CBSE Class 11 Maths: Analysis of Statistics - Exercise 15.2 (Measures of Dispersion)

Why Measures of Dispersion are Essential

Imagine two cricket teams scoring an average of 150 runs in five matches. Does this mean their performance is identical? Not necessarily! One team might consistently score around 150 runs (e.g., 145, 155, 150, 140, 160), while the other might have highly fluctuating scores (e.g., 50, 250, 150, 100, 200). Both have the same mean, but their consistency is vastly different. This is where measures of dispersion come into play. They quantify the degree to which individual data points vary from the central value, giving us a complete picture of the data distribution. Exercise 15.2 primarily focuses on Variance and Standard Deviation, which are the most commonly used and mathematically robust measures of dispersion. They help us understand the homogeneity or heterogeneity of a dataset, which is vital in making informed decisions and drawing reliable conclusions from statistical data.

Key Concepts: Variance and Standard Deviation

Variance (σ²)

Variance is the average of the squared differences from the mean. It quantifies how much the data points deviate from the mean. A larger variance indicates that data points are more spread out from the mean, while a smaller variance indicates they are clustered closer to the mean. It's expressed in squared units of the original data. For Ungrouped Data: σ² = (Σ(xᵢ - μ)²) / N or σ² = (Σxᵢ² / N) - μ² For Grouped Data: σ² = (Σfᵢ(xᵢ - μ)²) / N or σ² = (Σfᵢxᵢ² / N) - μ², where N = Σfᵢ

Standard Deviation (σ)

Standard Deviation is the square root of the variance. It measures the typical amount of variation or deviation of data points from the mean. It's expressed in the same units as the original data, making it easier to interpret than variance. A low standard deviation means data points are generally close to the mean, while a high standard deviation means data points are spread out over a wider range of values. For Ungrouped Data: σ = √[ (Σ(xᵢ - μ)²) / N ] or σ = √[ (Σxᵢ² / N) - μ² ] For Grouped Data: σ = √[ (Σfᵢ(xᵢ - μ)²) / N ] or σ = √[ (Σfᵢxᵢ² / N) - μ² ]

Mean (μ or x̄)

The arithmetic mean is the sum of all values in a dataset divided by the number of values. It represents the central value of the dataset. For ungrouped data, μ = Σxᵢ / N. For grouped data, μ = Σfᵢxᵢ / N, where N = Σfᵢ.

Step-by-Step Calculation of Variance and Standard Deviation

1. For Ungrouped Data — 1. Calculate the Mean (μ): Sum all the observations (Σxᵢ) and divide by the total number of observations (N). μ = Σxᵢ / N. 2. Calculate Deviations from the Mean (xᵢ - μ): Subtract the mean from each observation. 3. Square the Deviations (xᵢ - μ)²: Square each deviation calculated in step 2. 4. Sum the Squared Deviations (Σ(xᵢ - μ)²): Add up all the squared deviations. 5. Calculate Variance (σ²): Divide the sum of squared deviations by the total number of observations (N). σ² = Σ(xᵢ - μ)² / N. 6. Calculate Standard Deviation (σ): Take the square root of the variance. σ = √σ². Alternatively, using the shortcut method: 1. Calculate the Mean (μ): Same as above. 2. Square each observation (xᵢ²): Find the square of every data point. 3. Sum the squared observations (Σxᵢ²): Add all the squared observations. 4. Calculate Variance (σ²): Use the formula: σ² = (Σxᵢ² / N) - μ². 5. Calculate Standard Deviation (σ): Take the square root of the variance. σ = √σ².
2. For Grouped Data (Frequency Distribution) — 1. Calculate the Mean (μ): For grouped data, first find the midpoint (xᵢ) of each class interval. Then, multiply each midpoint by its corresponding frequency (fᵢxᵢ). Sum these products (Σfᵢxᵢ) and divide by the total number of observations (N = Σfᵢ). μ = Σfᵢxᵢ / N. 2. Calculate Deviations from the Mean (xᵢ - μ): Subtract the mean from each class midpoint. 3. Square the Deviations (xᵢ - μ)²: Square each deviation. 4. Multiply by Frequency (fᵢ(xᵢ - μ)²): Multiply each squared deviation by its corresponding frequency. 5. Sum the Products (Σfᵢ(xᵢ - μ)²): Add up all the products from step 4. 6. Calculate Variance (σ²): Divide the sum from step 5 by the total number of observations (N). σ² = Σfᵢ(xᵢ - μ)² / N. 7. Calculate Standard Deviation (σ): Take the square root of the variance. σ = √σ². Alternatively, using the shortcut method: 1. Calculate the Mean (μ): Same as above. 2. Multiply square of midpoint by frequency (fᵢxᵢ²): For each class, find xᵢ², then multiply by fᵢ. 3. Sum the products (Σfᵢxᵢ²): Add all the fᵢxᵢ² values. 4. Calculate Variance (σ²): Use the formula: σ² = (Σfᵢxᵢ² / N) - μ². 5. Calculate Standard Deviation (σ): Take the square root of the variance. σ = √σ².

