Class 11: Statistics
Measures of Dispersion, Mean Deviation, Variance & Standard Deviation
1. Basics of Dispersion
Dispersion measures how scattered the data is around a central value.
| Range | Difference between the maximum and minimum values. $$Range = Maximum Value – Minimum Value$$ |
2. Mean Deviation (M.D.)
The arithmetic mean of the absolute deviations of observations from a central value (Mean or Median).
| Data Type | About Mean ($$\bar{x}$$) | About Median ($$M$$) |
|---|---|---|
| Ungrouped Data | $$M.D.(\bar{x}) = \frac{\sum |x_i – \bar{x}|}{n}$$ | $$M.D.(M) = \frac{\sum |x_i – M|}{n}$$ |
| Grouped Data (Discrete & Continuous) |
$$M.D.(\bar{x}) = \frac{\sum f_i |x_i – \bar{x}|}{N}$$ (Where $$N = \sum f_i$$) |
$$M.D.(M) = \frac{\sum f_i |x_i – M|}{N}$$ |
Note for Median (Continuous Data):
$$Median = l + \frac{\frac{N}{2} – C}{f} \times h$$
Where $$l$$= lower limit of median class,$$N$$= total frequency,$$C$$= cumulative frequency of preceding class,$$f$$= frequency of median class,$$h$$ = class width.
$$Median = l + \frac{\frac{N}{2} – C}{f} \times h$$
Where $$l$$= lower limit of median class,$$N$$= total frequency,$$C$$= cumulative frequency of preceding class,$$f$$= frequency of median class,$$h$$ = class width.
3. Variance & Standard Deviation
Measures the average squared deviation from the mean. Standard Deviation ($$\sigma$$) is the square root of Variance ($$\sigma^2$$).
| Ungrouped Data | |
| Variance ($$\sigma^2$$) | $$\sigma^2 = \frac{1}{n} \sum (x_i – \bar{x})^2$$ |
| Standard Deviation ($$\sigma$$) | $$\sigma = \sqrt{\frac{1}{n} \sum (x_i – \bar{x})^2}$$ |
| Discrete Frequency Distribution | |
| Variance ($$\sigma^2$$) | $$\sigma^2 = \frac{1}{N} \sum f_i (x_i – \bar{x})^2$$ |
| Alternate Formula | $$\sigma^2 = \frac{1}{N^2} [N \sum f_i x_i^2 – (\sum f_i x_i)^2]$$ |
| Continuous Frequency Distribution (Shortcut Method) | |
| Step-Deviation Formula | $$\sigma = \frac{h}{N} \sqrt{N \sum f_i y_i^2 – (\sum f_i y_i)^2}$$ Where $$y_i = \frac{x_i – A}{h}$$($$A$$ is assumed mean,$$h$$ is class width) |