NDA Mathematics · Formula sheet

Statistics formulas

29 formulas and 38 common traps for NDA Mathematics Statistics, grouped by subtopic.

Foundations + Measures of Central Tendency

Frequency and tabulation

Total frequency

N = \sum_{i=1}^{k} f_i

$k$ number of distinct values or class intervals
$f_i$ frequency of the $i$ -th value/class
$N$ total number of observations

Class marks and class width (grouped data)

Class mark and class width

x_{\text{mark}} = \dfrac{L + U}{2} \qquad h = U - L

Summation notation Σ

Definition + two identities

\sum_{i=1}^{n} x_i = x_1 + \cdots + x_n,\quad \sum_{i=1}^{n} c = nc,\quad \sum (a x_i + b) = a\sum x_i + nb

Arithmetic Mean (raw data)

Arithmetic Mean

\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i = \dfrac{x_1 + x_2 + \cdots + x_n}{n}

$\bar{x}$ the arithmetic mean
$x_i$ the $i$ -th observation
$n$ the total number of observations

Arithmetic Mean (frequency / grouped data)

Frequency-weighted Mean

\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}

$x_i$ value (or class mark for grouped data)
$f_i$ frequency of $x_i$
$\sum f_i$ total frequency = total observations

Linear Transformation of the Mean

Linear transformation rule

y_i = a\,x_i + b \quad\Longrightarrow\quad \bar{y} = a\,\bar{x} + b

$a$ scale factor (multiplied)
$b$ shift (added)

Replacement and Wrong-Value Correction of the Mean

Replacement rule (single observation, n unchanged)

M_{\text{new}} = M + \dfrac{y - x}{n}

$M$ original mean
$n$ number of observations (unchanged in pure replacement)
$x$ the value being removed (or wrongly recorded)
$y$ the value taking its place (or the correct one)

Special-Case Means — Consecutive Integers, Squares, AP, Binomial

Closed-form means for common sequences

\bar{x}_{a..b} = \dfrac{a+b}{2} \qquad \overline{k^2}\big|_{1}^{n} = \dfrac{(n+1)(2n+1)}{6} \qquad \bar{x}_{\text{AP}} = \dfrac{a_1 + a_n}{2}

$a, b$ first and last integer of an arithmetic run
$n$ number of terms (for the squares formula, the upper index)
$a_1, a_n$ first and last term of an AP

Combined mean of two groups

M_{12} = \dfrac{n_1 M_1 + n_2 M_2}{n_1 + n_2}

$n_1, n_2$ sizes of the two groups
$M_1, M_2$ means of the two groups
$M_{12}$ combined mean of the pooled dataset

Median — Middle Value

Median (raw and grouped)

\text{Raw: } M = \begin{cases} x_{(n+1)/2} & n \text{ odd} \\[4pt] \dfrac{x_{n/2} + x_{n/2+1}}{2} & n \text{ even} \end{cases} \qquad \text{Grouped: } M = L + \dfrac{\tfrac{n}{2} - F}{f}\,h

$L$ lower bound of the median class
$F$ cumulative frequency before the median class
$f$ frequency of the median class
$h$ class width

Mode — Most Frequent Value

Mode (grouped data)

M_0 = L + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h

$L$ lower bound of the modal class
$f_1$ frequency of the modal class
$f_0$ frequency of the class before
$f_2$ frequency of the class after
$h$ class width

Geometric Mean (GM)

Geometric Mean

\text{GM} = \sqrt[n]{x_1 \, x_2 \, \cdots \, x_n} = \left(\prod_{i=1}^{n} x_i\right)^{1/n}

$n$ number of observations (all positive)

Harmonic Mean (HM)

Harmonic Mean

\text{HM} = \dfrac{n}{\displaystyle\sum_{i=1}^{n} \dfrac{1}{x_i}} = \dfrac{n}{\dfrac{1}{x_1} + \dfrac{1}{x_2} + \cdots + \dfrac{1}{x_n}}

$n$ number of observations (all positive)

Sum of Deviations & Empirical Relations

Identities to memorise

\sum_{i=1}^{n}(x_i - \bar{x}) = 0, \quad \text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}, \quad \text{MD} \approx \tfrac{4}{5}\,\text{SD}

Common traps

Outliers move the mean — sometimes a lot

A single very large or very small value shifts the mean noticeably. If you suspect skew, ask the question whether mean or median is the right choice.

