Understanding variance is crucial in statistics. It measures how spread out a dataset is, indicating how far individual data points deviate from the mean (average). A high variance signifies a wide spread of data, while a low variance indicates data points clustered closely around the mean. This guide will walk you through calculating variance, covering both population variance and sample variance.
Understanding the Concepts
Before diving into calculations, let's clarify some key terms:
- Population: The entire group you're interested in studying.
- Sample: A subset of the population used to make inferences about the entire population.
- Mean (Average): The sum of all values divided by the number of values.
- Variance: The average of the squared differences from the mean.
Calculating Population Variance
The population variance (σ²) uses all data points from the entire population. The formula is:
σ² = Σ(xᵢ - μ)² / N
Where:
- σ² represents the population variance.
- Σ denotes the sum.
- xᵢ represents each individual data point.
- μ represents the population mean.
- N represents the total number of data points in the population.
Step-by-Step Example:
Let's say our population data is: 2, 4, 6, 8, 10
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Calculate the mean (μ): (2 + 4 + 6 + 8 + 10) / 5 = 6
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Subtract the mean from each data point (xᵢ - μ):
- 2 - 6 = -4
- 4 - 6 = -2
- 6 - 6 = 0
- 8 - 6 = 2
- 10 - 6 = 4
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Square each difference [(xᵢ - μ)²]:
- (-4)² = 16
- (-2)² = 4
- 0² = 0
- 2² = 4
- 4² = 16
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Sum the squared differences [Σ(xᵢ - μ)²]: 16 + 4 + 0 + 4 + 16 = 40
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Divide by the number of data points (N): 40 / 5 = 8
Therefore, the population variance (σ²) is 8.
Calculating Sample Variance
The sample variance (s²) is an estimate of the population variance, using only a subset of the data. The formula is slightly different:
s² = Σ(xᵢ - x̄)² / (n - 1)
Where:
- s² represents the sample variance.
- x̄ represents the sample mean.
- n represents the total number of data points in the sample.
Notice the denominator is (n - 1) instead of n. This is known as Bessel's correction, which provides a less biased estimate of the population variance.
Step-by-Step Example:
Let's use the same data (2, 4, 6, 8, 10) but treat it as a sample this time.
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Calculate the sample mean (x̄): (2 + 4 + 6 + 8 + 10) / 5 = 6
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Subtract the mean from each data point (xᵢ - x̄): (Same as above)
-
Square each difference [(xᵢ - x̄)²]: (Same as above)
-
Sum the squared differences [Σ(xᵢ - x̄)²]: (Same as above) = 40
-
Divide by (n - 1): 40 / (5 - 1) = 10
Therefore, the sample variance (s²) is 10.
Why the Difference Between Population and Sample Variance?
The difference in formulas stems from the fact that a sample is only a representation of a larger population. Using (n-1) in the sample variance calculation corrects for the fact that a sample tends to underestimate the true population variance. This correction leads to a more accurate estimation of the population variance from the sample data.
Variance and Standard Deviation
The square root of the variance is the standard deviation, another crucial measure of data dispersion. Standard deviation is expressed in the same units as the original data, making it easier to interpret than variance.
By understanding how to calculate variance, you gain valuable insights into data variability, which is fundamental for various statistical analyses and decision-making processes. Remember to choose the appropriate formula (population or sample) based on whether you're working with the entire population or a sample.
