Find The Correlation Coefficient, $r$, Of The Data Described Below.A Bumper Sticker Company Is Developing A New Line Of Stickers Promoting Local Sports Teams. To Estimate Demand, The Company's Marketing Department Conducted A Phone Survey Of

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Introduction

In statistics, the correlation coefficient is a measure of the relationship between two variables. It is a numerical value that ranges from -1 to 1, indicating the strength and direction of the linear relationship between the variables. In this article, we will explore how to find the correlation coefficient, denoted as $r$, using a real-world example.

Problem Statement

A bumper sticker company is developing a new line of stickers promoting local sports teams. To estimate demand, the company's marketing department conducted a phone survey of 10 cities, asking how many stickers each city would buy if they were available. The results are shown in the table below:

City Number of Stickers
A 100
B 120
C 90
D 110
E 130
F 80
G 105
H 125
I 95
J 115

Step 1: Calculate the Mean of Each Variable

To find the correlation coefficient, we need to calculate the mean of each variable. The mean is the average value of the data.

Calculating the Mean of the Number of Stickers

To calculate the mean, we add up all the values and divide by the number of observations.

# Define the data
sticker_data <- c(100, 120, 90, 110, 130, 80, 105, 125, 95, 115)

mean_stickers <- mean(sticker_data)
print(paste("Mean number of stickers:", mean_stickers))

Calculating the Mean of the City Names

Since the city names are not numerical values, we will use them as labels for the data.

# Define the city names
city_names <- c("A", "B", "C", "D", "E", "F", "G", "H", "I", "J")


print(paste("City names:", paste(city_names, collapse = ", ")))

Step 2: Calculate the Deviations from the Mean

Next, we need to calculate the deviations from the mean for each variable. The deviation is the difference between each value and the mean.

Calculating the Deviations from the Mean for the Number of Stickers

# Calculate the deviations from the mean
deviations_stickers <- sticker_data - mean_stickers


print(paste("Deviations from the mean number of stickers:", paste(deviations_stickers, collapse = ", ")))

Calculating the Deviations from the Mean for the City Names

Since the city names are not numerical values, we will not calculate the deviations for them.

Step 3: Calculate the Covariance

The covariance is a measure of how much the two variables change together. It is calculated by multiplying the deviations from the mean for each variable and summing them up.

Calculating the Covariance

# Calculate the covariance
covariance <- sum(deviations_stickers * deviations_stickers)


print(paste("Covariance:", covariance))

Step 4: Calculate the Correlation Coefficient

The correlation coefficient is calculated by dividing the covariance by the product of the standard deviations of the two variables.

Calculating the Standard Deviation of the Number of Stickers

# Calculate the standard deviation
std_dev_stickers <- sqrt(var(sticker_data))


print(paste("Standard deviation of number of stickers:", std_dev_stickers))

Calculating the Correlation Coefficient

# Calculate the correlation coefficient
correlation_coefficient <- covariance / (std_dev_stickers * std_dev_stickers)


print(paste("Correlation coefficient (r):", correlation_coefficient))

Conclusion

In this article, we have learned how to find the correlation coefficient, denoted as $r$, using a real-world example. We have calculated the mean, deviations from the mean, covariance, and standard deviation of the number of stickers. Finally, we have calculated the correlation coefficient by dividing the covariance by the product of the standard deviations. The correlation coefficient is a measure of the relationship between two variables and can be used to make predictions and understand the underlying relationships between the variables.

Discussion

The correlation coefficient is a widely used statistical measure that can be used in various fields, including finance, economics, and social sciences. It is a useful tool for understanding the relationships between variables and making predictions. However, it is essential to note that correlation does not imply causation, and there may be other factors that influence the relationship between the variables.

Limitations

One of the limitations of the correlation coefficient is that it only measures the linear relationship between the variables. It does not account for non-linear relationships, which may be present in the data. Additionally, the correlation coefficient is sensitive to outliers and may not accurately reflect the relationship between the variables if there are extreme values in the data.

Future Work

In future work, we can explore other statistical measures, such as regression analysis, to understand the relationships between the variables. We can also use other data visualization techniques, such as scatter plots and heat maps, to visualize the relationships between the variables.

References

Q&A: Frequently Asked Questions About Correlation Coefficient

Q: What is the correlation coefficient?

A: The correlation coefficient is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It is a numerical value that ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.

Q: How is the correlation coefficient calculated?

A: The correlation coefficient is calculated using the following formula:

r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])

where xi and yi are the individual data points, x̄ and ȳ are the means of the two variables, and Σ denotes the sum.

Q: What is the difference between correlation and causation?

A: Correlation does not imply causation. Just because two variables are related, it does not mean that one variable causes the other. There may be other factors that influence the relationship between the variables.

Q: What are the limitations of the correlation coefficient?

A: The correlation coefficient only measures the linear relationship between the variables. It does not account for non-linear relationships, which may be present in the data. Additionally, the correlation coefficient is sensitive to outliers and may not accurately reflect the relationship between the variables if there are extreme values in the data.

Q: How can I use the correlation coefficient in real-world applications?

A: The correlation coefficient can be used in various fields, including finance, economics, and social sciences. It can be used to:

  • Identify relationships between variables
  • Make predictions
  • Understand the underlying relationships between variables
  • Identify potential causes of relationships between variables

Q: What are some common mistakes to avoid when using the correlation coefficient?

A: Some common mistakes to avoid when using the correlation coefficient include:

  • Assuming correlation implies causation
  • Failing to account for non-linear relationships
  • Ignoring outliers and extreme values
  • Using the correlation coefficient without considering other statistical measures

Q: How can I interpret the correlation coefficient?

A: The correlation coefficient can be interpreted as follows:

  • A correlation coefficient of 1 indicates a perfect positive linear relationship
  • A correlation coefficient of -1 indicates a perfect negative linear relationship
  • A correlation coefficient of 0 indicates no linear relationship
  • A correlation coefficient between -1 and 1 indicates a linear relationship, but the strength and direction of the relationship are not perfect

Q: What are some common applications of the correlation coefficient?

A: Some common applications of the correlation coefficient include:

  • Finance: to identify relationships between stock prices and other economic indicators
  • Economics: to understand the relationships between economic indicators, such as GDP and inflation
  • Social sciences: to identify relationships between variables, such as education and income

Conclusion

In this article, we have answered some frequently asked questions about the correlation coefficient. We have discussed its definition, calculation, limitations, and applications. We have also provided some tips on how to interpret and use the correlation coefficient in real-world applications.