Divide by $\sum f_i$ , not by the number of classes

If marks are 20, 30, 30, 20 students across four classes, the divisor is 100 — not 4. This is the single most common arithmetic error on grouped-mean PYQs.

Shift moves the mean, but not the SD

Adding a constant

b

shifts

\bar{x}

b

but leaves the standard deviation unchanged. Multiplying by

a

scales both. Don't apply the mean rule to dispersion questions.

Divide by $n$ , not by 1

Students often subtract

x - y

directly from

M

. The mistake: only ONE of the

n

terms changed, so the shift in the average is the change in that one term divided by

n

— not the full change.

Discards: work with totals $nM$ , not the rule directly

When

k

observations are discarded,

n

itself changes. Don't try to force the single-replacement formula. Instead: original total

= nM

, new total

= (n-k)M_{\text{new}}

, the difference is the sum of the discarded values.

AP shortcut fails for GPs and other non-uniform spacings

The mean

(a_1 + a_n)/2

works only because in an AP every term sits at equal distance around the centre. For

1, 2, 4, 8, \ldots

(GP) the shortcut gives the wrong answer — you must sum properly or use the GP sum formula.

Binomial-weighted means use $\sum \binom{n}{k} = 2^n$

When asked the mean of

1, 2, \ldots, n+1

with frequencies

\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}

, the denominator is

2^n

(sum of one row of Pascal's triangle) — not the number of distinct values. Use

\sum k \binom{n}{k} = n \cdot 2^{n-1}

for the numerator.

Plain average of the two means is wrong unless $n_1 = n_2$

Students average

M_1

and

M_2

directly. That gives the correct combined mean ONLY when both groups are the same size. For unequal sizes the larger group pulls the combined mean toward its own mean — which is exactly what the weighted formula encodes.

Reverse-solve: combined + group means give the size ratio

M_{12},\ M_1,\ M_2

are given and you need

n_1 : n_2

, rearrange the formula to

\dfrac{n_1}{n_2} = \dfrac{M_2 - M_{12}}{M_{12} - M_1}

. PYQs use this shape with concrete totals (150 students, combined 60 kg, boys 70, girls 55) to test whether you recognise it as one equation in one unknown.

Always sort before reading off the middle

The median of an unsorted list is not the middle of the original order. PYQs sometimes hand you data in random order to catch this.

Mode can be undefined or multimodal — don't force one answer

If every value occurs exactly once, there is no mode. If two values tie for highest frequency, the data is bimodal and the answer is both values. PYQs use this to test understanding.

GM is only defined for positive numbers

Zero or negative observations break the geometric mean — the product vanishes or the root becomes imaginary. If a PYQ throws a zero or negative into the set, GM is not the right measure.

Order is always $\text{AM} \geq \text{GM} \geq \text{HM}$

For any set of positive numbers, this inequality is strict unless every observation is equal. If your computed HM exceeds GM or AM, you made an arithmetic error.

$\text{GM}^2 = \text{AM} \times \text{HM}$ for two numbers

For exactly two positive numbers, the geometric mean is the geometric mean of the arithmetic and harmonic means:

\text{GM}^2 = \text{AM} \cdot \text{HM}

. When a PYQ gives you two of

\{\text{AM}, \text{GM}, \text{HM}\}

for a pair (e.g.

5\,\text{HM} = 4\,\text{GM}

), use this identity to recover the third without solving for the original numbers — much faster than setting up two equations in

m, n

Sum of deviations is zero only about the mean

About any other reference point

c

, the sum equals

\sum x_i - nc = n(\bar{x} - c)

— non-zero unless

c = \bar{x}

. PYQs often plant a non-mean reference point to test exactly this.