Worked Examples from Exercise 15.2 Context

• Example 1: Variance and Standard Deviation for Ungrouped Data Calculate the variance and standard deviation for the following data: 6, 7, 10, 12, 13, 4, 8, 12. Solution: Step 1: Calculate the Mean (μ). N = 8 Σxᵢ = 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72 μ = Σxᵢ / N = 72 / 8 = 9 Step 2: Create a table for deviations and squared deviations. | xᵢ | xᵢ - μ (xᵢ - 9) | (xᵢ - μ)² | |----|-----------------|-----------| | 6 | -3 | 9 | | 7 | -2 | 4 | | 10 | 1 | 1 | | 12 | 3 | 9 | | 13 | 4 | 16 | | 4 | -5 | 25 | | 8 | -1 | 1 | | 12 | 3 | 9 | Step 3: Sum the squared deviations. Σ(xᵢ - μ)² = 9 + 4 + 1 + 9 + 16 + 25 + 1 + 9 = 74 Step 4: Calculate Variance (σ²). σ² = Σ(xᵢ - μ)² / N = 74 / 8 = 9.25 Step 5: Calculate Standard Deviation (σ). σ = √σ² = √9.25 ≈ 3.041 Final Answer: Variance = 9.25, Standard Deviation ≈ 3.041
• Example 2: Variance and Standard Deviation for Grouped Data (Discrete Frequency Distribution) Find the variance and standard deviation for the following frequency distribution: | xᵢ | 2 | 4 | 6 | 8 | 10 | | fᵢ | 3 | 5 | 7 | 8 | 2 | Solution: Step 1: Create a table to calculate fᵢxᵢ and fᵢxᵢ². | xᵢ | fᵢ | fᵢxᵢ | xᵢ² | fᵢxᵢ² | |----|----|------|-----|-------| | 2 | 3 | 6 | 4 | 12 | | 4 | 5 | 20 | 16 | 80 | | 6 | 7 | 42 | 36 | 252 | | 8 | 8 | 64 | 64 | 512 | | 10 | 2 | 20 | 100 | 200 | Step 2: Calculate N, Σfᵢxᵢ, and Σfᵢxᵢ². N = Σfᵢ = 3 + 5 + 7 + 8 + 2 = 25 Σfᵢxᵢ = 6 + 20 + 42 + 64 + 20 = 152 Σfᵢxᵢ² = 12 + 80 + 252 + 512 + 200 = 1056 Step 3: Calculate the Mean (μ). μ = Σfᵢxᵢ / N = 152 / 25 = 6.08 Step 4: Calculate Variance (σ²) using the shortcut formula. σ² = (Σfᵢxᵢ² / N) - μ² σ² = (1056 / 25) - (6.08)² σ² = 42.24 - 36.9664 σ² = 5.2736 Step 5: Calculate Standard Deviation (σ). σ = √σ² = √5.2736 ≈ 2.2964 Final Answer: Variance = 5.2736, Standard Deviation ≈ 2.2964

Exam Tips and Common Pitfalls

To ace questions on variance and standard deviation, keep these points in mind:

