Sample Variance Demystified: Mastering Your Analysis with Essential Insights 101

What is Sample Variance?

In statistics, sample variance (s²) plays a crucial role in understanding how spread out data points are within a sample. It reflects the average squared distance of each data point in the sample from the sample mean (x̄). A higher variance indicates a wider spread of data points, while a lower variance suggests the data points are clustered closer to the mean.

Why is this Important?

Sample variance provides valuable insights into data variability within a sample. This information is beneficial for:

• Understanding data distribution: Sample variance helps visualize how data is distributed around the mean, offering valuable insights compared to just the mean alone.
• Statistical analysis: It serves as the foundation for calculating other statistical measures like sample standard deviation (which is the square root of sample variance).
• Comparing datasets from different samples: Sample variances can be used to compare the variability of data between different samples drawn from the same population.

Sample Variance vs. Population Variance (σ²)

It’s important to distinguish between sample variance (s²) and population variance (σ²). Population variance reflects the variability within the entire population, while sample variance estimates the population variance using data from a subset (sample) of the population. Obtaining data for the entire population can be impractical or impossible in many real-world scenarios. Sample variance plays a significant role when complete population data isn’t available.

How to calculate ?

Here’s the mathematical formula for calculating sample variance:

s² = Σ(xi - x̄)² / (n - 1)

Explanation of the Formula:

• Σ (sigma) represents the sum of all elements.
• xi (x-subscript-i) represents each individual value in the sample data set.
• x̄ (x-bar) represents the sample mean (calculated by summing the values in the sample and dividing by the number of elements in the sample – n).
• n represents the total number of elements in the sample.

Steps for Calculating :

1. Calculate the sample mean (x̄): Sum the values in your sample and divide by the number of elements (n).
2. Compute squared deviations from the mean: For each data point (xi) in the sample, subtract the sample mean (x̄) and square the result. This represents the squared deviation of that particular element from the mean.
3. Sum the squared deviations: Add up the squared deviations calculated for all elements in the sample.
4. Calculate : Divide the sum of squared deviations from step 3 by the number of elements in the sample minus one (n – 1). This gives you the average squared deviation from the mean, representing the sample variance (s²).

Examples

Consider a sample data set representing the test scores of 5 students: {85, 72, 90, 88, 65}.

1. Sample Mean (x̄):

x̄ = (85 + 72 + 90 + 88 + 65) / 5
x̄ = 400 / 5
x̄ = 80

2. Squared Deviations from the Mean:

3. Sum of Squared Deviations:

25 + 64 + 100 + 64 + 225 = 478

4. Sample Variance (s²):

s² = 478 / (5 - 1) s² = 478 / 4 s² = 119.5

Therefore, the sample variance (s²) for this example is 119.5, indicating a moderate spread of test scores within this sample.

How to calculate It in Python?

Here’s a Python code example demonstrating how to calculate sample variance:

sample_data = [85, 72, 90, 88, 65]

sample_mean = sum(sample_data) / len(sample_data)

squared_deviations = [(x - sample_mean) ** 2 for x in sample_data]

sample_variance = sum(squared_deviations) / (len(sample_data) - 1)

print("Sample Mean:", sample_mean)
print("Sample Variance:", sample_variance)

Explanation:

1. Sample Data: This line defines a list sample_data containing the test scores (replace with your actual data).
2. Sample Mean: The sample_mean is calculated by summing the elements in sample_data and dividing by the total number of elements (using len(sample_data)).
3. Squared Deviations: The list comprehension squared_deviations = [(x - sample_mean) ** 2 for x in sample_data] iterates through each element x in sample_data. It subtracts the sample_mean from each element, squares the result using the exponent **2, and creates a new list containing the squared deviations from the mean for all elements.
4. Sample Variance: The sample_variance is calculated by summing the squared deviations in squared_deviations and dividing by the total number of elements in the sample minus one (len(sample_data) - 1). This adjustment (n-1) is used to obtain an unbiased estimate of the population variance.
5. Print Results: The calculated sample_mean and sample_variance are printed.

Sample Mean: 80.0
Sample Variance: 119.5

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Reference: Variance | Brilliant Math & Science Wiki

Also Read:

• Understanding Population Variance
• Central Tendency: Sample Mean and Population Mean
• Difference between percentage and percentile
• 3 Measures of Central Tendency: Mean, Media, Mode

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