Statistics & Probability — NDA Mathematics

Two CBSE chapters share this 6% slice of the paper, and they reward opposite habits. Statistics is long division done right — mean, median, mode and dispersion from a frequency table, where the formula is rarely in doubt but a mis-added column or a forgotten division wrecks the answer. Probability is classification done right — addition vs multiplication, mutually exclusive vs independent, conditional vs unconditional — where the arithmetic is short but picking the wrong rule gives a confidently wrong answer. Neither sub-area needs a calculator; both need discipline.

1. What NDA actually asks

Statistics & Probability carries weightPct 6 of NDA Mathematics — the lightest topic on the syllabus, tied with Vector Algebra — roughly 7–8 of the 120 questions, worth about 18–20 of the 300 marks, at +2.5 for a correct answer and −0.8333 for a wrong one (the 1/3rd penalty). The syllabus is exactly CBSE Class 11 Statistics plus Class 11–12 Probability; nothing beyond that is asked. Questions cluster into two halves:

Statistics (roughly half the marks here):

1. Mean of raw or grouped data — direct method, assumed-mean (shortcut) method, or step-deviation method for a continuous frequency distribution.
2. Median — of raw data (odd/even count) or from a grouped frequency table via the cumulative-frequency formula.
3. Mode — most frequent raw value, or the grouped-data mode formula built from the modal class.
4. Dispersion — range, mean deviation (about mean or median), variance and standard deviation (direct and shortcut methods), and the coefficient of variation for comparing two distributions.

Probability (the other half): 5. Classical probability — counting favourable outcomes over total equally-likely outcomes (cards, dice, coins, balls from a bag). 6. Addition theorem — $P(A\cup B)$ for general events and the simpler mutually-exclusive case. 7. Conditional probability and the multiplication theorem — including the independent-events special case. 8. Bayes' theorem — "given the effect, find the probable cause," always via the total-probability-theorem denominator. 9. Discrete random variables and the binomial distribution — probability distribution tables, expectation, and $\text{mean}=np$, $\text{variance}=npq$.

No calculator is allowed, so every dataset in this chapter is built from small, clean numbers — the skill being tested is running the right procedure without an arithmetic slip, not handling ugly numbers.

2. Measures of central tendency

Raw (individual) data. For $n$ observations $x_1,\dots ,x_n$:

$\bar{x}=\frac{\sum x_i}{n}$

Median of raw data (arrange in ascending order first): if $n$ is odd, the median is the $\left(\frac{n+1}{2}\right)$th value; if $n$ is even, it is the average of the $\left(\frac{n}{2}\right)$th and $\left(\frac{n}{2}+1\right)$th values. Mode is simply the most frequently occurring value — read straight off the data, no formula needed.

Ungrouped (discrete) frequency distribution. With distinct values $x_i$ occurring $f_i$ times:

$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

Median uses the cumulative frequency column: locate the smallest $x_i$ whose cumulative frequency is $\ge \frac{N}{2}$ (where $N=\sum f_i$) — that $x_i$ is the median. Mode is the $x_i$ with the highest $f_i$.

Continuous (grouped) frequency distribution. Let $x_i$ denote class midpoints, $h$ the (uniform) class width, and $A$ any assumed mean.

• Direct method: $\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}$
• Assumed-mean (shortcut) method: with $d_i=x_i-A$, $\bar{x}=A+\frac{\sum f_i{d}_i}{\sum f_i}$
• Step-deviation method: with $u_i=\frac{x_i-A}{h}$, $\bar{x}=A+\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)\times h$

All three give identical answers — step-deviation is simply the fastest by hand, since $u_i$ stays a small integer instead of a multiple of $h$. Median (grouped):

$\text{Median}=l+\left(\frac{\frac{N}{2}-cf}{f}\right)\times h$

where $l$ = lower boundary of the median class (the class whose cumulative frequency first reaches or exceeds $\frac{N}{2}$), $cf$ = cumulative frequency of the class immediately before the median class, $f$ = frequency of the median class, $h$ = class width. Mode (grouped):

$\text{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$

where $l$ = lower boundary of the modal class (highest-frequency class), $f_1$ = its frequency, $f_0$ = frequency of the class before it, $f_2$ = frequency of the class after it. A useful cross-check when all three measures are computed from the same data: the empirical relationship $\text{Mode}=3\text{Median}-2\text{Mean}$ holds approximately for moderately skewed distributions — NDA occasionally asks for one of the three given the other two via this identity.

