How to Calculate Variance: Stepwise Guide with Example
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.
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FAQs on Variance in Mathematics: Meaning, Formula & Calculation
1. What is variance in Maths?
Variance, in mathematics and statistics, is a measure of how spread out a set of numbers is. It quantifies the average squared deviation of each number from the mean (average) of the set. A high variance indicates that the numbers are far from the mean, while a low variance indicates they are clustered closely around the mean.
2. What is the formula for variance?
The formula for population variance (σ²) is: σ² = Σ(xi - μ)² / N, where xi represents each individual value, μ is the population mean, and N is the total number of values in the population. For sample variance (s²), the formula is slightly different: s² = Σ(xi - x̄)² / (n - 1), where x̄ represents the sample mean and n is the sample size. The (n-1) is used to provide an unbiased estimate of the population variance.
3. How do I calculate variance step-by-step?
1. **Calculate the mean (average):** Add all the numbers and divide by the total count. 2. **Find the squared differences:** Subtract the mean from each number, then square the result. 3. **Average the squared differences:** Add up all the squared differences and divide by N (for population variance) or (n-1) (for sample variance).
4. What is the difference between variance and standard deviation?
Standard deviation is simply the square root of the variance. While variance gives a measure of spread in squared units, standard deviation returns a measure in the original units, making it easier to interpret in the context of the data.
5. What are the symbols used for variance?
The symbol for population variance is typically σ² (sigma squared), while the symbol for sample variance is usually s².
6. Can variance be negative?
No, variance cannot be negative. This is because the formula involves squaring the differences from the mean, and squares are always non-negative.
7. How is variance used in real life?
Variance is used across many fields. In finance, it helps assess investment risk. In manufacturing, it measures the consistency of a production process. In science, it describes the variability of experimental results. It’s a key concept in understanding data spread and variability.
8. What is the impact of outliers on variance?
Outliers (extreme values) significantly inflate variance because they contribute large squared differences. This makes variance sensitive to extreme values, potentially misrepresenting the typical spread of the data.
9. How does variance relate to the mean?
Variance measures the dispersion of data *around* the mean. A larger variance indicates greater dispersion, while a smaller variance indicates that the data points are clustered more tightly around the mean.
10. What is the difference between population variance and sample variance?
Population variance describes the spread of an entire population of data. Sample variance estimates the population variance based on a smaller sample, using a slightly different formula (dividing by n-1 instead of n) to correct for bias.
11. Why is the denominator (n-1) used in sample variance?
Using (n-1) instead of n in the sample variance formula provides an unbiased estimator of the population variance. Dividing by n would underestimate the population variance.
12. How is variance calculated for grouped data?
For grouped data, the variance calculation involves using the midpoint of each class interval as a representative value. The formula remains similar, but the individual data points are replaced by the midpoints and their corresponding frequencies. The calculation is more complex and often done using a statistical calculator or software.
