Sample Variance

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Sample Variance

In statistics, sample variance is a key measure of how spread out the values in a dataset are around the mean. It quantifies the variability or dispersion of the data, providing insights into the consistency and reliability of measurements. In this blog, we’ll explore what sample variance is, how to calculate it, and its practical applications.

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1. What is Sample Variance?

Sample variance measures the average squared deviation of each data point from the sample mean. It is a fundamental concept in descriptive and inferential statistics, helping us understand the spread of data.

Key Points:

• Variance is always non-negative.
• A higher variance indicates greater variability in the data.
• A lower variance indicates that the data points are closer to the mean.

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2. Formula for Sample Variance

The formula for sample variance is:
[
s^2 = \frac{\sum_{i=1}^{n} (x_i – \bar{x})^2}{n-1}
]
Where:

• ( s^2 ): Sample variance.
• ( x_i ): Individual data points.
• ( \bar{x} ): Sample mean.
• ( n ): Sample size.
• ( n-1 ): Degrees of freedom (used to correct bias in the estimation of the population variance).

Steps to Calculate Sample Variance:

1. Calculate the sample mean (( \bar{x} )).
2. Subtract the mean from each data point and square the result.
3. Sum the squared differences.
4. Divide the sum by ( n-1 ) to get the sample variance.

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3. Why Use ( n-1 ) in the Denominator?

The use of ( n-1 ) (instead of ( n )) in the denominator is called Bessel’s correction. It corrects the bias in the estimation of the population variance when using a sample. Specifically:

• Using ( n ) tends to underestimate the population variance.
• Using ( n-1 ) provides an unbiased estimate.

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4. Practical Applications of Sample Variance

Sample variance is widely used in various fields to analyze data variability:

1. Quality Control:

• Monitor the consistency of product measurements (e.g., weight, length).

2. Finance:

• Measure the risk or volatility of investment returns.

3. Healthcare:

• Analyze the variability in patient outcomes or treatment effects.

4. Social Sciences:

• Study the spread of survey responses or behavioral data.

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5. Example: Calculating Sample Variance

Scenario:

Suppose we have the following dataset of exam scores for 5 students:
[
{85, 90, 78, 92, 88}
]

Steps:

1. Calculate the Sample Mean (( \bar{x} )):
   [
   \bar{x} = \frac{85 + 90 + 78 + 92 + 88}{5} = \frac{433}{5} = 86.6
   ]
2. Calculate Squared Deviations:
   [
   (85 – 86.6)^2 = (-1.6)^2 = 2.56
   ]
   [
   (90 – 86.6)^2 = (3.4)^2 = 11.56
   ]
   [
   (78 – 86.6)^2 = (-8.6)^2 = 73.96
   ]
   [
   (92 – 86.6)^2 = (5.4)^2 = 29.16
   ]
   [
   (88 – 86.6)^2 = (1.4)^2 = 1.96
   ]
3. Sum the Squared Deviations:
   [
   2.56 + 11.56 + 73.96 + 29.16 + 1.96 = 119.2
   ]
4. Calculate the Sample Variance:
   [
   s^2 = \frac{119.2}{5-1} = \frac{119.2}{4} = 29.8
   ]

Interpretation:

• The sample variance is 29.8, indicating moderate variability in the exam scores.

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6. Sample Variance vs. Population Variance

• Sample Variance (( s^2 )):
• Used when working with a sample of data.
• Formula: ( s^2 = \frac{\sum (x_i – \bar{x})^2}{n-1} ).
• Population Variance (( \sigma^2 )):
• Used when working with the entire population.
• Formula: ( \sigma^2 = \frac{\sum (x_i – \mu)^2}{N} ).
• ( \mu ): Population mean.
• ( N ): Population size.

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7. Key Takeaways

• Sample variance measures the spread of data points around the sample mean.
• It is calculated using the formula ( s^2 = \frac{\sum (x_i – \bar{x})^2}{n-1} ).
• The use of ( n-1 ) corrects bias in estimating the population variance.
• Sample variance is widely used in quality control, finance, healthcare, and social sciences.

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Conclusion

Sample variance is a fundamental concept in statistics that helps us understand the variability in data. By calculating and interpreting sample variance, we can gain valuable insights into the consistency and reliability of measurements, enabling better decision-making in various fields.

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