How to Calculate Variance and Standard Deviation with Step-by-Step Examples
The concept of variance and standard deviation plays a key role in mathematics, especially in statistics, and helps you understand how spread out or close together data values are in a set. Knowing how to calculate and interpret variance and standard deviation is essential for exams, projects, and understanding data in real life.
What Is Variance and Standard Deviation?
Variance and standard deviation are two important measures of dispersion in maths. Variance tells us the average of the squared differences from the mean, giving an idea of how far data values spread from their average. Standard deviation is the square root of the variance and describes how much the values typically differ from the mean. You’ll find these concepts applied in data analysis, probability, and real-world decision-making.
Variance and Standard Deviation Formulas
Here are the key formulas for both population and sample data sets:
| Measure | Population Formula | Sample Formula |
|---|---|---|
| Variance (\(\sigma^2\), \(s^2\)) | \(\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\) | \(s^2 = \frac{\sum (x_i - \overline{x})^2}{n-1}\) |
| Standard Deviation (\(\sigma\), \(s\)) | \(\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}\) | \(s = \sqrt{\frac{\sum (x_i - \overline{x})^2}{n-1}}\) |
Where: \(x_i\) = each data value, \(\mu\) = population mean, \(\overline{x}\) = sample mean, \(N\) = population size, \(n\) = sample size.
Relationship Between Variance and Standard Deviation
Standard deviation is always the square root of variance. While variance shows the average squared distance from the mean, standard deviation brings it back to the original unit of the data, making it easier to understand.
| Aspect | Variance | Standard Deviation |
|---|---|---|
| Formula | Average squared distance from mean | Square root of variance |
| Unit | Squared unit (e.g., cm2) | Same as data (e.g., cm) |
Step-by-Step Calculation with Example
Let’s calculate variance and standard deviation using a simple data set: 4, 7, 8, 12, 15
Steps:1. Find the mean (\(\overline{x}\)):
\((4 + 7 + 8 + 12 + 15) / 5 = 46 / 5 = 9.2\)
2. Find the squared differences from the mean:
(4 - 9.2)2 = 27.04
(7 - 9.2)2 = 4.84
(8 - 9.2)2 = 1.44
(12 - 9.2)2 = 7.84
(15 - 9.2)2 = 33.64
3. Sum the squared differences:
27.04 + 4.84 + 1.44 + 7.84 + 33.64 = 74.8
4. For sample variance (\(n-1 = 5-1 = 4\)):
\(s^2 = 74.8 / 4 = 18.7\)
5. For sample standard deviation:
\(s = \sqrt{18.7} \approx 4.32\)
Final Answers: Variance ≈ 18.7, Standard Deviation ≈ 4.32
Calculator & Tools
To save time in calculations, you can use the Vedantu Standard Deviation Calculator:
- Enter your data set (use commas or spaces).
- Choose population or sample as needed.
- Press ‘Calculate’ to get variance and standard deviation instantly.
- Review the calculation steps shown below the result for practice.
Properties & Uses
- Variance is always zero or positive.
- Units of variance are always squared, while standard deviation has the same unit as the data.
- If all data points are same, both variance and standard deviation are zero.
- Low standard deviation means data are clustered closely; high standard deviation signals wider spread.
- These measures help compare consistency and reliability of data in science, sports, and economics.
Understanding variance and standard deviation is vital for exams, project work, and analyzing data trends in real life.
Common Mistakes & Tips
- For samples, always divide by (n-1), not n.
- Don’t confuse variance (squared unit) with standard deviation (original unit).
- Never skip squaring differences; it avoids positive and negative values cancelling each other.
- Remember: SD = √variance, and variance = (SD)2.
- Make sure to use the right formula for population or sample as per the question.
Practice Questions + Solutions
- Find the variance and standard deviation of 2, 4, 6, 8, 10.
- If the mean is 20 and data values are 15, 20, 25, what is the variance?
- Which has higher spread: data set A (5, 5, 5, 5) or set B (2, 8, 5, 10)? Why?
- MCQ: What does a standard deviation of zero indicate?
- Prove that adding the same number to all values does not change SD.
Solution Example for (1):
1. Mean = (2+4+6+8+10)/5 = 30/5 = 62. Squared differences:
(2-6)2=16, (4-6)2=4, (6-6)2=0, (8-6)2=4, (10-6)2=16
3. Sum = 16+4+0+4+16=40
4. Sample variance (n-1=4): 40/4=10
5. Sample SD: √10 ≈ 3.16
Answers: Variance = 10, SD ≈ 3.16
Interlinks to Related Concepts
- Compare with Mean, Median, and Mode for more on data averages.
