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.
σ² = Σ(xi - μ)² / N
Explanation:
Steps:
Consider a population data set representing the weights (in kg) of 5 individuals: {65, 72, 80, 78, 68}.
μ = (65 + 72 + 80 + 78 + 68) / 5
μ = 363 / 5
μ = 72.6 kg
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 |
57.76 + 0.36 + 54.76 + 29.16 + 21.16 = 163.2
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.
Population variance helps us gauge the variability of data within a population. This information is valuable for:
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.
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:
population_data
list represents the scores of all 1000 students.population_mean
is calculated by summing the scores and dividing by the total number of students.population_variance
is 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
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