Statistics - Variance and Standard Deviation

Measures of Dispersion: While the mean (central tendency) tells us where the center of the data is, dispersion tells us how spread out the data points are around that center. Visually, if you plot data on a number line, a high dispersion means the points are widely scattered, while low dispersion means they are clustered tightly around the mean.

Variance ($\sigma^2$): This is the arithmetic mean of the squares of deviations of all observations from their mean. By squaring the differences, we ensure that negative and positive deviations don't cancel each other out. Conceptually, imagine a square drawn for every data point where the side length is its distance from the mean; the variance represents the average area of these squares.

Standard Deviation ($\sigma$): Defined as the positive square root of the variance. It is the most widely used measure of dispersion because it is expressed in the same units as the original data. On a normal distribution curve (bell curve), the standard deviation determines the 'width' of the bell; a larger $\sigma$ creates a flatter, wider curve.

Discrete Frequency Distribution: When data is given with frequencies ($f_i$), the variance is calculated by weighting each squared deviation by its corresponding frequency. Visually, this is like looking at a bar chart where the distance of each bar from the central mean line is weighted by the height of that bar.

Continuous Frequency Distribution: For grouped data (classes), we use the mid-point (class mark) as the representative value ($x_i$) for each interval. The distribution is visualized using a histogram, and the variance calculation assumes all values within a class are concentrated at the mid-point.

Step-Deviation Method: This is a computational shortcut used to simplify calculations by shifting the origin to an assumed mean ($A$) and scaling the data by the class width ($h$). Visually, this is equivalent to a linear transformation (shifting and shrinking) of the x-axis to make the numbers smaller and easier to manage.

Coefficient of Variation (C.V.): A relative measure of dispersion calculated as the ratio of standard deviation to the mean, expressed as a percentage. It is used to compare the variability or consistency of two different data sets, even if they have different units or means. In a visual comparison of two frequency polygons, the one with the higher C.V. is considered more variable or less consistent.