Syllabus Unit 5 – Statistics Measures of Dispersion Range, Mean Deviation, Standard Deviation and Variance

Range, Mean Deviation, Standard Deviation and Variance

Master Range, Mean Deviation, Standard Deviation, and Variance for CBSE Class 11 Applied Mathematics with solved examples and practice problems.

Range and Mean Deviation

Range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset. Mean Deviation provides a more robust measure by calculating the average of the absolute differences between each data point and the mean (or median). It effectively describes how far, on average, the data points are from the central value.

Range = x_max − x_min; M.D.(x̄) = Σ|xᵢ − x̄| / n
Example: Solved Example: Find the Range and Mean Deviation of {4, 8, 12, 16, 20}
Show Step-by-Step Solution

Step 1: Range = 20 − 4 = 16.
Step 2: Mean (x̄) = (4+8+12+16+20)/5 = 60/5 = 12.
Step 3: M.D. = (|4-12| + |8-12| + |12-12| + |16-12| + |20-12|) / 5 = (8+4+0+4+8)/5 = 24/5 = 4.8.
Answer: Range = 16, M.D. = 4.8.

Variance

Variance measures the spread of a set of numbers from their mean value. It is defined as the average of the squared differences from the mean. A higher variance indicates that the data points are spread out over a wider range of values.

σ² = Σ(xᵢ − x̄)² / n
Example: Solved Example: Calculate Variance for {2, 4, 6}
Show Step-by-Step Solution

Step 1: Mean (x̄) = (2+4+6)/3 = 4.
Step 2: Squared differences: (2-4)²=4, (4-4)²=0, (6-4)²=4.
Step 3: Variance = (4+0+4)/3 = 8/3 ≈ 2.67.
Answer: 2.67.

Standard Deviation

Standard Deviation is the positive square root of the variance, providing a measure of dispersion in the same units as the original data. It is the most widely used measure of dispersion in statistical analysis due to its mathematical properties.

σ = √[Σ(xᵢ − x̄)² / n]
Example: Solved Example: Find Standard Deviation for {3, 7, 11}
Show Step-by-Step Solution

Step 1: Mean (x̄) = (3+7+11)/3 = 7.
Step 2: Variance = [(3-7)² + (7-7)² + (11-7)²] / 3 = (16+0+16)/3 = 32/3.
Step 3: Standard Deviation = √(32/3) ≈ √10.67 ≈ 3.27.
Answer: 3.27.