• Formula Recall: Memorize both the direct and shortcut formulas for variance and standard deviation for both ungrouped and grouped data. Often, one method might be significantly easier for a given dataset.
• Tabular Approach: Always use a table to organize your calculations, especially for grouped data. Columns for xᵢ, fᵢ, fᵢxᵢ, (xᵢ - μ), (xᵢ - μ)², and fᵢ(xᵢ - μ)² (or xᵢ² and fᵢxᵢ²) will prevent errors and make your steps clear.
• Mean First: The mean (μ) is the cornerstone of variance and standard deviation calculations. Ensure its calculation is accurate before proceeding.
• Units: Remember that standard deviation has the same units as the original data, while variance has squared units. This helps in interpretation.
• Rounding Off: Pay attention to the number of decimal places required in the final answer. Avoid premature rounding in intermediate steps to maintain accuracy.
• Negative Deviations: When squaring deviations, any negative sign will become positive. A common mistake is carrying forward negative signs incorrectly.
• Interpretation: Understand what a high or low standard deviation signifies about the spread of data. This might be asked in theory-based questions.

Practice Questions with Solutions

• Q: Calculate the variance and standard deviation for the following marks obtained by 5 students: 15, 20, 25, 30, 35. A: Step 1: Calculate the Mean (μ). N = 5 Σxᵢ = 15 + 20 + 25 + 30 + 35 = 125 μ = 125 / 5 = 25 Step 2: Calculate (xᵢ - μ) and (xᵢ - μ)². | xᵢ | xᵢ - 25 | (xᵢ - 25)² | |----|---------|------------| | 15 | -10 | 100 | | 20 | -5 | 25 | | 25 | 0 | 0 | | 30 | 5 | 25 | | 35 | 10 | 100 | Σ(xᵢ - μ)² = 100 + 25 + 0 + 25 + 100 = 250 Step 3: Calculate Variance (σ²). σ² = Σ(xᵢ - μ)² / N = 250 / 5 = 50 Step 4: Calculate Standard Deviation (σ). σ = √50 ≈ 7.07 Final answer: Variance = 50, Standard Deviation ≈ 7.07
• Q: Find the standard deviation of the first 7 natural numbers. A: Step 1: List the data and calculate the Mean (μ). Data: 1, 2, 3, 4, 5, 6, 7. N = 7 Σxᵢ = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 μ = 28 / 7 = 4 Step 2: Calculate (xᵢ - μ) and (xᵢ - μ)². | xᵢ | xᵢ - 4 | (xᵢ - 4)² | |----|--------|-----------| | 1 | -3 | 9 | | 2 | -2 | 4 | | 3 | -1 | 1 | | 4 | 0 | 0 | | 5 | 1 | 1 | | 6 | 2 | 4 | | 7 | 3 | 9 | Σ(xᵢ - μ)² = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28 Step 3: Calculate Variance (σ²). σ² = Σ(xᵢ - μ)² / N = 28 / 7 = 4 Step 4: Calculate Standard Deviation (σ). σ = √4 = 2 Final answer: Standard Deviation = 2
• Q: For the following data, calculate the variance: | Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | |----------------|------|-------|-------|-------|-------| | Frequency (fᵢ) | 2 | 3 | 5 | 4 | 1 | A: Step 1: Find midpoints (xᵢ), calculate N, Σfᵢxᵢ, and μ. | C.I. | fᵢ | xᵢ | fᵢxᵢ | xᵢ² | fᵢxᵢ² | |--------|----|----|------|-----|-------| | 0-10 | 2 | 5 | 10 | 25 | 50 | | 10-20 | 3 | 15 | 45 | 225 | 675 | | 20-30 | 5 | 25 | 125 | 625 | 3125 | | 30-40 | 4 | 35 | 140 | 1225| 4900 | | 40-50 | 1 | 45 | 45 | 2025| 2025 | N = Σfᵢ = 2+3+5+4+1 = 15 Σfᵢxᵢ = 10+45+125+140+45 = 365 μ = Σfᵢxᵢ / N = 365 / 15 = 24.33 (approx) Step 2: Calculate Σfᵢxᵢ². Σfᵢxᵢ² = 50+675+3125+4900+2025 = 10775 Step 3: Calculate Variance (σ²). σ² = (Σfᵢxᵢ² / N) - μ² σ² = (10775 / 15) - (24.33)² σ² = 718.3333 - 591.9489 = 126.3844 Final answer: Variance ≈ 126.38

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