Range and Its Significance

Oct 13, 2025 by recoveryshake.com 132 views

Given a set of data: ${$ X, 9, 11, 13, Y, 20$}$, we are provided with various statistical measures such as the range, median, mean, and interquartile range. Our goal is to determine the values of ${$ X$}$ and ${$ Y$}$ using the given information.

Range and Its Significance

The range of a data set is the difference between the highest and lowest values. In this case, the range is given as ${$ = 17$}$. This means that the difference between the highest and lowest values in the data set is 17.

Calculating the Range

To calculate the range, we need to find the highest and lowest values in the data set. Let's assume that ${$ X$}$ is the lowest value and ${$ Y$}$ is the highest value. We can write the equation as:

${$ Y - X = 17$}$

Median and Its Significance

The median of a data set is the middle value when the data is arranged in ascending order. In this case, the median is given as ${$ = 10$}$. This means that the middle value of the data set is 10.

Calculating the Median

To calculate the median, we need to arrange the data in ascending order. Let's assume that the data set is: ${$ X, 9, 11, 13, Y, 20$}$. Since the median is 10, we can conclude that ${$ X$}$ is less than or equal to 10.

Mean and Its Significance

The mean of a data set is the average of all the values. In this case, the mean is given as ${$ = 12$}$. This means that the average of all the values in the data set is 12.

Calculating the Mean

To calculate the mean, we need to add up all the values and divide by the total number of values. Let's assume that the sum of all the values is ${$ S$}$. We can write the equation as:

${$ S / 6 = 12$}$

Solving for ${$ S$}$, we get:

${$ S = 72$}$

Now, let's add up the given values: ${ $9 + 11 + 13 + 20 = 53\$}$ . We can write the equation as:

${$ X + Y = 72 - 53$}$

${$ X + Y = 19$}$

Interquartile Range and Its Significance

The interquartile range (IQR) of a data set is the difference between the third quartile (Q3) and the first quartile (Q1). In this case, the IQR is given as ${$ = 7$}$. This means that the difference between Q3 and Q1 is 7.

Calculating the Interquartile Range

To calculate the IQR, we need to find Q3 and Q1. Let's assume that Q3 is the third value in the data set when arranged in ascending order. We can write the equation as:

${$ Q3 - Q1 = 7$}$

Since the median is 10, we can conclude that Q1 is less than or equal to 10.

Finding the Values of ${$ X$}$ and ${$ Y$}$

Now that we have the equations:

${$ Y - X = 17$}{ $X + Y = 19\$}$

We can solve for ${$ X$}$ and ${$ Y$}$. Subtracting the first equation from the second equation, we get:

${ $2X = 2\$}$

Dividing both sides by 2, we get:

${$ X = 1$}$

Now, let's substitute ${$ X = 1$}$ into the first equation:

${$ Y - 1 = 17$}$

Adding 1 to both sides, we get:

${$ Y = 18$}$

Therefore, the values of ${$ X$}$ and ${$ Y$}$ are 1 and 18, respectively.

Conclusion

In this article, we used the given data set and statistical measures to determine the values of ${$ X$}$ and ${$ Y$}$. We calculated the range, median, mean, and interquartile range, and used the equations to solve for the missing values. The values of ${$ X$}$ and ${$ Y$}$ are 1 and 18, respectively.

References

[1] Wikipedia. (n.d.). Range (statistics). Retrieved from https://en.wikipedia.org/wiki/Range_(statistics)
[2] Wikipedia. (n.d.). Median. Retrieved from https://en.wikipedia.org/wiki/Median
[3] Wikipedia. (n.d.). Mean. Retrieved from https://en.wikipedia.org/wiki/Mean
[4] Wikipedia. (n.d.). Interquartile range. Retrieved from https://en.wikipedia.org/wiki/Interquartile_range
Frequently Asked Questions (FAQs) About the Given Data Set ================================================================

In the previous article, we discussed how to determine the values of ${$ X$}$ and ${$ Y$}$ using the given data set and statistical measures. Here are some frequently asked questions (FAQs) about the given data set:

Q: What is the range of the data set?

A: The range of the data set is 17, which means that the difference between the highest and lowest values is 17.

Q: How is the median calculated?

A: The median is the middle value when the data is arranged in ascending order. In this case, the median is 10, which means that the middle value of the data set is 10.

Q: What is the significance of the mean?

A: The mean is the average of all the values in the data set. In this case, the mean is 12, which means that the average of all the values in the data set is 12.

Q: How is the interquartile range (IQR) calculated?

A: The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). In this case, the IQR is 7, which means that the difference between Q3 and Q1 is 7.

Q: What are the values of ${$ X$}$ and ${$ Y$}$?

A: The values of ${$ X$}$ and ${$ Y$}$ are 1 and 18, respectively.

Q: How were the values of ${$ X$}$ and ${$ Y$}$ determined?

A: The values of ${$ X$}$ and ${$ Y$}$ were determined using the given data set and statistical measures. We calculated the range, median, mean, and IQR, and used the equations to solve for the missing values.

Q: What are some common applications of statistical measures?

A: Statistical measures such as range, median, mean, and IQR are commonly used in various fields such as business, economics, and social sciences. They are used to describe and analyze data, make predictions, and identify trends.

Q: How can I apply statistical measures to my own data set?

A: To apply statistical measures to your own data set, you can follow these steps:

Collect and organize your data.
Calculate the range, median, mean, and IQR.
Use the equations to solve for the missing values.
Interpret the results and make conclusions.

Q: What are some common mistakes to avoid when working with statistical measures?

A: Some common mistakes to avoid when working with statistical measures include:

Not checking for outliers and anomalies.
Not considering the distribution of the data.
Not using the correct formula for the statistical measure.
Not interpreting the results correctly.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the given data set. We discussed the range, median, mean, and IQR, and how to determine the values of ${$ X$}$ and ${$ Y$}$. We also provided some tips and common mistakes to avoid when working with statistical measures.