What are the differences between population and sample variance?

In statistics, understanding the concepts of population and sample variance is crucial for data analysis. Variance measures how much the values in a set of data differ from the average, or mean, of the data. When you're dealing with the entire group that you're interested in, that's your population. Conversely, a sample is a subset of the population, which you might analyze to make inferences about the whole. These two types of variance are calculated differently and serve different purposes in the realm of statistical analysis.

1 Population Variance

Population variance is a measure of how data points in a full population are spread out from the mean. It is calculated by taking the sum of the squared differences between each data point and the mean, and then dividing by the number of data points in the population. This value gives you a sense of the overall variability within the entire group. Since you're considering every single possible data point, population variance provides a very accurate measure of dispersion.

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Dr. Jayashree Rajesh Prasad

CertNexus Certified AI Practitioner, IBM Data Analyst, SAP Technology Consultant, Senior member IEEE, Chair IEEE CIS, Advisor CIS-SB, Member IEEE EXECOM, Professor & Head, CSE AIA, SoC MIT ADT University
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• Population Variance: Uses the entire population with N in the denominator. • Sample Variance: Uses a sample from the population with n-1 in the denominator to correct for bias. This adjustment (Bessel’s correction) ensures that the sample variance is an unbiased estimator of the population variance.

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Abhishek Tiwari

Data Scientist | DL | GenAI | NLP, LLM | Computer Vision | ML | Python | Trainer
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Understanding the differences between population and sample variance is crucial in data analysis. Population variance measures the dispersion of all data points in a dataset, using the formula: sum of squared deviations from the mean divided by the number of data points (N). Sample variance, however, estimates the population variance from a sample. It uses a similar formula but divides by the number of data points minus one (N-1) to account for sample bias. This adjustment, known as Bessel's correction, provides a more accurate estimate of the population variance when only a sample is available. #DataAnalysis #Statistics #Variance

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Sakshi Choube

Top Data Analysis Voice | Mathematician | Data Science | Machine learning | Statistics | Python | SQL | Power Bi | Seeking Opportunities
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Population variance measures the average squared deviation from the mean for every member of an entire population. It uses the formula σ² = Σ(X - μ)² / N, where N is the population size. Sample variance estimates this for a subset of the population, adjusting for bias with the formula s² = Σ(X - x̄)² / (n - 1), where n is the sample size. This adjustment (n - 1) is called Bessel's correction.

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🔥Ali Khreiss🔥

Founder @ Bands Consultancy |Statistician| Statistics Specialist|Trainer|Consultant|Machine Learning Master Thesis 2010
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The population variance is represented by sigma square and it helps us stating how the data points are spread out in the specific population, usually the population size is N, so the population variance is the the sum square of the difference between the observations and the population mean divided by the population size N.

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2 Sample Variance

Sample variance, on the other hand, deals with a sample—a smaller group selected from the population. The calculation is similar to population variance, but with a key difference: you divide by the number of data points in the sample minus one. This adjustment, known as Bessel's correction, accounts for the fact that you're working with only a portion of the data, which introduces additional uncertainty. Sample variance tends to be higher than population variance because it compensates for the variability that might exist beyond the sample.

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🔥Ali Khreiss🔥

Founder @ Bands Consultancy |Statistician| Statistics Specialist|Trainer|Consultant|Machine Learning Master Thesis 2010
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Sample variance is the same as the population variance but instead the population size N, we use the sample size n, we use a sample data from a population and the denominator is n-1

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3 Understanding n-1

The term "n-1" in the context of sample variance refers to the denominator of the variance formula: the sample size minus one. This is crucial because it corrects the bias in the estimation of the population variance from a sample. Without this adjustment, you would likely underestimate the true variability of the population. The "n-1" ensures that the sample variance is an unbiased estimator of the population variance.

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🔥Ali Khreiss🔥

Founder @ Bands Consultancy |Statistician| Statistics Specialist|Trainer|Consultant|Machine Learning Master Thesis 2010
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The sample variance divided by n-1 and not n is related to Bessel's correction, this method corrects the bias in the estimation of the population variance and it also partially corrects the bias in the estimation of the population standard deviation

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4 Practical Implications

The choice between using population or sample variance has practical implications. If you have access to all the data from a population, you would use population variance to get an exact measure of variability. However, in most real-world scenarios, gathering data from an entire population is impractical or impossible. Therefore, you would collect a sample and calculate the sample variance, keeping in mind that it's an estimate that includes a margin for potential error.

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5 Variance in Research

In research, the distinction between population and sample variance is pivotal. Sample variance is often used in hypothesis testing, confidence interval construction, and regression analysis. Researchers must be aware that using sample data requires adjustments to ensure their conclusions about the population are valid. Understanding these differences helps prevent errors and improves the reliability of statistical inferences.

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6 Choosing Correctly

Choosing between population and sample variance depends on your data set and research goals. If you're looking at a complete set of data with no elements left out, use population variance. For studies involving only a segment of your target group, sample variance is the correct choice. Always remember that using the wrong variance calculation can lead to inaccurate results, so it's essential to understand these differences thoroughly.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are the differences between population and sample variance?

1

2

3

4

5

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7

1 Population Variance

2 Sample Variance

3 Understanding n-1

4 Practical Implications

5 Variance in Research

6 Choosing Correctly

7 Here’s what else to consider

Statistics

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What are the differences between population and sample variance?

1

2

3

4

5

6

7

1 Population Variance

2 Sample Variance

3 Understanding n-1

4 Practical Implications

5 Variance in Research

6 Choosing Correctly

7 Here’s what else to consider

Statistics

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