The Formula for Unbiased Estimation of Population Variance

Oct 13, 2025 by talynt.org 100 views

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Introduction

In statistics, the population variance is a measure of the spread or dispersion of a population. It is an essential concept in descriptive statistics and is used as a building block for more advanced statistical analyses. However, in many real-world scenarios, we do not have access to the entire population, and we are left with a random sample of data. In such cases, we need to estimate the population variance using the sample data. In this article, we will discuss how to find the unbiased estimate of the population variance using the given data.

What is Unbiased Estimation?

Unbiased estimation is a statistical concept that refers to the process of estimating a population parameter using a sample statistic in such a way that the expected value of the estimator is equal to the true population parameter. In other words, an unbiased estimator is one that, on average, gives the correct value of the population parameter.

The Formula for Unbiased Estimation of Population Variance

The formula for the unbiased estimation of the population variance is given by:

σ² = (n-1) * s² / (n-1)

where σ² is the population variance, n is the sample size, and s² is the sample variance.

Calculating the Sample Variance

The sample variance (s²) is calculated using the following formula:

s² = Σ(xi - x̄)² / (n-1)

where xi is each individual data point, x̄ is the sample mean, and n is the sample size.

Calculating the Sample Mean

The sample mean (x̄) is calculated using the following formula:

x̄ = Σxi / n

where xi is each individual data point and n is the sample size.

Example Data

Let's consider the given data for the time taken (in days) for a random sample of letters to be delivered:

Number of days for letter to be delivered	Number of letters
1	24
2	32
3	29
4	9
5	6

Calculating the Sample Mean

Using the formula for the sample mean, we get:

x̄ = (1 * 24 + 2 * 32 + 3 * 29 + 4 * 9 + 5 * 6) / 5 = (24 + 64 + 87 + 36 + 30) / 5 = 241 / 5 = 48.2

Calculating the Sample Variance

Using the formula for the sample variance, we get:

s² = [(1-48.2)² * 24 + (2-48.2)² * 32 + (3-48.2)² * 29 + (4-48.2)² * 9 + (5-48.2)² * 6] / (5-1) = [(-47.2)² * 24 + (-46.2)² * 32 + (-45.2)² * 29 + (-44.2)² * 9 + (-43.2)² * 6] / 4 = [2236.64 * 24 + 2136.64 * 32 + 2048.64 * 29 + 1956.64 * 9 + 1864.64 * 6] / 4 = [53809.76 + 68368.48 + 59015.36 + 17598.96 + 11188.64] / 4 = 251081.2 / 4 = 62770.3

Calculating the Unbiased Estimate of Population Variance

Using the formula for the unbiased estimate of the population variance, we get:

σ² = (n-1) * s² / (n-1) = (5-1) * 62770.3 / (5-1) = 4 * 62770.3 / 4 = 251080.2

Conclusion

In this article, we discussed how to find the unbiased estimate of the population variance using the given data. We calculated the sample mean and sample variance using the provided data and then used these values to calculate the unbiased estimate of the population variance. The unbiased estimate of the population variance is a crucial concept in statistics and is used as a building block for more advanced statistical analyses.

References

[1] Wikipedia. (2023). Unbiased estimation. Retrieved from https://en.wikipedia.org/wiki/Unbiased_estimation
[2] Khan, A. (2020). Statistics for Dummies. John Wiley & Sons.
[3] Moore, D. S. (2017). Statistics: Concepts and Controversies. W.H. Freeman and Company.

Code

import numpy as np
data = np.array([1, 2, 3, 4, 5])
counts = np.array([24, 32, 29, 9, 6])
sample_mean = np.sum(data * counts) / np.sum(counts)
print("Sample mean:", sample_mean)
sample_variance = np.sum((data - sample_mean) ** 2 * counts) / (np.sum(counts) - 1)
print("Sample variance:", sample_variance)
unbiased_estimate = (np.sum(counts) - 1) * sample_variance / (np.sum(counts) - 1)
print("Unbiased estimate of population variance:", unbiased_estimate)

Note: The code provided is a simple implementation of the calculations performed in this article. It is not intended to be a production-ready code and should be used for educational purposes only.

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Introduction

In our previous article, we discussed how to find the unbiased estimate of the population variance using the given data. In this article, we will answer some frequently asked questions related to the topic.

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

A: The sample variance is an estimate of the population variance, which is a measure of the spread or dispersion of a population. The sample variance is calculated using a random sample of data, while the population variance is calculated using the entire population.

Q: Why do we need to use the unbiased estimate of the population variance?

