Variance: Quantitative Ability Study Note for CMAT
Introduction to Variance
In statistics, Variance is a key measure of dispersion that quantifies how much the data points in a set spread out from their arithmetic mean. While measures of central tendency like the mean or median tell us where the middle of the data lies, variance tells us how scattered the data points are around that middle value.
For the CMAT exam, understanding the properties of variance is highly critical because direct conceptual questions, scaling property problems, and computational questions are frequently asked under the Quantitative Ability section.
Core Formulas
There are two main formulas for calculating variance depending on whether you are analyzing an entire population or a representative sample.
1. Population Variance ($\sigma^2$)
Used when the dataset represents the entire group under study:
$$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$$
Where:
- $x_i$ represents each individual value in the dataset.
- $\mu$ is the population arithmetic mean.
- $N$ is the total number of observations in the population.
- $\sigma$ is the lowercase Greek letter sigma (so $\sigma^2$ represents population variance).
2. Sample Variance ($s^2$)
Used when the dataset represents a smaller sample taken from a larger population (Bessel's correction is applied by dividing by $n - 1$ instead of $n$ to avoid bias):
$$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$$
Where:
- $\bar{x}$ is the sample mean.
- $n$ is the total number of observations in the sample.
Step-by-Step Calculation Guide
Let's calculate the population variance for a small, typical dataset of CMAT standard: $3, 5, 7, \text{ and } 9$.
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Step 1: Find the Mean ($\mu$) $$\mu = \frac{3 + 5 + 7 + 9}{4} = \frac{24}{4} = 6$$
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Step 2: Find the Deviation of Each Value from Mean ($x_i - \mu$)
- For $3$: $3 - 6 = -3$
- For $5$: $5 - 6 = -1$
- For $7$: $7 - 6 = 1$
- For $9$: $9 - 6 = 3$
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Step 3: Square Each Deviation ($(x_i - \mu)^2$)
- $(-3)^2 = 9$
- $(-1)^2 = 1$
- $(1)^2 = 1$
- $(3)^2 = 9$
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Step 4: Sum the Squared Deviations $$\sum (x_i - \mu)^2 = 9 + 1 + 1 + 9 = 20$$
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Step 5: Divide by the Number of Observations ($N$) $$\sigma^2 = \frac{20}{4} = 5$$
Thus, the population variance of this dataset is $5$.
Essential Properties of Variance
CMAT often features logical or conceptual questions based on these mathematical properties of variance:
- Non-negativity: Variance is always greater than or equal to zero ($"\text{Var}(X) \ge 0"$). It cannot be negative because it is a sum of squared numbers.
- Effect of Adding/Subtracting a Constant: If a constant $k$ is added to or subtracted from every data point in the set, the variance remains unchanged. $$\text{Var}(X \pm k) = \text{Var}(X)$$
- Effect of Multiplying/Dividing by a Constant: If every data point in the set is multiplied by a constant $k$, the new variance is multiplied by $k^2$ (the square of that constant). $$\text{Var}(kX) = k^2 \cdot \text{Var}(X)$$
- Variance of a Constant: The variance of a constant sequence (where all elements are equal, e.g., $5, 5, 5, 5$) is always $0$ because there is no variation.
Variance vs. Standard Deviation
| Feature | Variance | Standard Deviation |
|---|---|---|
| Definition | Average of the squared deviations from the mean. | The square root of the variance. |
| Symbol | $\sigma^2$ (Population) or $s^2$ (Sample) | $\sigma$ (Population) or $s$ (Sample) |
| Units | Measured in squared units of the original data (e.g., $\text{cm}^2$). | Measured in the same unit as the original data (e.g., $\text{cm}$). |
| Use Case | Useful in mathematical proofs and algebraic manipulations. | Best for intuitive interpretation of data spread. |
Common CMAT Exam Traps
- Trap 1: Forgetting to square the scale factor Example: If the variance of $10$ items is $4$ and each item is multiplied by $3$, students often answer the new variance as $4 \times 3 = 12$. Correct Approach: You must square the multiplier. The new variance is $4 \times 3^2 = 4 \times 9 = 36$.
- Trap 2: Using the wrong denominator for sample variance Always read the question carefully. If the question explicitly mentions "sample variance", divide the sum of squared deviations by $n-1$, not $n$.
- Trap 3: Mixing up Variance and Standard Deviation (SD) Remember: $\text{Variance} = (\text{SD})^2$. If the question asks for SD but you calculated Variance, you must take the square root to find the correct option.
Quick Revision Recap
- Variance measures the dispersion or spread of a dataset.
- Formula for population variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$.
- Formula for sample variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$.
- Shift of origin (adding/subtracting a value) does not change variance.
- Change of scale (multiplying/dividing by a value) scales the variance by $k^2$.
Practise these as timed MCQs to lock the pattern in.