What is Variance of a Random Variable?

S7-SA3-0031

Grade Level:

Class 12

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

Definition

What is it?

Variance of a random variable tells us how much the values of a random variable are spread out from its average (mean). It measures the average of the squared differences from the mean, showing how much the data points vary from each other.

Simple Example

Quick Example

Imagine you have two cricket players. Player A scores 50, 52, 48, 51, 49 runs in 5 matches. Player B scores 10, 90, 30, 70, 50 runs in 5 matches. Both have an average score of 50 runs. But Player B's scores are much more spread out than Player A's. Variance helps us measure this 'spread' or 'consistency'.

Worked Example

Step-by-Step

Let's find the variance for the number of samosas sold per hour at a small chai shop: 5, 7, 3, 9, 6.

1. Find the Mean (Average): Add all values and divide by the count.
Mean = (5 + 7 + 3 + 9 + 6) / 5 = 30 / 5 = 6

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2. Subtract the Mean from each value and square the result (this is the squared difference).
(5 - 6)^2 = (-1)^2 = 1
(7 - 6)^2 = (1)^2 = 1
(3 - 6)^2 = (-3)^2 = 9
(9 - 6)^2 = (3)^2 = 9
(6 - 6)^2 = (0)^2 = 0

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3. Sum all the squared differences.
Sum = 1 + 1 + 9 + 9 + 0 = 20

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4. Divide the sum of squared differences by the total number of values (n). This is the Variance.
Variance = 20 / 5 = 4

Answer: The variance for the number of samosas sold is 4.

Why It Matters

Understanding variance helps scientists and engineers predict outcomes and manage risks. For example, in medicine, doctors use it to see how effective a new drug is by checking the variation in patient recovery times. In finance, it helps predict how much a stock price might fluctuate, which is super important for investors.

Common Mistakes

MISTAKE: Forgetting to square the differences from the mean. | CORRECTION: Always square the difference (value - mean) before adding them up. This ensures positive values and gives more weight to larger deviations.

MISTAKE: Calculating standard deviation instead of variance. | CORRECTION: Variance is the average of squared differences. Standard deviation is the square root of the variance. Don't take the square root if the question asks for variance.

MISTAKE: Using the wrong denominator for sample variance vs. population variance. | CORRECTION: For population variance (all possible data), divide by 'n'. For sample variance (a part of the data), divide by 'n-1'. In Class 12, usually 'n' is used for simplicity unless specified.

Practice Questions

Try It Yourself

QUESTION: Find the variance for the number of missed calls a student received in a week: 2, 4, 3, 1, 5. | ANSWER: Mean = 3. Variance = 2.

QUESTION: A local store recorded daily sales (in thousands of Rupees) for 4 days as: 10, 12, 8, 14. Calculate the variance of daily sales. | ANSWER: Mean = 11. Variance = 5.

QUESTION: The scores of a student in 5 science tests were 70, 75, 80, 70, 85. If a new test score of 90 is added, how does the variance change? (Calculate old and new variance). | ANSWER: Old Mean = 76, Old Variance = 34. New Mean = 78.33, New Variance approx = 66.89. The variance increases.

MCQ

Quick Quiz

What does a high variance indicate about a set of data?

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 observations.

The Correct Answer Is:

A high variance means the individual data points are far from the average, showing a large spread. Options A and C are incorrect as they describe the opposite or a different property. Option D is unrelated to the definition of variance.

Real World Connection

In the Real World

In cricket analytics, companies like Star Sports or ESPNcricinfo use variance to assess a batsman's consistency. A batsman with low variance in scores is considered more consistent and reliable, while one with high variance might have brilliant days but also very poor ones. This helps team selectors make informed decisions.

Key Vocabulary

Key Terms

RANDOM VARIABLE: A variable whose value is determined by the outcome of a random event. | MEAN: The average of a set of numbers. | SPREAD: How far apart the values in a dataset are. | DEVIATION: The difference between an individual data point and the mean.

What's Next

What to Learn Next

Now that you understand variance, you should learn about 'Standard Deviation'. Standard deviation is simply the square root of the variance and is often easier to interpret because it's in the same units as your original data. It's the next logical step to master data spread!