Variance formula with steps and solved examples
The concept of Variance plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Variance measures how much the values in a set differ from their average (mean), helping us understand how spread out data is. Whether in maths exams, statistics, or daily life, knowing how to calculate and use variance is a valuable skill!
What Is Variance?
Variance is defined as the average of the squared differences between each value and the mean (average) of the dataset. You’ll find this concept applied in areas such as statistics, data science, and probability. Variance helps us understand the spread or "dispersion" of data, and is a building block to understanding Standard Deviation, another important topic in Math.
Key Formula for Variance
Here’s the standard formula for the variance of a population:
| Formula | Symbols Used | Meaning |
|---|---|---|
| \( \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \) | σ2, μ, xi, N | σ2: Variance μ: Mean xi: Each value N: Number of values |
For a sample, use:
\( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \)
Cross-Disciplinary Usage
Variance is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE, NEET, or Olympiads will see its relevance in questions about probability, data analysis, and statistical interpretation. In business, variance helps compare differences in profit or marks; in science, it detects how much values differ from the norm.
Step-by-Step Illustration
Let’s calculate the variance of this set: 2, 4, 6, 8, 10
1. Find the mean (average):2. Subtract the mean from each value and square it:
(4-6)2 = 4
(6-6)2 = 0
(8-6)2 = 4
(10-6)2 = 16
3. Add up the squared differences:
4. Divide by the number of values (for population):
So, the variance of 2, 4, 6, 8, 10 is 8. Remember: for a sample, divide by (n-1) instead of n at the last step.
Speed Trick or Vedic Shortcut
Here’s a quick tip: When all numbers are spaced evenly (like 4, 8, 12, 16), their variance can be found using the formula d² × [n²−1]/12 (where d is the difference between numbers, n is the number of terms). Many students use this shortcut in exams for arithmetic sequences.
Example Trick: For the numbers 5, 8, 11, 14:
1. The difference (d) is 3,2. There are 4 numbers (n=4)
3. Variance = 3² × (16−1)/12 = 9×15/12 = 135/12 = 11.25
Vedantu live classes cover more such speed tricks and techniques to help you solve math questions faster!
Try These Yourself
- Find the variance of: 3, 6, 9, 12.
- If the mean of 4, 5, 6, 7 is 5.5, what is their variance?
- Calculate the variance for the scores: 13, 17, 15, 20.
- Using the shortcut, what’s the variance for 2, 4, 6, 8, 10, 12?
Frequent Errors and Misunderstandings
- Confusing variance with standard deviation (which is its square root).
- Forgetting to square differences (using just differences will always sum to zero).
- Dividing by n for a sample instead of (n−1).
Variance vs. Standard Deviation
| Variance | Standard Deviation |
|---|---|
| Average of squared differences from the mean | Square root of variance |
| Unit is square of original | Unit is same as original |
| Shows spread, not easy to interpret directly | Directly shows average distance from mean |
For more on this, visit variance vs standard deviation.
Relation to Other Concepts
The idea of Variance connects closely with topics such as Mean, Dispersion, and Probability. Mastering variance will help you solve problems in statistics and understand more complex distributions.
Classroom Tip
A quick way to remember variance: “Square, then mean!” Always subtract the mean, square the result, add them, then average. Many teachers at Vedantu call this the “three-step variance rule.”
We explored Variance—from definition, formula, examples, mistakes, to its relation with other subjects. Continue practicing with Vedantu to become confident in solving statistics and data problems using this concept.
Useful Vedantu Links
FAQs on Variance in Statistics Explained Clearly
1. What is variance in mathematics?
Variance is a measure of how spread out a set of numbers is from their mean. It tells you how far, on average, each value in a data set lies from the mean.
- If variance is small, the data points are close to the mean.
- If variance is large, the data points are more spread out.
- Variance is measured in squared units of the original data.
2. What is the formula for variance?
The formula for variance is σ² = Σ(x − μ)² / N for a population and s² = Σ(x − x̄)² / (n − 1) for a sample.
- σ² = population variance
- s² = sample variance
- μ = population mean
- x̄ = sample mean
- N = population size
- n = sample size
3. How do you calculate variance step by step?
To calculate variance, find the mean, compute squared deviations, and divide by N or (n − 1).
- Step 1: Find the mean of the data set.
- Step 2: Subtract the mean from each value.
- Step 3: Square each deviation.
- Step 4: Add all squared deviations.
- Step 5: Divide by N (population) or n − 1 (sample).
4. What is the difference between population variance and sample variance?
The key difference is that population variance divides by N while sample variance divides by (n − 1).
- Population variance: σ² = Σ(x − μ)² / N
- Sample variance: s² = Σ(x − x̄)² / (n − 1)
5. Can you give an example of calculating variance?
Yes, for the data set 2, 4, 6, the population variance is 2.67 (approx).
- Mean = (2 + 4 + 6) / 3 = 4
- Deviations: −2, 0, 2
- Squared deviations: 4, 0, 4
- Sum = 8
- Variance = 8 / 3 = 2.67
6. What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. In symbols, σ = √σ² and s = √s².
- Variance measures spread in squared units.
- Standard deviation measures spread in the original units.
7. Why is variance always non-negative?
Variance is always non-negative because it is based on squared differences, and squares cannot be negative.
- Each deviation (x − mean) may be positive or negative.
- Squaring removes the negative sign.
- The sum of squared values is zero or positive.
8. What does a high or low variance mean?
A high variance means data points are widely spread out, while a low variance means they are closely clustered around the mean.
- Low variance: values are similar to each other.
- High variance: values differ greatly from the mean.
9. What is the shortcut formula for variance?
The shortcut formula for population variance is σ² = (Σx² / N) − μ².
- Step 1: Find Σx² (sum of squares).
- Step 2: Divide by N.
- Step 3: Subtract μ² (square of the mean).
10. What are the main properties of variance?
The main properties of variance describe how it behaves under addition and scaling of data.
- Var(X + c) = Var(X) (adding a constant does not change variance).
- Var(aX) = a² Var(X) (multiplying by a constant multiplies variance by a²).
- Variance is always ≥ 0.
- If all values are equal, variance is 0.
