Dispersion: quick revision, 13 bites

Standard Deviation: The Ruler of Spread

Measure how far numbers stray.

**Standard deviation** ($\sigma$) is the most important measure of dispersion. It tells you the typical distance of each observation from the mean. A small $\sigma$ means data points cluster tightly around the average; a large $\sigma$ means they are spread out.

Takeaway. Standard deviation is the square root of the average squared deviation from the mean.

Computing SD: Step by Step

Follow the recipe for $\sigma$.

Takeaway. Always square, sum, divide, then root.

When Dispersion Vanishes

All equal? Spread is zero.

If every observation is the same constant $k$ (i.e., $x_1 = x_2 = \ldots = x_n = k$), then the mean is also $k$. Every deviation $(x_i - \bar{x})$ is zero, so the standard deviation is **zero**. This is the only case where dispersion is zero.

Takeaway. No variation means zero standard deviation.

Coefficient of Variation: Comparing Apples and Oranges

Relative spread, not absolute.

The **coefficient of variation** (CV) expresses standard deviation as a percentage of the mean. It lets you compare variability between datasets with different units or very different means.

Takeaway. CV is unit-free and perfect for comparing consistency.

SD vs. CV: Absolute vs. Relative

One measures spread, one compares.

Takeaway. Use SD for one set, CV to compare sets.

Who Is More Consistent? Batsman A or B?

Lower CV wins the consistency game.

From the illustration: Batsman A has mean $\bar{x}_A = 44$, $\sigma_A = \sqrt{1710/10} \approx 13.08$. Batsman B has mean $\bar{x}_B = 53$, $\sigma_B = \sqrt{5928/10} \approx 24.35$. CV for A = $13.08/44 \times 100 \approx 29.7\%$; CV for B = $24.35/53 \times 100 \approx 45.9\%$. Since A has lower CV, **A is more consi…

Takeaway. Lower CV means more consistency.

Don't Forget the Square Root

Variance is not SD.

Takeaway. SD = $\sqrt{\text{Variance}}$.

Stability of Salaries: Company A vs. B

Which company pays more steadily?

Company A: mean = ₹600, SD = ₹30 → CV = $30/600 \times 100 = 5\%$. Company B: mean = ₹2100, SD = ₹84 → CV = $84/2100 \times 100 = 4\%$. Company B has a lower CV, so its salaries are **more stable** relative to the mean.

Takeaway. Lower CV indicates greater stability.

Combined Standard Deviation

Merge two groups correctly.

When combining two groups with sizes $n_1, n_2$, means $\bar{x}_1, \bar{x}_2$, and standard deviations $s_1, s_2$, the combined standard deviation $s$ is:

Takeaway. Combined SD accounts for both within-group and between-group variation.

Which Formula for Grouped Data?

Use midpoints for classes.

For grouped frequency distributions, use the class midpoints as $x_i$ and the frequencies as weights. The formula becomes $\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$.

Takeaway. Grouped data: replace each class with its midpoint.

CV: The Consistency Quotient

Lower CV, more consistent.

Remember: **C**oefficient of **V**ariation = **C**onsistency **V**alue. The smaller the CV, the more consistent (stable) the data. Think: less variation relative to mean.

Takeaway. Small CV = high consistency.

Why Square Deviations?

Squaring removes negatives.

If we simply summed deviations $(x_i - \bar{x})$, they would always add to zero. Squaring makes all terms positive and gives more weight to larger deviations, which is desirable when measuring spread.

Takeaway. Squaring ensures positive contributions and emphasizes outliers.

Solving a CV Problem: Step-by-Step

Follow the flow to compare.

Takeaway. Always compute both CVs before concluding.