Consider The Six Numbers: 9, 12, 16, 25, 29, 60.If We Want To Reduce The Range, The Inter-quartile Range And The Standard Deviation, Which Number Should Be Removed?A. 9B. 16C. 25D. 60Which Of The Following CANNOT Be Read Directly From A Box-and-whisker

Apr 8, 2025 by ADMIN 253 views

**Understanding the Problem: Reducing Range, Inter-Quartile Range, and Standard Deviation**

When dealing with a dataset, it's essential to understand the measures of central tendency and dispersion. The range, inter-quartile range (IQR), and standard deviation are three crucial measures that help us understand the spread of data. In this article, we'll explore how to reduce these measures by removing a single number from the dataset.

The Dataset: 9, 12, 16, 25, 29, 60

We're given a dataset of six numbers: 9, 12, 16, 25, 29, 60. Our goal is to reduce the range, IQR, and standard deviation by removing one number from the dataset.

Range: The Difference Between the Largest and Smallest Values

The range is the simplest measure of dispersion, calculated as the difference between the largest and smallest values in the dataset. In this case, the range is 60 - 9 = 51.

Inter-Quartile Range (IQR): The Difference Between the 75th and 25th Percentiles

The IQR is a measure of dispersion that's more robust than the range. It's calculated as the difference between the 75th percentile (Q3) and the 25th percentile (Q1). In this case, the IQR is Q3 - Q1.

Standard Deviation: A Measure of the Spread of Data

The standard deviation is a measure of the spread of data, calculated as the square root of the variance. It's a more sensitive measure of dispersion than the range or IQR.

Which Number Should Be Removed?

To reduce the range, IQR, and standard deviation, we need to remove a number that has the most significant impact on these measures. Let's analyze each option:

A. 9: Removing 9 would reduce the range, but it would also increase the IQR and standard deviation, as the dataset would become more skewed.
B. 16: Removing 16 would have a moderate impact on the range, IQR, and standard deviation. However, it's not the most significant number to remove.
C. 25: Removing 25 would have a moderate impact on the range, IQR, and standard deviation. However, it's not the most significant number to remove.
D. 60: Removing 60 would have the most significant impact on the range, IQR, and standard deviation. The range would decrease from 51 to 29, the IQR would decrease, and the standard deviation would decrease.

Conclusion

Based on our analysis, removing 60 would have the most significant impact on reducing the range, IQR, and standard deviation. Therefore, the correct answer is:

D. 60

Which of the Following CANNOT be Read Directly from a Box-and-Whisker Plot?

A box-and-whisker plot is a graphical representation of a dataset that shows the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. However, there are some measures that cannot be read directly from a box-and-whisker plot:

Mean: The mean is the average of all values in the dataset. It cannot be read directly from a box-and-whisker plot.
Standard Deviation: The standard deviation is a measure of the spread of data. It cannot be read directly from a box-and-whisker plot.
Range: The range is the difference between the largest and smallest values in the dataset. It can be read directly from a box-and-whisker plot.
Inter-Quartile Range (IQR): The IQR is the difference between the 75th percentile (Q3) and the 25th percentile (Q1). It can be read directly from a box-and-whisker plot.

Conclusion

In conclusion, a box-and-whisker plot is a useful tool for visualizing a dataset and understanding its distribution. However, some measures, such as the mean and standard deviation, cannot be read directly from a box-and-whisker plot.

References

Box-and-Whisker Plot: A box-and-whisker plot is a graphical representation of a dataset that shows the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
Range: The range is the difference between the largest and smallest values in the dataset.
Inter-Quartile Range (IQR): The IQR is the difference between the 75th percentile (Q3) and the 25th percentile (Q1).
Standard Deviation: The standard deviation is a measure of the spread of data.
Mean: The mean is the average of all values in the dataset.
Q&A: Understanding the Problem and Solution =============================================

Q: What is the problem we're trying to solve?

A: We're given a dataset of six numbers: 9, 12, 16, 25, 29, 60. Our goal is to reduce the range, inter-quartile range (IQR), and standard deviation by removing one number from the dataset.

Q: Why do we want to reduce the range, IQR, and standard deviation?

A: Reducing the range, IQR, and standard deviation can help us understand the spread of data and make it more manageable. It can also help us identify outliers and anomalies in the data.

Q: Which number should we remove to reduce the range, IQR, and standard deviation?

A: Based on our analysis, removing 60 would have the most significant impact on reducing the range, IQR, and standard deviation.

Q: Why is removing 60 the best option?

A: Removing 60 would decrease the range from 51 to 29, the IQR would decrease, and the standard deviation would decrease. This is because 60 is the largest value in the dataset, and removing it would reduce the spread of data.

Q: What are some other measures that cannot be read directly from a box-and-whisker plot?

A: Some other measures that cannot be read directly from a box-and-whisker plot include:

Mean: The mean is the average of all values in the dataset.
Standard Deviation: The standard deviation is a measure of the spread of data.
Range: While the range can be estimated from a box-and-whisker plot, it's not always possible to determine the exact range.

Q: What are some benefits of using a box-and-whisker plot?

A: Some benefits of using a box-and-whisker plot include:

Visualizing data: A box-and-whisker plot provides a visual representation of the data, making it easier to understand and analyze.
Identifying outliers: A box-and-whisker plot can help identify outliers and anomalies in the data.
Comparing datasets: A box-and-whisker plot can be used to compare datasets and identify differences.

Q: What are some limitations of using a box-and-whisker plot?

A: Some limitations of using a box-and-whisker plot include:

Limited information: A box-and-whisker plot only provides information about the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
Difficulty in estimating range: While the range can be estimated from a box-and-whisker plot, it's not always possible to determine the exact range.
Difficulty in estimating standard deviation: The standard deviation is a measure of the spread of data, and it's not always possible to determine the exact standard deviation from a box-and-whisker plot.

Conclusion

In conclusion, a box-and-whisker plot is a useful tool for visualizing a dataset and understanding its distribution. However, some measures, such as the mean standard deviation, cannot be read directly from a box-and-whisker plot. By understanding the problem and solution, we can make informed decisions and take action to reduce the range, IQR, and standard deviation.