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Chapter 5: Problem 14

An auction house released a list of 25 recently sold paintings. Eight artists were represented in these sales. The sale price of each painting appears on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist \((x)\) and sale price \((y)\) ? Why or why not?

Short Answer

Expert verified
No, the correlation coefficient would not be an appropriate way to summarize the relationship between artist and sale price. It's because the correlation coefficient is used to indicate the strength and direction of a linear relationship between two numeric variables, and one of the variables 'artist' in this scenario is not numeric but categorical.

Step by step solution

01

Understanding Correlation Coefficient

The correlation coefficient is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. The value ranges between -1.0 and 1.0. A correlation of -1.0 shows a perfect negative correlation, while a correlation of 1.0 shows a perfect positive correlation.
02

Application of Correlation Coefficient

Correlation coefficient is used when there is a quantitative (numerically measured) relationship between two variables, and both variables are numeric. In the exercise, 'artist' is a categorical variable and 'sale price' is a quantitative variable.
03

Inappropriateness of Correlation Coefficient

In this scenario, correlating a categorical variable (artist) with a numerical one (sale price) would be inappropriate using the correlation coefficient. It's because it cannot accurately represent the type of relationship that might exist between a categorical and a numerical variable.

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Key Concepts

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

Statistical Measure
A statistical measure is a quantitative evaluation of certain characteristics taken from a sample of data. It's used extensively in statistical analysis to condense large volumes of data into simpler, meaningful figures that can inform decision-making or further analysis. An essential statistical measure is the correlation coefficient.

When studying relationships between two continuous, numerical variables鈥攍ike a person's height and weight鈥攚e use the correlation coefficient to express how strongly related these variables are. Its calculation provides a value that indicates the nature and strength of the association between the variables. A key point to remember relating to our auction house scenario is that the correlation coefficient is best applied to numerical data; it loses its utility when working with non-numeric data, such as categorical variables.
Quantitative Relationship
A quantitative relationship in statistics denotes a connection between variables that can be measured and expressed numerically. In research or data analysis, discovering quantitative relationships helps to understand patterns and predict future outcomes based on numerical values. The correlation coefficient shines in these scenarios, offering a clear, concise measure of just how in-sync two quantitative variables are (whether they tend to increase together, decrease together, or have no discernible pattern at all).For instance, when we consider the sale prices of paintings from an auction, we're dealing with hard numbers that can be compared, plotted, and analyzed to find patterns. However, as mentioned in our exercise, pairing a quantitative variable like sale price with a categorical one such as artist's name skews the analysis, because we cannot quantify the category 'artist' in the same way we can with sale prices.
Categorical Variable
Categorical variables represent types, names, or other classifications that do not naturally carry a numerical value. They can include characteristics like color, type of cuisine, or, as with our auction house example, artists. These variables are not about mentioning 'how much' but about stating 'which kind' or 'what type.'

It is crucial to understand that statistical tools designed for numeric data will not work properly with categorical variables. This is because categories encapsulate qualitative differences, not the quantitative differences measured by tools like the correlation coefficient. In our exercise, artists' names are a categorical variable, and trying to calculate their correlation with the sale prices, which are numerical, would not yield meaningful results because we cannot assign a numerical degree of 'artist-ness' to each painting sold.

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Most popular questions from this chapter

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