Find The Sample Variance And Standard Deviation For The Data Set: 19 , 10 , 5 , 6 , 12 19, 10, 5, 6, 12 19 , 10 , 5 , 6 , 12 Choose The Correct Answer Below. Fill In The Answer Box To Complete Your Choice. (Type An Integer Or A Decimal. Round To One Decimal Place As Needed.)A.

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Introduction

In statistics, variance and standard deviation are two fundamental measures of dispersion that help us understand the spread of a dataset. Variance measures the average of the squared differences from the mean, while standard deviation is the square root of the variance. In this article, we will explore how to find the sample variance and standard deviation for a given dataset.

What is Sample Variance?

Sample variance is a measure of the spread of a dataset, calculated as the average of the squared differences from the sample mean. It is denoted by the symbol s^2 and is calculated using the following formula:

s^2 = Σ(xi - x̄)^2 / (n - 1)

where xi is each individual data point, x̄ is the sample mean, and n is the number of data points.

What is Sample Standard Deviation?

Sample standard deviation is the square root of the sample variance and is denoted by the symbol s. It is calculated using the following formula:

s = √s^2

Calculating Sample Variance and Standard Deviation

Let's apply the formulas to the given dataset: 19,10,5,6,1219, 10, 5, 6, 12. First, we need to calculate the sample mean:

x̄ = (19 + 10 + 5 + 6 + 12) / 5 x̄ = 52 / 5 x̄ = 10.4

Next, we calculate the squared differences from the sample mean:

(19 - 10.4)^2 = 8.6^2 = 73.96 (10 - 10.4)^2 = -0.4^2 = 0.16 (5 - 10.4)^2 = -5.4^2 = 29.16 (6 - 10.4)^2 = -4.4^2 = 19.36 (12 - 10.4)^2 = 1.6^2 = 2.56

Now, we calculate the sample variance:

s^2 = (73.96 + 0.16 + 29.16 + 19.36 + 2.56) / (5 - 1) s^2 = 125.2 / 4 s^2 = 31.3

Finally, we calculate the sample standard deviation:

s = √31.3 s = 5.6

Conclusion

In this article, we have learned how to calculate the sample variance and standard deviation for a given dataset. We have applied the formulas to the dataset 19,10,5,6,1219, 10, 5, 6, 12 and obtained the sample variance of 31.3 and the sample standard deviation of 5.6. Understanding sample variance and standard deviation is essential in statistics, as they provide valuable insights into the spread of a dataset.

Discussion

  • What is the difference between sample variance and population variance?
  • How does the sample size affect the calculation of sample variance and standard deviation?
  • Can you think of any real-world applications of sample variance and standard deviation?

Answer

The correct answer is:

5.6

References

  • "Statistics for Dummies" by Deborah J. Rumsey
  • "Mathematics for Dummies" by Mary Jane Sterling
  • "Statistics: A First Course" by James T. McClave
    Sample Variance and Standard Deviation Q&A =============================================

Introduction

In our previous article, we explored the concept of sample variance and standard deviation, and how to calculate them for a given dataset. In this article, we will answer some frequently asked questions about sample variance and standard deviation.

Q: What is the difference between sample variance and population variance?

A: Sample variance is a measure of the spread of a sample dataset, while population variance is a measure of the spread of the entire population. The formula for sample variance is:

s^2 = Σ(xi - x̄)^2 / (n - 1)

where xi is each individual data point, x̄ is the sample mean, and n is the number of data points.

The formula for population variance is:

σ^2 = Σ(xi - μ)^2 / N

where xi is each individual data point, μ is the population mean, and N is the total number of data points.

Q: How does the sample size affect the calculation of sample variance and standard deviation?

A: The sample size affects the calculation of sample variance and standard deviation in the following ways:

  • As the sample size increases, the sample variance and standard deviation tend to decrease.
  • As the sample size decreases, the sample variance and standard deviation tend to increase.
  • If the sample size is very small (less than 30), the sample variance and standard deviation may not be reliable estimates of the population variance and standard deviation.

Q: Can you think of any real-world applications of sample variance and standard deviation?

A: Yes, sample variance and standard deviation have many real-world applications, including:

  • Quality control: Sample variance and standard deviation can be used to monitor the quality of a product or process.
  • Finance: Sample variance and standard deviation can be used to calculate the risk of an investment.
  • Medicine: Sample variance and standard deviation can be used to analyze the results of a clinical trial.
  • Social sciences: Sample variance and standard deviation can be used to analyze the results of a survey or experiment.

Q: How do you calculate the sample variance and standard deviation for a dataset with missing values?

A: If a dataset has missing values, you can calculate the sample variance and standard deviation using the following steps:

  1. Remove the missing values from the dataset.
  2. Calculate the sample mean and sample variance using the remaining data points.
  3. Use the sample mean and sample variance to calculate the sample standard deviation.

Q: Can you explain the concept of degrees of freedom in the context of sample variance and standard deviation?

A: Yes, degrees of freedom is a concept that is used in the calculation of sample variance and standard deviation. It refers to the number of independent pieces of information that are used to calculate the sample variance and standard deviation.

In the formula for sample variance, the degrees of freedom is n - 1, where n is the number of data points. This is because the sample mean is used to calculate the sample variance, and the sample mean is a function of the data points.

Q: How do you calculate the sample variance and standard deviation for a dataset with outliers?

A: If a dataset has outliers, you can calculate the sample variance and standard deviation using the following steps:

  1. Remove the outliers from the dataset.
  2. Calculate the sample mean and sample variance using the remaining data points.
  3. Use the sample mean and sample variance to calculate the sample standard deviation.

Alternatively, you can use a robust method to calculate the sample variance and standard deviation, such as the median absolute deviation (MAD) or the interquartile range (IQR).

Conclusion

In this article, we have answered some frequently asked questions about sample variance and standard deviation. We have discussed the difference between sample variance and population variance, the effect of sample size on the calculation of sample variance and standard deviation, and the concept of degrees of freedom. We have also discussed how to calculate the sample variance and standard deviation for a dataset with missing values, outliers, and how to use robust methods to calculate the sample variance and standard deviation.