Statistics — NDA Mathematics

Formulas (14)

• Frequency and tabulation · Total frequency
  $$N = \sum_{i=1}^{k} f_i$$
• 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
  $$\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$$
• Arithmetic Mean (frequency / grouped data) · Frequency-weighted Mean
  $$\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$$
• Linear Transformation of the Mean · Linear transformation rule
  $$y_i = a\,x_i + b \quad\Longrightarrow\quad \bar{y} = a\,\bar{x} + b$$
• Replacement and Wrong-Value Correction of the Mean · Replacement rule (single observation, n unchanged)
  $$M_{\text{new}} = M + \dfrac{y - x}{n}$$
• Special-Case Means — Consecutive Integers, Squares, AP, Binomial · Closed-form means for common sequences
  $$\bar{x}_{a..b} = \dfrac{a+b}{2} \qquad \overline{k^2}\big|_{1}^{n} = \dfrac{(n+1)(2n+1)}{6} \qquad \bar{x}_{\text{AP}} = \dfrac{a_1 + a_n}{2}$$
• Combined Mean of Two Groups · Combined mean of two groups
  $$M_{12} = \dfrac{n_1 M_1 + n_2 M_2}{n_1 + n_2}$$
• Median — Middle Value · Median (raw and grouped)
  $$\text{Raw: } M = \begin{cases} x_{(n+1)/2} & n \text{ odd} \\[4pt] \dfrac{x_{n/2} + x_{n/2+1}}{2} & n \text{ even} \end{cases} \qquad \text{Grouped: } M = L + \dfrac{\tfrac{n}{2} - F}{f}\,h$$
• Mode — Most Frequent Value · Mode (grouped data)
  $$M_0 = L + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h$$
• Geometric Mean (GM) · Geometric Mean
  $$\text{GM} = \sqrt[n]{x_1 \, x_2 \, \cdots \, x_n} = \left(\prod_{i=1}^{n} x_i\right)^{1/n}$$
• Harmonic Mean (HM) · Harmonic Mean
  $$\text{HM} = \dfrac{n}{\displaystyle\sum_{i=1}^{n} \dfrac{1}{x_i}} = \dfrac{n}{\dfrac{1}{x_1} + \dfrac{1}{x_2} + \cdots + \dfrac{1}{x_n}}$$
• Sum of Deviations & Empirical Relations · Identities to memorise
  $$\sum_{i=1}^{n}(x_i - \bar{x}) = 0, \quad \text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}, \quad \text{MD} \approx \tfrac{4}{5}\,\text{SD}$$

Watch out for (16)

• Outliers move the mean — sometimes a lot
  → Arithmetic Mean (raw data)
• Divide by $\sum f_i$, not by the number of classes
  → Arithmetic Mean (frequency / grouped data)
• Shift moves the mean, but not the SD
  → Linear Transformation of the Mean
• Divide by $n$, not by 1
  → Replacement and Wrong-Value Correction of the Mean
• Discards: work with totals $nM$, not the rule directly
  → Replacement and Wrong-Value Correction of the Mean
• AP shortcut fails for GPs and other non-uniform spacings
  → Special-Case Means — Consecutive Integers, Squares, AP, Binomial
• Binomial-weighted means use $\sum \binom{n}{k} = 2^n$
  → Special-Case Means — Consecutive Integers, Squares, AP, Binomial
• Plain average of the two means is wrong unless $n_1 = n_2$
  → Combined Mean of Two Groups
• Reverse-solve: combined + group means give the size ratio
  → Combined Mean of Two Groups
• Always sort before reading off the middle
  → Median — Middle Value
• Mode can be undefined or multimodal — don't force one answer
  → Mode — Most Frequent Value
• GM is only defined for positive numbers
  → Geometric Mean (GM)
• Order is always $\text{AM} \geq \text{GM} \geq \text{HM}$
  → Harmonic Mean (HM)
• $\text{GM}^2 = \text{AM} \times \text{HM}$ for two numbers
  → Harmonic Mean (HM)
• Sum of deviations is zero only about the mean
  → Sum of Deviations & Empirical Relations
• Empirical relation is approximate, not exact
  → Sum of Deviations & Empirical Relations

Formulas (7)

