Class 11 Maths • Chapter 13 • Comprehensive Interactive Notes
Dispersion measures how "spread out" the data is. The mean tells us the center, but dispersion tells us about the variation.
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“Why is standard deviation better than mean deviation?”
| Measure | Formula |
|---|---|
| Mean (\(\bar{x}\)) | \( \frac{\sum x_i}{n} \) |
| Median (M) | \( \text{Odd } n: (\frac{n+1}{2})^{th} \text{ term} \) \( \text{Even } n: \text{Avg of } (\frac{n}{2})^{th}, (\frac{n}{2}+1)^{th} \) |
| Mean Deviation (\(\bar{x}\)) | \( \frac{\sum |x_i - \bar{x}|}{n} \) |
| Mean Deviation (M) | \( \frac{\sum |x_i - M|}{n} \) |
| Variance (\(\sigma^2\)) | \( \frac{\sum (x_i - \bar{x})^2}{n} \) |
| Standard Deviation (\(\sigma\)) | \( \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \) |
| Shortcut Method for Variance | \( \frac{\sum x_i^2}{n} - (\frac{\sum x_i}{n})^2 \) |
| Measure | Formula |
|---|---|
| Mean (\(\bar{x}\)) | \( \frac{\sum f_i x_i}{N} \) (Direct) \( A + \frac{\sum f_i d_i}{N} \) (Assumed Mean) |
| Median (M) | \( l + \frac{\frac{N}{2} - C}{f} \times h \) |
| Mean Deviation (\(\bar{x}\)) | \( \frac{\sum f_i |x_i - \bar{x}|}{N} \) |
| Mean Deviation (M) | \( \frac{\sum f_i |x_i - M|}{N} \) |
| Variance (\(\sigma^2\)) | \( \frac{\sum f_i (x_i - \bar{x})^2}{N} \) |
| Variance (Shortcut Method) | \( \frac{\sum f_i x_i^2}{N} - (\frac{\sum f_i x_i}{N})^2 \) |
| Variance (Step-Deviation) | \( h^2 \left[ \frac{\sum f_i d_i^2}{N} - \left( \frac{\sum f_i d_i}{N} \right)^2 \right] \), where \(d_i = \frac{x_i - A}{h}\) |
| Standard Deviation (\(\sigma\)) | \( \sqrt{\text{Variance}} \) |
Steps for Ungrouped Data:
Formula:
\[ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \]CBSE Tip: Always show steps — even if calculation is simple.
To compare the variability of two series with different units, we use the Coefficient of Variation (C.V.).
\( C.V. = \frac{\sigma}{\bar{x}} \times 100 \)
Rule: Lower C.V. = More Consistent.
Compare two datasets (e.g., Players A & B).
1. Which measure of dispersion is based on squared deviations?
2. If Standard Deviation is 4, Variance is:
3. A lower Coefficient of Variation indicates:
4. Mean deviation can be calculated about:
5. The variance of 5, 5, 5, 5, 5 is:
Self-Test: