What is Variance? | Simple Explanation for Class 12

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What is Variance (Dispersion)?

Grade Level:

Class 12

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

Definition

What is it?

Variance tells us how spread out a set of numbers is from their average (mean). It measures the average of the squared differences from the mean, showing if data points are close together or widely scattered.

Simple Example

Quick Example

Imagine two cricket teams. Team A scores 10, 50, 55, 60, 125 runs in 5 matches. Team B scores 50, 52, 55, 58, 60 runs. Both teams might have a similar average score, but Team A's scores are much more spread out (higher variance) than Team B's, which are very consistent (lower variance).

Worked Example

Step-by-Step

Let's find the variance for the number of samosas sold at a stall over 5 days: 10, 12, 8, 14, 11.

1. Find the Mean (Average): (10 + 12 + 8 + 14 + 11) / 5 = 55 / 5 = 11.

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2. Subtract the Mean from each data point and square the result:
(10 - 11)^2 = (-1)^2 = 1
(12 - 11)^2 = (1)^2 = 1
(8 - 11)^2 = (-3)^2 = 9
(14 - 11)^2 = (3)^2 = 9
(11 - 11)^2 = (0)^2 = 0

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3. Sum these squared differences: 1 + 1 + 9 + 9 + 0 = 20.

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4. Divide the sum by the total number of data points (N) if it's the whole population, or by (N-1) if it's a sample. For simplicity here, we'll divide by N (5): 20 / 5 = 4.

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The Variance is 4.

Why It Matters

Understanding variance helps scientists and engineers make better decisions. In finance, it helps predict how risky an investment is. In medicine, doctors use it to see how effective a new drug is by looking at the spread of patient responses. Data scientists use it to understand patterns in large datasets, which is key for AI and machine learning.

Common Mistakes

MISTAKE: Forgetting to square the differences from the mean. | CORRECTION: Always square the differences (data point - mean) before adding them up. This ensures negative and positive differences don't cancel out.

MISTAKE: Confusing variance with standard deviation. | CORRECTION: Variance is the average of the squared differences. Standard deviation is the square root of the variance, giving a value in the original units of the data, which is often easier to interpret.

MISTAKE: Using N-1 in the denominator when calculating population variance. | CORRECTION: For population variance (when you have data for *everyone* or *everything*), divide by N. For sample variance (when you have data for only *some*), divide by N-1. This is a subtle but important difference.

Practice Questions

Try It Yourself

QUESTION: Find the variance for the following daily temperatures in Celsius: 20, 22, 18, 20. | ANSWER: Mean = (20+22+18+20)/4 = 80/4 = 20. Differences squared: (20-20)^2=0, (22-20)^2=4, (18-20)^2=4, (20-20)^2=0. Sum = 8. Variance = 8/4 = 2.

QUESTION: A class of 5 students got marks: 60, 70, 80, 90, 100. Calculate the variance of their marks. | ANSWER: Mean = (60+70+80+90+100)/5 = 400/5 = 80. Differences squared: (60-80)^2=400, (70-80)^2=100, (80-80)^2=0, (90-80)^2=100, (100-80)^2=400. Sum = 1000. Variance = 1000/5 = 200.

QUESTION: Two auto-rickshaw drivers, Ram and Shyam, recorded their earnings (in Rupees) for 3 trips. Ram: 100, 150, 200. Shyam: 140, 150, 160. Which driver's earnings were more consistent (less variance)? | ANSWER: Ram's Mean = (100+150+200)/3 = 150. Ram's Variance = ((100-150)^2 + (150-150)^2 + (200-150)^2)/3 = (2500+0+2500)/3 = 5000/3 = 1666.67. Shyam's Mean = (140+150+160)/3 = 150. Shyam's Variance = ((140-150)^2 + (150-150)^2 + (160-150)^2)/3 = (100+0+100)/3 = 200/3 = 66.67. Shyam's earnings were more consistent (lower variance).

MCQ

Quick Quiz

What does a high variance value indicate about a dataset?

The data points are very close to the mean.

The data points are widely spread out from the mean.

The mean of the data is very high.

The data set has very few data points.

The Correct Answer Is:

Variance measures how spread out data points are. A high variance means the data points are far from the mean, indicating a wide spread. A low variance means they are close to the mean.

Real World Connection

In the Real World

When you use a weather app like AccuWeather or Google Weather, the 'chance of rain' or temperature range isn't just an average. Meteorologists use variance to understand how much the weather patterns might deviate from the average, helping them predict if it will be a consistently sunny day or if there's a high chance of sudden showers.

Key Vocabulary

Key Terms

MEAN: The average of a set of numbers | DISPERSION: The extent to which data is spread out | DATA POINT: An individual value in a dataset | SQUARED DIFFERENCE: The result of subtracting the mean from a data point and then multiplying the result by itself

What's Next

What to Learn Next

Now that you understand variance, the next step is to learn about Standard Deviation. It's closely related to variance and will help you interpret the spread of data in a more direct and understandable way, making it even easier to apply to real-world problems!