Statistics

1. Check this before you revise anything

Coefficient of Variation is not part of the current book — and not part of the syllabus either. Coaching material commonly teaches CV ($\text{CV}=\frac{\sigma }{\bar{x}}\times 100$) as this chapter's payoff, since it directly answers the "which of these two datasets is more consistent" question the chapter opens with.

But the word never appears anywhere in the book (confirmed by a full-text search — zero hits), and the syllabus line lists only "Range, Mean deviation, variance and standard deviation" — CV isn't named even as a formative-only topic. It simply isn't part of this chapter.

Quartile Deviation is named but explicitly excluded by the book's own words. The book lists four measures of dispersion — range, quartile deviation, mean deviation, standard deviation — then states directly: "we shall study all of these measures of dispersion except the quartile deviation." Don't expect a formula or a worked example for it here.

The old stub's exercise count was badly wrong. It listed four exercises (13.1–13.4, 34 questions total, with 13.3 and 13.4 entirely invented) and never mentioned the real Miscellaneous Exercise. The current book has exactly two numbered exercises plus a Miscellaneous Exercise — 28 questions in total.

2. What this chapter covers

3. Range and mean deviation

Range is the simplest measure of spread: $\text{Range}=\text{Maximum value}-\text{Minimum value}$. It uses only the two extreme values and ignores everything in between.

Mean deviation about a value $a$ is the average of the absolute deviations from $a$:

$\text{M.D.}(a)=\frac{\sum |x_i-a|}{n}$

Absolute values are essential here — deviations from the mean always sum to zero, so a plain average of signed deviations is useless as a measure of spread. In practice $a$ is either the mean $\bar{x}$ or the median $M$.

For grouped data (discrete or continuous, using class midpoints for continuous data), weight each deviation by its frequency:

$\text{M.D.}(\bar{x})=\frac{\sum f_i|x_i-\bar{x}|}{N},\text{M.D.}(M)=\frac{\sum f_i|x_i-M|}{N},N=\sum f_i$

For a continuous distribution's median, use the same interpolation formula as earlier classes: $M=l+\frac{\frac{N}{2}-C}{f}\times h$, where $l$ is the median class's lower boundary, $C$ the cumulative frequency before it, $f$ its own frequency, and $h$ its width.

Worked, mirroring the textbook's own Example 1. Find the mean deviation about the mean for $6,7,10,12,13,4,8,12$. Mean $=\frac{72}{8}=9$. Absolute deviations: $3,2,1,3,4,5,1,3$, summing to $22$. $\text{M.D.}(\bar{x})=22/8=2.75$.

4. Variance and standard deviation

Squaring the deviations instead of taking absolute values gives a measure that's easier to work with algebraically — this is variance, and its square root is the standard deviation $\sigma$ (taken in the original units, since variance is in squared units):

${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n},\sigma =\sqrt{{\sigma }^2}\text{(ungrouped data)}$

${\sigma }^2=\frac{\sum f_i(x_i-\bar{x}{)}^2}{N}\text{(grouped data, discrete or continuous via midpoints)}$

Shortcut (step-deviation) method, useful when the $x_i$ values or class midpoints are large. With assumed mean $A$, class width $h$, and $y_i=\frac{x_i-A}{h}$:

${\sigma }^2=\frac{h^2}{N^2}\left[N\sum f_i{y}_i^2-{\left(\sum f_i{y}_i\right)}^2\right]$

A useful scaling property, worth knowing even though the book states it only through a worked example, not a boxed formula. If every observation is multiplied by a constant $a$, the new mean is $a\bar{x}$ and the new variance is $a^2{\sigma }^2$ — squaring the deviations means the scale factor gets squared too.

Worked, mirroring the textbook's own Example 8. Find the variance of $6,8,10,\dots ,24$ (step $2$, ten terms). Using the step-deviation method with assumed mean $A=14$, $h=2$: the deviations $y_i$ run $-4,-3,\dots ,4,5$, giving $\sum f_i{y}_i=5$ and $\sum f_i{y}_i^2=165$. Mean $=14+2(5/10)=15$, and variance $=\frac{1}{10}\sum (x_i-\bar{x}{)}^2=33$, so $\sigma =\sqrt{33}\approx 5.74$.

Summary

• Range $=$ Maximum $-$ Minimum; the simplest but crudest measure of spread.
• Mean deviation $\text{M.D.}(a)=\frac{\sum |x_i-a|}{n}$ (or $\frac{\sum f_i|x_i-a|}{N}$ for grouped data), computed about the mean or the median.
• Variance ${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n}$ and standard deviation $\sigma =\sqrt{{\sigma }^2}$; for grouped data, weight by frequency and use class midpoints for continuous distributions.
• Shortcut method: ${\sigma }^2=\frac{h^2}{N^2}\left[N\sum f_i{y}_i^2-(\sum f_i{y}_i{)}^2\right]$ with $y_i=(x_i-A)/h$.
• Scaling every observation by a constant $a$ multiplies the mean by $a$ and the variance by $a^2$.
• Quartile Deviation is named as a measure of dispersion but explicitly excluded by the book's own words; Coefficient of Variation is absent from both the book and the syllabus entirely.