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.
Enter numbers separated by commas (e.g., 6, 7, 10, 12, 13, 4, 8, 12)
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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) \( A + \frac{\sum f_i u_i}{N} \times h \) (Step-Deviation) |
| 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 (\(\sigma\)) | \( \sqrt{\text{Variance}} \) |
This method simplifies calculations when class intervals are large and of equal size \( h \).
Note: Used heavily when numbers are large to reduce calculation errors.
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: