Statistics

The mean is the average, the median the middle value, and the mode the most frequent. The empirical relation is Mode = 3 Median − 2 Mean.

In Class 10 you summarised data with a single central value — mean, median or mode. But two classes can share the same average and still be wildly different: one steady, the other scattered. Measures of dispersion answer the question the average ignores — how spread out are the values?

The crudest measure is the range, the gap between the largest and smallest observation:

Range is quick but fragile — it depends only on the two extreme values and ignores everything in between, so a single outlier can blow it up. A more honest measure looks at how far every value sits from the centre. That is the idea behind mean deviation: average the distances of all observations from a central value $a$ (the mean or the median). Distances are taken as absolute values so that values below and above the centre do not cancel.

Mean deviation for ungrouped (raw) data about a central value $a$:

Taking $a=\bar{x}$ gives mean deviation about the mean; taking $a=M$ (the median) gives mean deviation about the median.

For grouped data with values (or class marks) $x_i$ and frequencies $f_i$, where $N=\sum f_i$:

For a continuous frequency distribution, $x_i$ is the class mark (midpoint) of each class. The summary below collects the dispersion measures of this chapter:

A useful fact worth remembering: among all choices of the central value $a$, the median minimises the sum of absolute deviations, so M.D. about the median is never larger than M.D. about the mean.

Deeper Insight — why absolute values, and why median is special: The whole point of a deviation is to record distance, and distance has no sign — a value $4$ below the mean is just as "far away" as a value $4$ above it. If we summed the raw signed deviations $\sum(x_i-\bar{x})$ we would always get exactly zero, because the mean is the precise balance point where positive and negative deviations cancel. That zero tells us nothing about spread, so we strip the signs with $|\cdot|$ before averaging. The absolute value also explains a subtler property: the function $\sum|x_i-a|$ is smallest when $a$ is the median, not the mean. Intuitively, the median is the point with as many observations on its left as on its right, so nudging $a$ in either direction increases as many distances as it decreases. This is exactly why mean deviation about the median is the natural "minimum total distance" measure, and it is the reason the NCERT singles it out.

Mean deviation uses absolute values, which are awkward to handle algebraically. Statisticians prefer to square the deviations instead — squaring also removes signs, but produces a smooth, differentiable measure that behaves beautifully under further analysis. This leads to the two most important measures of spread.

Variance $\sigma^2$ is the mean of the squared deviations from the mean. For raw (ungrouped) data of $n$ values:

Standard deviation $\sigma$ is the positive square root of the variance. The square root brings the measure back to the same units as the original data, which is why it is the quantity actually quoted:

For a frequency distribution (discrete or continuous, with class marks $x_i$, frequencies $f_i$, and $N=\sum f_i$):

Expanding the square gives an equivalent computational form that avoids first finding $\bar{x}$ and subtracting it from every value:

When the numbers are large, the step-deviation (short-cut) method rescales the data first. Choose an assumed mean $A$ and class width $h$, and set $y_i=\dfrac{x_i-A}{h}$. Then:

The table below contrasts the data types and the form you would use:

Deeper Insight — why we square instead of taking absolute values: Squaring deviations does the same sign-removing job as the absolute value, but it earns three decisive advantages that make variance the cornerstone of all statistics. First, the function $\sum(x_i-a)^2$ is smooth everywhere — it has a derivative at every point — whereas $\sum|x_i-a|$ has sharp corners; this smoothness lets us minimise it cleanly and prove that the minimising value is exactly the mean $\bar{x}$. Second, squaring deliberately gives more weight to large deviations: an observation twice as far from the mean contributes four times as much, so the measure is sensitive to outliers in a controlled, predictable way. Third, variances add for independent quantities, a property absolute deviations simply do not have, and that additivity is what makes the standard deviation the natural scale for the normal distribution, error analysis and the entire edifice of inferential statistics you will meet later. Taking the square root at the end is not cosmetic — it restores the original units (rupees, kilograms, marks), so a standard deviation is something you can actually interpret on the same axis as the data itself.