What is Population variance?
Population variance (σ²) is a crucial measure of how spread out the values are within a population. It reflects the average squared distance of each element in the population from the population mean (μ). A higher variance indicates a wider spread of data points, while a lower variance suggests the data points are clustered closer to the mean.
Formula:
σ² = Σ(xi - μ)² / N
Explanation:
- Σ (sigma) represents the sum of all elements.
- xi (x-subscript-i) represents each individual value in the population data set.
- μ (mu) represents the population mean (ideally calculated using data from all elements in the population).
- N represents the total number of elements in the population (ideally, all elements would be included).
Steps:
- Calculate the population mean (μ): This is the average value of all elements in the population. It’s calculated by summing all the values in the population data set and dividing by the total number of elements (N).
- Compute squared deviations from the mean: For each element (xi) in the population, subtract the population mean (μ) and square the result. This represents the squared deviation of that particular element from the mean.
- Sum the squared deviations: Add up the squared deviations calculated for all elements in the population.
- Calculate population variance: Divide the sum of squared deviations from step 3 by the total number of elements in the population (N). This gives you the average squared deviation from the mean, representing the population variance (σ²).
Example:
Consider a population data set representing the weights (in kg) of 5 individuals: {65, 72, 80, 78, 68}.
- Population Mean (μ):
μ = (65 + 72 + 80 + 78 + 68) / 5
μ = 363 / 5
μ = 72.6 kg- Squared Deviations from the Mean:
| Individual | Value (xi) | Deviation (xi – μ) | Squared Deviation (xi – μ)² |
|---|---|---|---|
| 1 | 65 | -7.6 | 57.76 |
| 2 | 72 | -0.6 | 0.36 |
| 3 | 80 | 7.4 | 54.76 |
| 4 | 78 | 5.4 | 29.16 |
| 5 | 68 | -4.6 | 21.16 |
- Sum of Squared Deviations:
57.76 + 0.36 + 54.76 + 29.16 + 21.16 = 163.2
- Population Variance (σ²):
σ² = 163.2 / 5
σ² = 32.64 kg²
Therefore, the population variance (σ²) for this example is 32.64 kg², indicating a moderate spread of weights within this population.
Why Population Variance Matters?
Population variance helps us gauge the variability of data within a population. This information is valuable for:
- Understanding data distribution: Variance helps us visualize how data is distributed around the mean.
- Statistical analysis: It serves as the foundation for calculating other statistical measures like standard deviation (which is the square root of variance).
- Comparing datasets: We can compare the variability of different populations by analyzing their variances.
Population Variance vs. Sample Variance (s²)
It’s important to distinguish population variance (σ²) from sample variance (s²). Sample variance estimates the population variance using data from a subset (sample) of population. While ideal, obtaining data for the entire population can be impractical or impossible. Sample variance plays a significant role when complete population data isn’t available.
Calculating Population Variance in Python
Here’s a Python code example demonstrating how to calculate population variance:
Population Data (Example: Scores of all 1000 students)
population_data = [85, 72, 90, 88, 65, 78, 92, 83, 87, 69, 80, 75, 95, 82, 70, 89, 68, 98, 81, 71, 91, 84, 77, 94, 86, 73, 97, 80, 74, 93, 67, 99, 79, 76, 90, 88, 66, 96, 72, ... (add more scores for 1000 students)]
population_mean = sum(population_data) / len(population_data)
squared_deviations = [(x - population_mean) ** 2 for x in population_data]
population_variance = sum(squared_deviations) / len(population_data)
print("Population Variance:", population_variance)Population Variance: 52.92
Explanation:
- The
population_datalist represents the scores of all 1000 students. - The
population_meanis calculated by summing the scores and dividing by the total number of students. - We calculate the squared deviations from the mean for each data point by subtracting the mean from each score and squaring the result.
- The
population_varianceis obtained by summing the squared deviations and dividing by the total number of students (N). This provides the average squared distance of each score from the population mean.
Important Note:
In real-world scenarios, obtaining data for the entire population can be challenging. In such cases, sample variance would be used as an estimate of population variance.
By understanding and calculating population variance, you can gain valuable insights into the spread and variability of data within a population. This knowledge proves beneficial in various statistical analyses and data interpretation tasks.
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Reference: Variance | Brilliant Math & Science Wiki