- Learn about Range in Statistics to see another way to measure spread.
- Dive into Mean Absolute Deviation for a different perspective on dispersion.
- See Difference Between Variance and Standard Deviation for detailed comparisons.
Quick Revision Tips
- V for Variance = “V for Very Squared” — it’s always in squared units!
- Standard deviation “undoes” the square, so its unit is same as data.
- Always check: Sample = denominator n-1; Population = denominator n.
We explored variance and standard deviation from definition, formulas, calculation steps, common mistakes, and connections to other maths concepts. Regular practice with Vedantu’s stepwise techniques and calculators can help you become confident for exams and real-life data problems!
FAQs on Variance and Standard Deviation in Maths: Concepts, Formulas & Solved Examples
1. What is the definition of variance? What are the properties of a variance?
Variance measures how spread out a data set is from its mean (average). It's the average of the squared differences from the mean, represented by σ2 (sigma squared). Key properties include:
- Always non-negative: Variance is always zero or positive because it involves squaring differences.
- Units are squared: The units of variance are the square of the original data's units (e.g., if data is in meters, variance is in square meters).
- Sensitive to outliers: Large deviations from the mean significantly inflate the variance.
2. What is the definition of standard deviation? What are the properties of standard deviation?
Standard deviation (σ) measures the spread of data around the mean. It's the square root of the variance, giving a value in the original data's units. Key properties are:
- Always non-negative: It cannot be negative because it's the square root of a squared value.
- Same units as data: The standard deviation's units are the same as the original data's units.
- Interpretable: It tells you the typical distance of data points from the mean. A small standard deviation indicates data points are clustered near the mean; a large standard deviation indicates they're spread far from the mean.
- Affected by outliers: Outliers can significantly affect the standard deviation.
3. What is the difference between variance and standard deviation?
Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance. Standard deviation is preferred for interpretation because it's in the same units as the original data, making it easier to understand how spread out the data is.
4. When should you use sample vs. population standard deviation?
Use population standard deviation when you have data for the entire population. Use sample standard deviation when your data is a sample from a larger population. The sample standard deviation formula uses n-1 in the denominator (instead of n for the population) to provide an unbiased estimate of the population standard deviation.
5. What is the standard deviation of 5, 5, 9, 9, 9, 10, 5, 10, 10?
To calculate this, follow these steps:
- Calculate the mean: (5+5+9+9+9+10+5+10+10)/9 = 8
- Find the differences from the mean: -3, -3, 1, 1, 1, 2, -3, 2, 2
- Square the differences: 9, 9, 1, 1, 1, 4, 9, 4, 4
- Sum the squared differences: 42
- Divide by n-1 (sample standard deviation): 42/8 = 5.25
- Take the square root: √5.25 ≈ 2.29
The sample standard deviation is approximately 2.29.
6. Why is variance expressed in squared units, but standard deviation is not?
Variance uses squared units because it's calculated using squared differences from the mean. Taking the square root to obtain the standard deviation returns the units to their original scale, making the result more easily interpretable in the context of the original data.
7. How do standard deviation and variance affect confidence intervals and probability?
Standard deviation is crucial in calculating confidence intervals. Larger standard deviations lead to wider confidence intervals, reflecting greater uncertainty about the population parameter being estimated. In probability distributions, standard deviation measures the spread or dispersion of data around the mean, impacting the likelihood of observing values within a certain range.
8. What are common mistakes in variance and standard deviation calculations on scientific calculators?
Common mistakes include incorrect order of operations (remember to square differences *before* averaging), using the wrong formula (population vs. sample), and misinterpreting the displayed result (units, significance).
9. Can two data sets have the same mean but different standard deviations?
Yes, absolutely! Two data sets can have identical means but significantly different standard deviations. This indicates that while the central tendency is the same, the spread or dispersion of data around that mean is different. One set might be tightly clustered around the mean, while the other is more spread out.
10. How are variance and standard deviation used in machine learning and data science?
Variance and standard deviation are fundamental in machine learning and data science for feature scaling, evaluating model performance (e.g., measuring prediction error), and identifying outliers in data. They provide insights into data distributions and inform model selection and parameter tuning.
11. What are some real-world applications of standard deviation?
Standard deviation is used in finance to measure investment risk (volatility), in manufacturing to control product quality, in healthcare to analyze patient outcomes, and in many other fields to understand data variability and make informed decisions.
