Key Points

Measures of dispersion like range, mean deviation, variance, and standard deviation describe the spread or variability of data, which measures of central tendency (mean, median, mode) alone cannot.

The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset. Range = Maximum Value - Minimum Value.

Mean deviation measures the average of the absolute deviations from a central value. About the mean $\bar{x}$, it is M.D.$(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$. About the median M, it is M.D.$(M) = \frac{1}{n} \sum_{i=1}^{n} |x_i - M|$.

For a discrete or continuous frequency distribution, the mean deviation about the mean $\bar{x}$ is M.D.$(\bar{x}) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - \bar{x}|$, where $N = \sum f_i$. A similar formula applies for deviation about the median.

The median for a continuous frequency distribution is found using the formula: $\text{Median} = l + \frac{\frac{N}{2} - C}{f} \times h$. Here, $l$ is the lower limit of the median class, $N$ is total frequency, $C$ is cumulative frequency of the preceding class, $f$ is frequency of the median class, and $h$ is the class width.

Variance, denoted by $\sigma^2$, is the mean of the squared deviations from the mean. The formula is $\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$.

Standard deviation, denoted by $\sigma$, is the positive square root of the variance. It is calculated as $\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$.

For a discrete or continuous frequency distribution, variance is calculated as $\sigma^2 = \frac{1}{N} \sum_{i=1}^{n} f_i (x_i - \bar{x})^2$, where $N = \sum f_i$.

Standard deviation for grouped data is the square root of the variance: $\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{n} f_i (x_i - \bar{x})^2}$.

A computationally simpler formula for variance is $\sigma^2 = \frac{1}{N} \sum_{i=1}^{n} f_i x_i^2 - (\frac{\sum_{i=1}^{n} f_i x_i}{N})^2$. This can also be written as $\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - \bar{x}^2$.

Using step-deviations $y_i = \frac{x_i - A}{h}$, where A is the assumed mean and h is the class width, variance is $\sigma_x^2 = h^2 \sigma_y^2$. The full formula is $\sigma^2 = \frac{h^2}{N^2} [N \sum f_i y_i^2 - (\sum f_i y_i)^2]$.

If a constant 'a' is added to each observation in a dataset, the mean changes by 'a', but the variance and standard deviation remain unchanged. This is because the spread of the data does not change.

If each observation is multiplied by a constant 'k', the new mean is k times the original mean. The new variance becomes $k^2$ times the original variance, and the new standard deviation becomes $|k|$ times the original standard deviation.

A series with a smaller standard deviation (or variance) is considered more consistent or less variable than a series with a larger standard deviation. A larger value indicates greater spread in the data.