Calculate The Sample Standard Deviation (using Your Calculator) Of The Data Shown, To Two Decimal Places.$\[ \begin{tabular}{|r|} \hline \multicolumn{1}{|c|}{$x$} \\ \hline 24.4 \\ \hline 1 \\ \hline 9.2 \\ \hline 27.2 \\ \hline 5.9 \\ \hline

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Understanding Sample Standard Deviation

Sample standard deviation is a measure of the amount of variation or dispersion from the average of a set of data. It is an essential statistical concept used in various fields, including mathematics, science, and finance. In this article, we will guide you through the process of calculating the sample standard deviation using a calculator.

Step 1: Calculate the Sample Mean

To calculate the sample standard deviation, we first need to find the sample mean. The sample mean is the average of all the data points in the set. To calculate the sample mean, we add up all the data points and divide by the number of data points.

Calculating the Sample Mean

Sample mean = (24.4 + 1 + 9.2 + 27.2 + 5.9) / 5
Sample mean = 67.7 / 5
Sample mean = 13.54

Step 2: Calculate the Deviations from the Mean

Next, we need to calculate the deviations from the mean. This involves subtracting the sample mean from each data point.

Calculating the Deviations

Deviations:
(24.4 - 13.54) = 10.86
(1 - 13.54) = -12.54
(9.2 - 13.54) = -4.34
(27.2 - 13.54) = 13.66
(5.9 - 13.54) = -7.64

Step 3: Calculate the Squared Deviations

Now, we need to calculate the squared deviations. This involves squaring each deviation.

Calculating the Squared Deviations

Squared Deviations:
(10.86)^2 = 117.69
(-12.54)^2 = 157.31
(-4.34)^2 = 18.85
(13.66)^2 = 186.51
(-7.64)^2 = 58.01

Step 4: Calculate the Sum of Squared Deviations

Next, we need to calculate the sum of squared deviations.

Calculating the Sum of Squared Deviations

Sum of Squared Deviations:
117.69 + 157.31 + 18.85 + 186.51 + 58.01 = 538.37

Step 5: Calculate the Sample Variance

Now, we need to calculate the sample variance. This involves dividing the sum of squared deviations by the number of data points minus one.

Calculating the Sample Variance

Sample Variance:
538.37 / (5 - 1)
Sample Variance:
538.37 / 4
Sample Variance:
134.59

Step 6: Calculate the Sample Standard Deviation

Finally, we need to calculate the sample standard deviation. This involves taking the square root of the sample variance.

Calculating the Sample Standard Deviation

Sample Standard Deviation:
√134.59
Sample Standard Deviation:
11.58

Conclusion

In this article, we have guided you through the process of calculating the sample standard deviation using a calculator. We have also provided the calculations for the sample mean, deviations, squared deviations, sum of squared deviations, sample variance, and sample standard deviation. By following these steps, you can calculate the sample standard deviation of any set of data.

Discussion

The sample standard deviation is an essential statistical concept used in various fields. It provides a measure of the amount of variation or dispersion from the average of a set of data. In this article, we have provided a step-by-step guide on how to calculate the sample standard deviation using a calculator. We have also provided the calculations for the sample mean, deviations, squared deviations, sum of squared deviations, sample variance, and sample standard deviation.

Real-World Applications

The sample standard deviation has various real-world applications. It is used in finance to calculate the risk of investments, in science to analyze data, and in engineering to design systems. By understanding the sample standard deviation, you can make informed decisions and analyze data effectively.

Common Mistakes

When calculating the sample standard deviation, there are several common mistakes to avoid. These include:

  • Not using the correct formula for the sample standard deviation
  • Not calculating the sample mean correctly
  • Not calculating the deviations correctly
  • Not calculating the squared deviations correctly
  • Not calculating the sum of squared deviations correctly
  • Not calculating the sample variance correctly
  • Not calculating the sample standard deviation correctly

By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.

Conclusion

Frequently Asked Questions

Q: What is the sample standard deviation?

A: The sample standard deviation is a measure of the amount of variation or dispersion from the average of a set of data. It is an essential statistical concept used in various fields, including mathematics, science, and finance.

Q: How do I calculate the sample standard deviation?

A: To calculate the sample standard deviation, you need to follow these steps:

  1. Calculate the sample mean.
  2. Calculate the deviations from the mean.
  3. Calculate the squared deviations.
  4. Calculate the sum of squared deviations.
  5. Calculate the sample variance.
  6. Calculate the sample standard deviation.

Q: What is the difference between the sample standard deviation and the population standard deviation?

A: The sample standard deviation is used when you are working with a sample of data, while the population standard deviation is used when you are working with the entire population of data.

Q: How do I calculate the sample standard deviation using a calculator?

A: To calculate the sample standard deviation using a calculator, you need to follow these steps:

  1. Enter the data points into the calculator.
  2. Use the calculator to calculate the sample mean.
  3. Use the calculator to calculate the deviations from the mean.
  4. Use the calculator to calculate the squared deviations.
  5. Use the calculator to calculate the sum of squared deviations.
  6. Use the calculator to calculate the sample variance.
  7. Use the calculator to calculate the sample standard deviation.

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

A: Some common mistakes to avoid when calculating the sample standard deviation include:

  • Not using the correct formula for the sample standard deviation
  • Not calculating the sample mean correctly
  • Not calculating the deviations correctly
  • Not calculating the squared deviations correctly
  • Not calculating the sum of squared deviations correctly
  • Not calculating the sample variance correctly
  • Not calculating the sample standard deviation correctly

Q: What are some real-world applications of the sample standard deviation?

A: The sample standard deviation has various real-world applications, including:

  • Finance: The sample standard deviation is used to calculate the risk of investments.
  • Science: The sample standard deviation is used to analyze data.
  • Engineering: The sample standard deviation is used to design systems.

Q: How do I interpret the sample standard deviation?

A: The sample standard deviation provides a measure of the amount of variation or dispersion from the average of a set of data. A small sample standard deviation indicates that the data points are close to the mean, while a large sample standard deviation indicates that the data points are far from the mean.

Q: What are some limitations of the sample standard deviation?

A: Some limitations of the sample standard deviation include:

  • It is sensitive to outliers.
  • It is sensitive to non-normal data.
  • It is not suitable for small sample sizes.

Conclusion

In conclusion, the sample standard deviation is an essential statistical concept used in various fields. It provides a measure of the amount of variation or dispersion from the average of a set of data. By understanding the sample standard deviation, you can make informed decisions and analyze data effectively.