A: The unbiased estimate of the population variance is used because it is an unbiased estimator of the population variance. This means that the expected value of the estimator is equal to the true population variance. Using the unbiased estimate ensures that our estimate is as close to the true population variance as possible.

Q: How do we calculate the sample variance?

A: The sample variance is calculated using the following formula:

s² = Σ(xi - x̄)² / (n-1)

where xi is each individual data point, x̄ is the sample mean, and n is the sample size.

Q: What is the formula for the unbiased estimate of the population variance?

A: The formula for the unbiased estimate of the population variance is given by:

σ² = (n-1) * s² / (n-1)

where σ² is the population variance, n is the sample size, and s² is the sample variance.

Q: Can we use the sample variance as an estimate of the population variance?

A: No, we cannot use the sample variance as an estimate of the population variance. The sample variance is an estimate of the population variance, but it is not an unbiased estimator. The unbiased estimate of the population variance is calculated using the formula:

σ² = (n-1) * s² / (n-1)

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

A: The sample standard deviation is an estimate of the population standard deviation, which is the square root of the population variance. The sample standard deviation is calculated using a random sample of data, while the population standard deviation is calculated using the entire population.

Q: How do we calculate the sample standard deviation?

A: The sample standard deviation is calculated using the following formula:

s = √(s²)

where s² is the sample variance.

Q: What is the formula for the unbiased estimate of the population standard deviation?

A: The formula for the unbiased estimate of the population standard deviation is given by:

σ = √(σ²)

where σ² is the population variance.

Q: Can we use the sample standard deviation as an estimate of the population standard deviation?

A: No, we cannot use the sample standard deviation as an estimate of the population standard deviation. The sample standard deviation is an estimate of the population standard deviation, but it is not an unbiased estimator. The unbiased estimate of the population standard deviation is calculated using the formula:

σ = √(σ²)

Q: What is the importance of the unbiased estimate of the population variance?

A: The unbiased estimate of the population variance is important because it is an unbiased estimator of the population variance. This means that the expected value of the estimator is equal to the true population variance. Using the unbiased estimate ensures that our estimate is as close to the true population variance as possible.

Q: Can we use the unbiased estimate of the population variance in hypothesis testing?

A: Yes, we can use the unbiased estimate of the population variance in hypothesis testing. The unbiased estimate of the population variance is used as a test statistic in many hypothesis tests, such as the t-test and the F-test.

Q: What is the difference between the t-test and the F-test?

A: The t-test is used to compare the means of two groups, while the F-test is used to compare the variances of two groups.

Q: Can we use the unbiased estimate of the population variance in regression analysis?

A: Yes, we can use the unbiased estimate of the population variance in regression analysis. The unbiased estimate of the population variance is used as a measure of the spread or dispersion of the residuals in a regression model.

Q: What is the importance of the unbiased estimate of the population variance in regression analysis?

A: The unbiased estimate of the population variance is important in regression analysis because it is used as a measure of the spread or dispersion of the residuals. This helps to determine the goodness of fit of the regression model and to identify any outliers or anomalies in the data.

Conclusion

In this article, we answered some frequently asked questions related to the unbiased estimate of the population variance. We discussed the importance of the unbiased estimate, how to calculate it, and its applications in hypothesis testing and regression analysis.

References

[1] Wikipedia. (2023). Unbiased estimation. Retrieved from https://en.wikipedia.org/wiki/Unbiased_estimation
[2] Khan, A. (2020). Statistics for Dummies. John Wiley & Sons.
[3] Moore, D. S. (2017). Statistics: Concepts and Controversies. W.H. Freeman and Company.

Code

import numpy as np
data = np.array([1, 2, 3, 4, 5])
counts = np.array([24, 32, 29, 9, 6])
sample_mean = np.sum(data * counts) / np.sum(counts)
sample_variance = np.sum((data - sample_mean) ** 2 * counts) / (np.sum(counts) - 1)
unbiased_estimate = (np.sum(counts) - 1) * sample_variance / (np.sum(counts) - 1)
sample_std = np.sqrt(sample_variance)
unbiased_std = np.sqrt(unbiased_estimate)
print("Sample mean:", sample_mean)
print("Sample variance:", sample_variance)
print("Unbiased estimate of population variance:", unbiased_estimate)
print("Sample standard deviation:", sample_std)
print("Unbiased estimate of population standard deviation:", unbiased_std)

Note: The code provided is a simple implementation of the calculations performed in this article. It is not intended to be a production-ready code and should be used for educational purposes only.