Exercise 12.3: Variance, Standard Deviation & Coefficient of Variation for Grouped Data

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Chapter Overview & SLOs

What is statistical dispersion and why is it important in data analysis? While measures of central tendency (mean, median, mode) tell us where data is centered, they don't reveal how spread out the data points are. Dispersion measures the extent to which numerical data points scatter or deviate from their central average value. This is crucial for understanding data reliability, consistency, and variability.

The Three Fundamental Absolute Dispersion Measures:

• Range ($R$): The simplest measure — the difference between maximum and minimum values: $R = x_{\text{max}} - x_{\text{min}}$
• Variance ($S^2$ or $\sigma^2$): The mean of the squared deviations from the mean. Squaring ensures positive and negative deviations don't cancel out.
• Standard Deviation ($S$ or $\sigma$): The positive square root of variance — brings the spread metric back to original data units.

How do you calculate the weighted arithmetic mean for grouped data?

For grouped frequency distributions, we use the weighted mean formula:

$$\bar{x} = \frac{\sum fx}{\sum f}$$

where $x$ = class midpoint, $f$ = frequency, and $\sum f = N$ (total number of observations).

Example: Medical Blood Pressure Data

A medical dataset monitoring 30 emergency operating theater patients yields a mean blood pressure of $\bar{x} \approx 127.33$ mmHg.

How do you calculate variance and standard deviation for grouped data?

Step 1: Find class midpoints ($x$)

• For interval $a-b$: midpoint $x = \frac{a+b}{2}$

Step 2: Calculate $\sum fx$ and $\sum fx^2$

Step 3: Apply the grouped variance formula:

$$S^2 = \frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2$$

Step 4: Calculate standard deviation:

$$S = \sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}$$

Alternative Method (Mean Deviation Method):

$$S^2 = \frac{\sum f(x-\bar{x})^2}{\sum f}$$

What is the Coefficient of Variation (CV) and why is it useful?

Absolute standard deviations cannot be directly compared if datasets use different units or scales. The Coefficient of Variation normalizes standard deviation against the mean, expressing it as a unitless percentage:

$$CV = \left(\frac{S}{\bar{x}}\right) \times 100\%$$

Interpreting CV Values:

• Higher CV = Greater relative volatility — data is more spread out relative to the mean
• Lower CV = More consistent data — data points are more tightly clustered around the mean

Example Comparison:

• Dataset A: Mean = 100, SD = 10 → CV = 10%
• Dataset B: Mean = 50, SD = 10 → CV = 20%
• Conclusion: Dataset A is more consistent (lower relative variability) despite having the same absolute standard deviation

What key concepts are tested in FBISE Exercise 12.3? This comprehensive exercise tests:

• Defining statistical dispersion and its importance
• Range, variance, and standard deviation formulas
• Weighted arithmetic mean calculation for grouped data: $\bar{x} = \frac{\sum fx}{\sum f}$
• Grouped variance using direct formula: $S^2 = \frac{\sum fx^2}{\sum f} - (\frac{\sum fx}{\sum f})^2$
• Grouped variance using mean deviation method: $S^2 = \frac{\sum f(x-\bar{x})^2}{\sum f}$
• Standard deviation: $S = \sqrt{S^2}$
• Coefficient of Variation: $CV = (\frac{S}{\bar{x}}) \times 100\%$
• Interpreting CV for comparing consistency across datasets

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do you define statistical dispersion and identify the core algebraic formulas for Range, Variance, and Standard Deviation? Students will distinguish between absolute and relative dispersion measures.
• How do you compute weighted arithmetic means and continuous class midpoints from grouped data tables using $\bar{x} = \frac{\sum fx}{\sum f}$? Students will process interval distributions into midpoints.
• How do you calculate absolute variance and standard deviation for grouped data using the direct formula $S^2 = \frac{\sum fx^2}{\sum f} - (\frac{\sum fx}{\sum f})^2$ and the mean deviation method $S^2 = \frac{\sum f(x-\bar{x})^2}{\sum f}$? Students will apply both methods correctly.
• How do you determine the Coefficient of Variation ($CV = \frac{S}{\bar{x}} \times 100\%$) to compare relative stability and consistency between different datasets? Students will interpret CV percentages to identify which dataset is more consistent.

Frequently Asked Questions (FAQ)