Empirical relation is approximate, not exact

It works for moderately skewed unimodal data. For symmetric data (mean = median = mode) it is trivially true. For multimodal or heavily skewed data it can be misleading.

Dispersion — Standard Deviation, Variance, Mean Deviation

Learn this subtopic in the notes

Mean Deviation

Mean Deviation about A

\text{MD}(A) = \dfrac{1}{n}\sum_{i=1}^{n}|x_i - A|

$A$ reference point (typically mean or median)
$|x_i - A|$ absolute deviation of $x_i$ from $A$

Variance

Variance — two equivalent forms

\sigma^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2

$\sigma^2$ variance
$\bar{x}$ arithmetic mean
$\sum x_i^2$ sum of squares of observations

Standard Deviation

\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}

$\sigma$ standard deviation (always $\geq 0$ )

Linear Transformation of SD and Variance

Variance and SD under Y = aX + b

\text{Var}(Y) = a^2\,\text{Var}(X) \qquad \sigma_Y = |a|\,\sigma_X

$a$ scale factor
$b$ shift — irrelevant for dispersion

Special-Case Variance & SD — Natural Numbers and AP

Variance of the first n natural numbers and of an AP

\sigma^2_{1..n} = \dfrac{n^2 - 1}{12} \qquad \sigma^2_{\text{AP}} = d^2\,\dfrac{n^2 - 1}{12}

$n$ number of terms
$d$ common difference of the AP ( $=1$ for natural numbers)

Coefficient of Variation (CV)

Coefficient of Variation

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

$\sigma$ standard deviation
$\bar{x}$ arithmetic mean

Computational Identity & Minimum-SSE Property

Two load-bearing identities

\dfrac{\sum x_i^2}{n} = \bar{x}^2 + \sigma^2 \qquad \text{and} \qquad \arg\min_{a}\sum_{i}(x_i - a)^2 = \bar{x}

Common traps

Mean deviation about median is always $\leq$ about the mean

Among all reference points

A

, the median minimises

\sum|x_i - A|

. If a PYQ asks for the minimum mean deviation, the answer uses the median, not the mean.

Computational form saves time on $\Sigma x_i^2$ -style PYQs

When you are given

\sum x_i

and

\sum x_i^2

directly, use

\sigma^2 = \overline{x^2} - \bar{x}^2

— not the original definition. NDA papers favour this shape because it tests whether you remember the identity.

SD and mean deviation share units; variance does not

If the data is measured in cm, SD and MD are also in cm but variance is in cm². A PYQ asks "which has the same unit as the mean?" — answer is SD or MD, not variance.

Squaring $a$ for variance, taking absolute value for SD

Students often write

\sigma_Y^2 = a\,\sigma_X^2

(forgetting the square) or

\sigma_Y = a\,\sigma_X

(forgetting the modulus). If

a

is negative,

|a|

is the correct scale for SD — SD is non-negative by definition.

It is $(n^2 - 1)/12$ , not $n^2/12$ or $(n^2 + 1)/12$

The exact constant is

\dfrac{n^2 - 1}{12}

— memorise the

-1

. A frequent slip is writing

n^2/12

(forgetting the

-1

) or confusing it with the mean-of-squares formula.

For an AP, only the common difference matters — not the starting value

Adding a constant shifts every term but not the spread, so

\{3, 6, \ldots, 60\}

and the natural numbers

\{1, \ldots, 20\}

scaled by 3 have the SAME variance

\left(3^2 \times \dfrac{20^2-1}{12} = 299.25\right)

. Only

d

and

n

enter the formula.

CV is unitless — that's the entire point

Some answers give CV with a unit attached. Wrong. CV is a percentage. If the question compares two datasets with different units (e.g. height in cm vs weight in kg), only CV makes a fair comparison — not SD.