3. Measures of dispersion

Range $=\text{Maximum}-\text{Minimum}$ — the crudest measure, but occasionally asked directly.

Mean deviation. About the mean, for raw data:

$\text{MD}(\bar{x})=\frac{\sum |x_i-\bar{x}|}{n}$

and about the median, replace $\bar{x}$ with the median $M$. For a frequency distribution, weight each term by $f_i$: $\text{MD}(\bar{x})=\frac{\sum f_i|x_i-\bar{x}|}{\sum f_i}$. The absolute value is essential — deviations from the mean always sum to zero unsigned, so skipping the modulus silently collapses every mean-deviation question to zero.

Variance and standard deviation. For raw data, the defining and shortcut forms:

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

NDA (following CBSE Class 11) divides by $n$ throughout — this is the population variance, not the sample variance with an $(n-1)$ divisor used in some statistics courses. For a grouped frequency distribution, the equivalent shortcut and step-deviation forms:

${\sigma }^2=\frac{\sum f_i{x}_i^2}{\sum f_i}-{\bar{x}}^2=h^2\left[\frac{\sum f_i{u}_i^2}{\sum f_i}-{\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)}^2\right],u_i=\frac{x_i-A}{h}$

The step-deviation variance needs the $h^2$ multiplier outside the bracket — this is the single most-forgotten factor in the whole chapter, since the mean's step-deviation formula only multiplies by $h$ once (not squared).

Coefficient of variation. To compare the relative spread of two datasets with different means or units:

$\text{CV}=\frac{\sigma }{\bar{x}}\times 100\%$

The dataset with the smaller CV is the more consistent one — a rule NDA tests directly by giving two mean/SD pairs and asking which is "more consistent" or "less variable."

4. Probability — classical definition and the two core theorems

Classical (equally-likely) definition. For a sample space $S$ of equally likely outcomes and an event $E$:

$P(E)=\frac{n(E)}{n(S)},0\le P(E)\le 1,P(S)=1,P(E^{\prime })=1-P(E)$

where $E^{\prime }$ is the complement of $E$ — "at least one" questions are almost always faster via $1-P(\text{none})$ than by direct counting.

Addition theorem (general form). For any two events $A,B$:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Mutually exclusive events cannot occur together, so $A\cap B=\varnothing$ and $P(A\cap B)=0$, which collapses the formula to the simpler $P(A\cup B)=P(A)+P(B)$. The single most common NDA trap in this chapter is applying this simplified addition when the events are not actually mutually exclusive — e.g. "a king or a heart" from a deck: King and Heart overlap at the king of hearts, so the $P(A\cap B)$ term must be subtracted; skipping it overcounts that one card.

Conditional probability. The probability of $A$ given that $B$ has already occurred:

$P(A\mid B)=\frac{P(A\cap B)}{P(B)},P(B)>0$

Multiplication theorem (rearranging the definition above):

$P(A\cap B)=P(A)\cdot P(B\mid A)=P(B)\cdot P(A\mid B)$

Independent events are the special case where knowing one occurred doesn't change the other's probability: $P(A\mid B)=P(A)$ and $P(B\mid A)=P(B)$, which simplifies the multiplication theorem to the direct product

$P(A\cap B)=P(A)\cdot P(B)$

Independent is not the same as mutually exclusive — in fact, two mutually exclusive events with nonzero probabilities can never be independent, since $P(A\cap B)=0$ but $P(A)\cdot P(B)\ne 0$ whenever both events are possible. NDA sets this up as a direct true/false trap regularly.

