An Insurance Company Crashed Four Cars In Succession At 5 Miles Per Hour. The Cost Of Repair For Each Of The Four Crashes Was $\$417, \$459, \$415, \$219$. Compute The Range, Sample Variance, And Sample Standard Deviation

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Introduction

In this article, we will be discussing the computation of the range, sample variance, and sample standard deviation of a given dataset. The dataset consists of the cost of repair for four cars that were crashed in succession at 5 miles per hour. The costs are $$417, $459, $415, $219219. We will be using these values to calculate the range, sample variance, and sample standard deviation.

Calculating the Range

The range is the difference between the highest and lowest values in a dataset. To calculate the range, we need to find the highest and lowest values in the dataset.

  • The highest value in the dataset is $$459.
  • The lowest value in the dataset is $$219.

Now, we can calculate the range by subtracting the lowest value from the highest value.

Range = Highest Value - Lowest Value Range = $$459 - $$219 Range = $$240

Calculating the Sample Variance

The sample variance is a measure of the spread of a dataset. It is calculated by finding the average of the squared differences between each value and the mean of the dataset.

First, we need to calculate the mean of the dataset.

Mean = (Sum of all values) / (Number of values) Mean = ($$417 + $$459 + $$415 + $$219) / 4 Mean = $$1510 / 4 Mean = $$377.5

Next, we need to calculate the squared differences between each value and the mean.

Squared differences:

  • ($$417 - $377.5)2=(377.5)^2 = ($39.5)^2 = $$1552.25
  • ($$459 - $377.5)2=(377.5)^2 = ($81.5)^2 = $$6642.25
  • ($$415 - $377.5)2=(377.5)^2 = ($37.5)^2 = $$1406.25
  • ($$219 - $377.5)2=(377.5)^2 = ($-158.5)^2 = $$25100.25

Now, we can calculate the sample variance by finding the average of the squared differences.

Sample Variance = (Sum of squared differences) / (Number of values - 1) Sample Variance = ($$1552.25 + $$6642.25 + $$1406.25 + $$25100.25) / 3 Sample Variance = $$38401 / 3 Sample Variance = $$12800.33

Calculating the Sample Standard Deviation

The sample standard deviation is the square root of the sample variance. It is a measure of the spread of a dataset.

Sample Standard Deviation = √(Sample Variance) Sample Standard Deviation = √($$12800.33) Sample Standard Deviation = $$113.01

Conclusion

In this article, we calculated the range, sample variance, and sample standard deviation of a given dataset. The dataset consisted of the cost of repair for four cars that were crashed in succession at 5 miles per hour. The costs were $$417, $$459, $$415, $$219. We found that the range was $$240, the sample variance was $$12800.33, and the sample standard deviation was $$113.01. These values provide a measure of the spread of the dataset and can be used in a variety of statistical analyses.

References

  • "Statistics for Dummies". John Wiley & Sons. 2019.
  • "Mathematics for Dummies". John Wiley & Sons. 2018.
  • "Statistics and Probability with Technology". McGraw-Hill. 2017.

Q&A

Q: What is the range of the dataset?

A: The range of the dataset is the difference between the highest and lowest values in the dataset. In this case, the highest value is $$459 and the lowest value is $$219. Therefore, the range is $$240.

Q: How is the sample variance calculated?

A: The sample variance is calculated by finding the average of the squared differences between each value and the mean of the dataset. First, we need to calculate the mean of the dataset, which is $$377.5. Then, we need to calculate the squared differences between each value and the mean. Finally, we need to find the average of these squared differences.

Q: What is the sample variance of the dataset?

A: The sample variance of the dataset is $$12800.33.

Q: How is the sample standard deviation calculated?

A: The sample standard deviation is the square root of the sample variance. Therefore, the sample standard deviation of the dataset is $\sqrt{$12800.33} = $$113.01.

Q: What is the significance of the range, sample variance, and sample standard deviation?

A: The range, sample variance, and sample standard deviation are all measures of the spread of a dataset. The range is the difference between the highest and lowest values in the dataset, while the sample variance and sample standard deviation are measures of the average squared difference between each value and the mean of the dataset. These values can be used in a variety of statistical analyses, such as hypothesis testing and confidence intervals.

Q: Can you provide an example of how to use the range, sample variance, and sample standard deviation in a real-world scenario?

A: Yes, here is an example. Suppose we are a car insurance company and we want to know how much variation there is in the cost of repairs for cars that are crashed at 5 miles per hour. We collect a dataset of the cost of repairs for four cars that were crashed in succession at 5 miles per hour, and we find that the range is $$240, the sample variance is $$12800.33, and the sample standard deviation is $$113.01. This information can be used to inform our pricing and risk assessment strategies.

Q: What are some common applications of the range, sample variance, and sample standard deviation?

A: The range, sample variance, and sample standard deviation are commonly used in a variety of fields, including statistics, data analysis, and research. They are used to describe the spread of a dataset and to make inferences about the population from which the dataset was sampled. They are also used in hypothesis testing and confidence intervals.

Q: Can you provide some tips for calculating the range, sample variance, and sample standard deviation?

A: Yes, here are some tips:

  • Make sure to calculate the mean of the dataset correctly.
  • Make sure to calculate the squared differences between each value and the mean correctly.
  • Make sure to find the average of the squared differences correctly.
  • Make sure to take the square root of the sample variance to find the sample standard deviation.
  • Use a calculator or computer software to help with the calculations, if possible.

Q: What are some common mistakes to avoid when calculating the range, sample variance, and sample standard deviation?

A: Some common mistakes to avoid include:

  • Calculating the mean incorrectly.
  • Calculating the squared differences between each value and the mean incorrectly.
  • Finding the average of the squared differences incorrectly.
  • Not taking the square root of the sample variance to find the sample standard deviation.
  • Not using a calculator or computer software to help with the calculations, if possible.

Q: Can you provide some resources for learning more about the range, sample variance, and sample standard deviation?

A: Yes, here are some resources:

  • "Statistics for Dummies". John Wiley & Sons. 2019.
  • "Mathematics for Dummies". John Wiley & Sons. 2018.
  • "Statistics and Probability with Technology". McGraw-Hill. 2017.
  • Online tutorials and videos on statistics and data analysis.
  • Online courses and degree programs in statistics and data analysis.