Statistics, Probability

In the DSSSB AE Civil syllabus under Arithmetical & Numerical Ability · 2 parts

Elementary Statistics · Permutation, Combination & Probability

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📑 Contents (12 sections)

Part 1 of 2

Elementary Statistics

Last reviewed 22 Sept 2026 · 6 min read

Frequency distributions

Raw data is grouped into classes with frequencies. For a class such as 20–30: the class mark (midpoint) is 25, the class size is 10, and the cumulative frequency is the running total of the frequencies up to that class. In an exclusive distribution (10–20, 20–30) the upper limit belongs to the next class; in an inclusive one (10–19, 20–29) it does not, and the limits must be corrected by half the gap before any formula is used.

The three averages

FormulaMean

Ungrouped: $\overset{x}{ˉ} = \frac{\sum x}{n}$
Grouped: $\overset{x}{ˉ} = \frac{\sum f x}{\sum f}$ with $x$ the class mark
Assumed mean: $\overset{x}{ˉ} = A + \frac{\sum f d}{\sum f}$ , where $d = x - A$
Step deviation: $\overset{x}{ˉ} = A + h \frac{\sum f u}{\sum f}$ , where $u = \frac{x - A}{h}$
Combined mean of two groups: $\frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}$

FormulaMedian and mode

Ungrouped (data arranged in order):

$n$ odd → the $(\frac{n + 1}{2})$ th value; $n$ even → the mean of the two middle values.
Mode is the most frequent value; a set may have none or several.

Grouped:

Median = l + \frac{\frac{N}{2} - c f}{f} \times h, Mode = l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

where $l$ is the lower limit of the median or modal class, $N = \sum f$ , $c f$ the cumulative frequency before it, $f$ its frequency, $h$ the class size, and $f_{0}, f_{1}, f_{2}$ the frequencies of the classes before, at and after the modal class.

Empirical relation: $Mode = 3 Median - 2 Mean$ .

Which one to use: the mean uses every value but is dragged by extremes; the median ignores extremes and suits skewed data such as incomes; the mode is the only average for a non-numeric category.

Part 2 of 2

Permutation, Combination & Probability

Last reviewed 22 Sept 2026 · 5 min read

Counting first

FormulaFundamental principle

If one job can be done in $m$ ways and, independently, a second in $n$ ways, then both together can be done in $m \times n$ ways (AND → multiply). If a job can be done by either of two exclusive methods, in $m$ or in $n$ ways, the total is $m + n$ (OR → add).

$n! = n (n - 1) \dots 2 \cdot 1$ , with $0! = 1$ .

Permutations — order matters

FormulaArrangements

^{n} P_{r} = \frac{n !}{( n - r )!}

All $n$ objects: $n!$
With repetition allowed, $r$ places from $n$ types: $n^{r}$
$n$ letters of which one letter repeats $p$ times and another $q$ times: $\frac{n !}{p ! q !}$
Circular arrangements of $n$ objects: $(n - 1)!$ , and $\frac{( n - 1 )!}{2}$ when clockwise and anticlockwise are the same (a garland or necklace)
Objects that must stay together: tie them into one block, arrange the blocks, then arrange within the block

Combinations — order does not matter

FormulaSelections

^{n} C_{r} = \frac{n !}{r ! ( n - r )!},^{n} C_{r} =^{n} C_{n - r},^{n} C_{r} = \frac{^{n} P _{r}}{r !}

$^{n} C_{0} =^{n} C_{n} = 1$ ; $^{n} C_{1} = n$
$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$
Total subsets of an $n$ -set: $2^{n}$
"At least one" is usually easiest as total $-$ none

A quick test: a committee is a combination (Ram and Shyam is the same committee as Shyam and Ram), while a president-and-secretary pair is a permutation.

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