5. Bayes' theorem, random variables & the binomial distribution

Total probability theorem. If $A_1,A_2,\dots ,A_n$ partition the sample space (mutually exclusive, exhaustive) and $B$ is any event:

$P(B)={\sum }_{i=1}^{n}P(A_i)P(B\mid A_i)$

Bayes' theorem reverses a conditional probability — given that the effect $B$ has occurred, it finds the probability that a particular cause $A_i$ produced it:

$P(A_i\mid B)=\frac{P(A_i)P(B\mid A_i)}{{\sum }_{j=1}^{n}P(A_j)P(B\mid A_j)}$

The denominator is exactly the total-probability-theorem expression for $P(B)$ — compute it once, then reuse it for the probability of each cause. NDA-level Bayes' questions almost always involve exactly two or three causes (machines, boxes, factories) producing a defective/success outcome at different rates.

Random variable & probability distribution (discrete). A random variable $X$ assigns a real number to every outcome; its probability distribution is a table of values $x_i$ with probabilities $p_i=P(X=x_i)$, satisfying $p_i\ge 0$ and $\sum p_i=1$ — this normalisation condition is frequently used to solve for an unknown constant in the table. Its mean (expectation) and variance:

$E(X)=\sum x_i{p}_i,\text{Var}(X)=\sum x_i^2{p}_i-(E(X){)}^2=E(X^2)-[E(X){]}^2$

Binomial distribution. For $n$ independent Bernoulli trials, each with success probability $p$ (failure probability $q=1-p$), the number of successes $X$ follows:

$P(X=r)={}^n{C}_r{p}^r{q}^{n-r},r=0,1,2,\dots ,n$

with

$\text{Mean}=np,\text{Variance}=npq,\text{SD}=\sqrt{npq}$

NDA keeps this introductory — expect a fixed $n$ and $p$ (typically a coin, die, or a stated success rate), a request for $P(X=r)$ at one specific $r$, or a direct mean/variance question, never the full distribution table for large $n$.

7. Common traps

• Forgetting the modulus in mean deviation. $\sum (x_i-\bar{x})$ is always exactly zero by definition of the mean — mean deviation requires absolute values, or the answer silently comes out zero every single time.
• Population vs sample variance. NDA/CBSE variance divides by $n$, not $n-1$. Applying the sample-variance (Bessel-corrected) formula from a different course gives a systematically inflated wrong answer.
• Step-deviation variance missing the $h^2$. The mean's step-deviation formula multiplies by $h$ once; the variance's step-deviation formula multiplies the whole bracket by $h^2$. Copying the mean's single-$h$ pattern into a variance question is a very common slip.
• Misidentifying the median/modal class. The median class is the one where cumulative frequency first reaches $\frac{N}{2}$ — using the cumulative frequency of that class itself (instead of the class before it) for $cf$ gives a badly wrong answer.
• Addition theorem misapplied to non-mutually-exclusive events. If the two events can happen together (e.g. "king or heart," "even or greater than 4"), the intersection must be subtracted. Applying $P(A)+P(B)$ directly over-counts the overlap exactly once.
• Confusing independent with mutually exclusive. Independent events can occur together ($P(A\cap B)=P(A)P(B)\ne 0$); mutually exclusive events cannot ($P(A\cap B)=0$). Two nonzero-probability events can be one or the other, never both.
• Bayes' theorem without the total-probability denominator. Writing $P(A_i)P(B\mid A_i)$ alone as the final answer (skipping the division by $\sum P(A_j)P(B\mid A_j)$) gives an unnormalised number that isn't even guaranteed to be $\le 1$.
• Treating unequal-probability outcomes as equally likely. With two dice, the 11 possible sums (2 through 12) are not equally likely — always count over the 36 elementary outcomes, never over the sums directly.

8. Revision protocol

Split practice cleanly along the chapter's two halves, since they fail in different ways. For statistics, build one clean frequency table and compute mean (all three methods), median, mode, variance and CV from it in a single sitting — the goal is catching column-arithmetic errors (a wrong $cf$, a dropped $f_i{u}_i$ term) before the exam, since the formulas themselves are rarely the problem. For probability, drill the classification step in isolation: given fifteen one-line scenarios, write only "addition (mutually exclusive)," "addition (general)," "conditional," "multiplication (independent)," or "Bayes'" next to each — without solving anything — until that fifteen-second read of the wording is instant. At 6% weight and roughly 7–8 questions, this chapter won't decide the paper, but with no calculator and a −0.8333 penalty for a rushed guess, it is one of the most reliably gettable few marks on offer if the two halves are drilled separately rather than blurred together.