$\overline{x^2} \neq (\bar{x})^2$ — they differ by exactly $\sigma^2$

Mean of the squares is NOT the square of the mean. Their difference is the variance:

\overline{x^2} - \bar{x}^2 = \sigma^2 \geq 0

. PYQs plant this trap by asking for "mean of squares" or "

M^2 + \sigma^2

" and expecting you to recognise it as

\sum x_i^2/n

Scaling inside the deviation moves the minimiser too

For

S(a) = \sum (c\,x_i - a)^2

, expand and minimise: the minimum is at

a = c\,\bar{x}

, NOT

a = \bar{x}

. PYQs commonly use

c = 2

(e.g.

S = \sum (2x_i - a)^2

) and expect you to identify the minimiser as

2\bar{x}

— twice the mean, not the mean.

Regression and Correlation

Learn this subtopic in the notes

Correlation Coefficient and Its Properties

Correlation Coefficient and Invariance Rule

r = \dfrac{\text{Cov}(X,Y)}{\sigma_X\,\sigma_Y} \qquad r_{(aX+b,\,cY+d)} = \text{sign}(ac)\,r_{XY}

$\text{Cov}(X,Y)$ covariance of X and Y
$\sigma_X,\sigma_Y$ standard deviations of X and Y
$\text{sign}(ac)$ $+1$ if $a, c$ have same sign, $-1$ otherwise

Lines of Regression

Lines of Regression (point-slope form)

y - \bar{y} = b_{yx}(x - \bar{x}) \qquad x - \bar{x} = b_{xy}(y - \bar{y})

$b_{yx}$ slope of $y$ on $x$ line $= r\,\sigma_y/\sigma_x$
$b_{xy}$ slope of $x$ on $y$ line $= r\,\sigma_x/\sigma_y$
$(\bar{x},\bar{y})$ the only point on BOTH regression lines

Regression Coefficients and Their Link to r

Product Identity

b_{yx} \cdot b_{xy} = r^2, \qquad r = \pm\sqrt{b_{yx}\,b_{xy}}

Sign of $r$ same as the common sign of $b_{yx}$ and $b_{xy}$

Identifying Which Regression Line is Which

Sieve Inequality

\text{Correct pairing satisfies } b_{yx}\,b_{xy} \leq 1; \text{ wrong pairing gives } > 1.

Angle Between the Two Regression Lines

Angle between two lines (applied to regression)

\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1\,m_2}\right|

$m_1, m_2$ slopes of the two regression lines in the $(x,y)$ plane
$\theta$ acute angle between the lines

Common traps

$r$ is bounded by $-1$ and $+1$ — always

If a calculation gives

|r| > 1

, the arithmetic is wrong. Use this as a sanity check at the end of any correlation computation.

Shift does not change $r$ ; only scale-with-negative-sign flips it

Adding constants to either variable is invisible to

r

. Multiplying by a positive constant is also invisible. Only a negative multiplier flips the sign — and even then, the magnitude is preserved.

Both regression lines always pass through $(\bar{x}, \bar{y})$

If a PYQ gives you two regression lines and asks for the means, solve the two equations simultaneously — their intersection is exactly

(\bar{x}, \bar{y})

. No need to compute anything from raw data.

From raw bivariate data, compute $b_{yx}$ via the Pearson form

When given

n

raw paired points (e.g. four

(x_i, y_i)

values), use the computational formula

b_{yx} = \dfrac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}

with

\bar{x}, \bar{y}

read straight from the column sums. The regression line is then

y - \bar{y} = b_{yx}(x - \bar{x})

. Faster than the deviation-from-mean form because it works directly off the column totals.

$b_{yx} \cdot b_{xy} \leq 1$ is non-negotiable

If your computed product exceeds 1, you have assigned the wrong line to

y

x

. Swap the assignment and recompute — the inequality

b_{yx} b_{xy} = r^2 \leq 1

picks the correct pairing every time.

Both slopes share the sign of $r$

You cannot have

b_{yx} > 0

and

b_{xy} < 0

— if a problem seems to suggest this, the lines have been labelled wrong.

