What is Sample Variance? | Simple Explanation for Class 12

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What is a Sample Variance?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Sample variance tells us how spread out the numbers in a sample dataset are from their average value. It measures the average of the squared differences from the mean, helping us understand the variability within a small group of observations.

Simple Example

Quick Example

Imagine you're checking the delivery times of 5 food orders from your favourite biryani place: 20, 25, 22, 28, 20 minutes. The sample variance would tell you how much these delivery times typically differ from the average delivery time. A high variance means times vary a lot, while a low variance means they are quite consistent.

Worked Example

Step-by-Step

Let's find the sample variance for the marks of 4 students in a surprise quiz: 8, 10, 6, 12.

1. **Calculate the mean (average):**
Mean = (8 + 10 + 6 + 12) / 4 = 36 / 4 = 9

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2. **Subtract the mean from each data point and square the result:**
(8 - 9)^2 = (-1)^2 = 1
(10 - 9)^2 = (1)^2 = 1
(6 - 9)^2 = (-3)^2 = 9
(12 - 9)^2 = (3)^2 = 9

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3. **Sum these squared differences:**
Sum = 1 + 1 + 9 + 9 = 20

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4. **Divide the sum by (n - 1), where n is the number of data points (n=4):**
Sample Variance = Sum / (n - 1) = 20 / (4 - 1) = 20 / 3 = 6.67 (approximately)

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**Answer:** The sample variance for the quiz marks is approximately 6.67.

Why It Matters

Understanding sample variance is crucial for data scientists and engineers to assess risk and consistency. For example, in AI/ML, it helps evaluate model performance, and in FinTech, it's used to analyze stock price volatility. This knowledge can lead to exciting careers in data science, finance, or even climate research.

Common Mistakes

MISTAKE: Dividing by 'n' instead of 'n-1' when calculating sample variance. | CORRECTION: Always divide by (n-1) for sample variance. Dividing by 'n' is for population variance, which is different.

MISTAKE: Forgetting to square the differences from the mean. | CORRECTION: The formula specifically requires squaring each difference (x - mean) before summing them up. This ensures positive values and emphasizes larger deviations.

MISTAKE: Calculating the mean incorrectly at the first step. | CORRECTION: Double-check your mean calculation. An incorrect mean will lead to errors throughout the rest of the variance calculation.

Practice Questions

Try It Yourself

QUESTION: Calculate the sample variance for the number of runs scored by a batsman in 3 matches: 20, 30, 25. | ANSWER: Mean = (20+30+25)/3 = 25. Differences squared: (20-25)^2=25, (30-25)^2=25, (25-25)^2=0. Sum = 50. Sample Variance = 50 / (3-1) = 50 / 2 = 25.

QUESTION: A small shop recorded daily sales (in thousands of rupees) for 4 days: 12, 15, 10, 13. Find the sample variance of their daily sales. | ANSWER: Mean = (12+15+10+13)/4 = 50/4 = 12.5. Differences squared: (12-12.5)^2=0.25, (15-12.5)^2=6.25, (10-12.5)^2=6.25, (13-12.5)^2=0.25. Sum = 13. Sample Variance = 13 / (4-1) = 13 / 3 = 4.33 (approx).

QUESTION: Five friends measured their heights in cm: 150, 155, 160, 150, 165. Calculate the sample variance. If another friend with height 150 cm joins, how would the variance change? (You don't need to recalculate, just predict if it would increase, decrease, or stay similar). | ANSWER: For 5 friends: Mean = (150+155+160+150+165)/5 = 156. Differences squared: (150-156)^2=36, (155-156)^2=1, (160-156)^2=16, (150-156)^2=36, (165-156)^2=81. Sum = 170. Sample Variance = 170 / (5-1) = 170 / 4 = 42.5. If another friend of 150 cm joins, the variance would likely decrease or stay similar because 150 cm is close to the current average and adds another data point that isn't an extreme outlier, making the data appear less spread out relative to the mean.

MCQ

Quick Quiz

Why do we divide by (n-1) instead of 'n' when calculating sample variance?

To make the calculation simpler

Because it gives a more accurate estimate of the true population variance

To avoid negative results

It's a historical convention with no mathematical reason

The Correct Answer Is:

Dividing by (n-1) is known as Bessel's correction. It provides an unbiased estimate of the population variance from a sample, meaning it's a more accurate way to guess the spread of the larger group. Options A, C, and D are incorrect.

Real World Connection

In the Real World

In cricket, analysts use sample variance to evaluate a player's consistency. If a batsman has a high variance in scores, it means their performance varies a lot (sometimes very high, sometimes very low), making them unpredictable. Low variance means they consistently score around their average. This helps coaches and team strategists make informed decisions.

Key Vocabulary

Key Terms

VARIANCE: A measure of how spread out a set of data is | MEAN: The average of a set of numbers | SAMPLE: A small, representative group taken from a larger population | POPULATION: The entire group of individuals or items being studied | DEVIATION: The difference between a data point and the mean

What's Next

What to Learn Next

Great job understanding sample variance! Next, you should explore 'Standard Deviation'. It's closely related to variance and is often preferred because it's in the same units as your original data, making it easier to interpret the spread in real-world terms.