealistic negative or excessively high values, which doesn't align with possible real-world scenarios.
Median
The median is a central measure of data located at the midpoint of the dataset when it is ordered from lowest to highest. The significance of the median is that it is less affected by outliers or extremely skewed data compared to the mean.

In the context of housing market analysis, the median price of homes provides a more accurate picture of the typical home price in the market, because it is not as heavily influenced by extremely high or low prices as the mean might be. Here are some reasons why the median is often used in housing data:
  • The median home price is the value at which half the homes sell for more and half sell for less, making it a robust indicator of a typical property price.
  • It is especially useful in markets with significant price fluctuations, ensuring that single unusual sales don't skew the data interpretation.
  • The median offers clear insights into the housing market trend by focusing on the central point, reducing the impact of statistical noise present due to outliers.
Housing Market Analysis
Housing market analysis involves understanding various metrics that help determine the state of the real estate market. This type of analysis frequently uses statistical tools like median and standard deviation to assess the distribution of home values.

Conducting housing market analysis helps in making informed decisions whether you're buying, selling, or investing in real estate. Here are some key elements:
  • Utilizing the median home price gives a clearer view of pricing, indicating the market's mid-point rather than the average which could be skewed by high-end sales.
  • Standard deviation helps identify the price variability across different homes, providing insight into how uniform or dispersed the market is.
  • By examining both standard deviation and median, analysts can assess factors like affordability, market growth potential, and economic health, aiding in strategic planning.
Through these measures, participants in the housing market can understand trends and variations, crucial for addressing both short and long-term real estate goals.

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