Try the inequality before doing anything else

Always compute both candidate pairings of

b_{yx} \cdot b_{xy}

. The pairing that satisfies the

\leq 1

condition is the correct one. Don't try to reason geometrically from the slopes — the inequality is mechanical and unambiguous.

Slope of the $x$ -on- $y$ line is NOT $b_{xy}$ in the $(x,y)$ plane

The slope of the

y

-on-

x

line in the

(x,y)

plane is

b_{yx}

. But the slope of the

x

-on-

y

line (written

x = a + b_{xy} y

) in the

(x,y)

plane is

1/b_{xy}

, NOT

b_{xy}

. When reading slopes off the line equation directly (solve for

y

, take the coefficient of

x

), you sidestep this trap.

Acute angle only — take absolute value

A negative tangent would correspond to the obtuse supplement. The formula's absolute value guarantees

\theta \leq 90^\circ

. PYQs always ask the acute angle.

Frequency Distributions and Graphical Representation

Learn this subtopic in the notes

Histograms, Frequency Polygons & Ogives

Frequency Density (for unequal class widths)

\text{Density} = \dfrac{\text{Frequency}}{\text{Class width}}

Class widthupper bound − lower bound of the class

Pie Charts

Sector Angle in a Pie Chart

\theta_i = \dfrac{f_i}{N} \times 360^\circ \qquad \sum_i \theta_i = 360^\circ

$f_i$ frequency / count of category $i$
$N$ total frequency $\sum f_i$

Reading Frequency Tables — Mode, Cumulative, Median

Median from a Grouped Frequency Distribution

M = L + \dfrac{\tfrac{n}{2} - F}{f}\,h

$L$ lower bound of the median class
$F$ cumulative frequency BEFORE the median class
$f$ frequency of the median class
$h$ class width

Common traps

Bar height $\neq$ frequency when class widths differ

If two classes have the same frequency but different widths, the wider class has the SHORTER bar — because density (height) divides frequency by width. Students draw bars of equal height for equal frequencies; correct histograms make AREAS equal, not heights.

All angles MUST sum to $360^\circ$

If your computed angles don't add to 360, you have an arithmetic error. PYQs that give angle relations ("

9p = 3q = 2r = 6s

") use

p + q + r + s = 360

as the closing equation — without that, the system is underdetermined.

Cumulative frequency is RUNNING total, not class total

The cumulative frequency at class

k

is the sum of frequencies from class 1 through class

k

— not the frequency of class

k

alone. Tripping on this turns every median-from-grouped-data question into nonsense.

Identify the median class FIRST, then plug into the formula

The median class is the class where the cumulative frequency first reaches or exceeds

n/2

. Don't pick the class with the highest frequency (that's the modal class) or the middle row of the table.

Statistics formulas

Foundations + Measures of Central Tendency

Frequency and tabulation

Class marks and class width (grouped data)

Summation notation Σ

Arithmetic Mean (raw data)

Arithmetic Mean (frequency / grouped data)

Linear Transformation of the Mean

Replacement and Wrong-Value Correction of the Mean

Special-Case Means — Consecutive Integers, Squares, AP, Binomial

Median — Middle Value

Mode — Most Frequent Value

Geometric Mean (GM)

Harmonic Mean (HM)

Sum of Deviations & Empirical Relations

Common traps

Dispersion — Standard Deviation, Variance, Mean Deviation

Mean Deviation

Variance

Linear Transformation of SD and Variance

Special-Case Variance & SD — Natural Numbers and AP

Coefficient of Variation (CV)

Computational Identity & Minimum-SSE Property

Common traps

Regression and Correlation

Correlation Coefficient and Its Properties

Lines of Regression

Regression Coefficients and Their Link to r

Identifying Which Regression Line is Which

Angle Between the Two Regression Lines

Common traps

Frequency Distributions and Graphical Representation

Histograms, Frequency Polygons & Ogives

Pie Charts

Reading Frequency Tables — Mode, Cumulative, Median

Common traps

More NDA Mathematics formula sheets