• Mean Deviation · Mean Deviation about A
  $$\text{MD}(A) = \dfrac{1}{n}\sum_{i=1}^{n}|x_i - A|$$
• Variance · Variance — two equivalent forms
  $$\sigma^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2$$
• Standard Deviation · Standard Deviation
  $$\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}$$
• Linear Transformation of SD and Variance · Variance and SD under Y = aX + b
  $$\text{Var}(Y) = a^2\,\text{Var}(X) \qquad \sigma_Y = |a|\,\sigma_X$$
• Special-Case Variance & SD — Natural Numbers and AP · Variance of the first n natural numbers and of an AP
  $$\sigma^2_{1..n} = \dfrac{n^2 - 1}{12} \qquad \sigma^2_{\text{AP}} = d^2\,\dfrac{n^2 - 1}{12}$$
• Coefficient of Variation (CV) · Coefficient of Variation
  $$\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100 \%$$
• Computational Identity & Minimum-SSE Property · Two load-bearing identities
  $$\dfrac{\sum x_i^2}{n} = \bar{x}^2 + \sigma^2 \qquad \text{and} \qquad \arg\min_{a}\sum_{i}(x_i - a)^2 = \bar{x}$$

Watch out for (9)

• Mean deviation about median is always $\leq$ about the mean
  → Mean Deviation
• Computational form saves time on $\Sigma x_i^2$-style PYQs
  → Variance
• SD and mean deviation share units; variance does not
  → Standard Deviation
• Squaring $a$ for variance, taking absolute value for SD
  → Linear Transformation of SD and Variance
• It is $(n^2 - 1)/12$, not $n^2/12$ or $(n^2 + 1)/12$
  → Special-Case Variance & SD — Natural Numbers and AP
• For an AP, only the common difference matters — not the starting value
  → Special-Case Variance & SD — Natural Numbers and AP
• CV is unitless — that's the entire point
  → Coefficient of Variation (CV)
• $\overline{x^2} \neq (\bar{x})^2$ — they differ by exactly $\sigma^2$
  → Computational Identity & Minimum-SSE Property
• Scaling inside the deviation moves the minimiser too
  → Computational Identity & Minimum-SSE Property

Formulas (5)

• Correlation Coefficient and Its Properties · Correlation Coefficient and Invariance Rule
  $$r = \dfrac{\text{Cov}(X,Y)}{\sigma_X\,\sigma_Y} \qquad r_{(aX+b,\,cY+d)} = \text{sign}(ac)\,r_{XY}$$
• Lines of Regression · Lines of Regression (point-slope form)
  $$y - \bar{y} = b_{yx}(x - \bar{x}) \qquad x - \bar{x} = b_{xy}(y - \bar{y})$$
• Regression Coefficients and Their Link to r · Product Identity
  $$b_{yx} \cdot b_{xy} = r^2, \qquad r = \pm\sqrt{b_{yx}\,b_{xy}}$$
• 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)
  $$\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1\,m_2}\right|$$

Watch out for (9)

• $r$ is bounded by $-1$ and $+1$ — always
  → Correlation Coefficient and Its Properties
• Shift does not change $r$; only scale-with-negative-sign flips it
  → Correlation Coefficient and Its Properties
• Both regression lines always pass through $(\bar{x}, \bar{y})$
  → Lines of Regression
• From raw bivariate data, compute $b_{yx}$ via the Pearson form
  → Lines of Regression
• $b_{yx} \cdot b_{xy} \leq 1$ is non-negotiable
  → Regression Coefficients and Their Link to r
• Both slopes share the sign of $r$
  → Regression Coefficients and Their Link to r
• Try the inequality before doing anything else
  → Identifying Which Regression Line is Which
• Slope of the $x$-on-$y$ line is NOT $b_{xy}$ in the $(x,y)$ plane
  → Angle Between the Two Regression Lines
• Acute angle only — take absolute value
  → Angle Between the Two Regression Lines

Formulas (3)

• Histograms, Frequency Polygons & Ogives · Frequency Density (for unequal class widths)
  $$\text{Density} = \dfrac{\text{Frequency}}{\text{Class width}}$$
• Pie Charts · Sector Angle in a Pie Chart
  $$\theta_i = \dfrac{f_i}{N} \times 360^\circ \qquad \sum_i \theta_i = 360^\circ$$
• Reading Frequency Tables — Mode, Cumulative, Median · Median from a Grouped Frequency Distribution
  $$M = L + \dfrac{\tfrac{n}{2} - F}{f}\,h$$

Watch out for (4)

• Bar height $\neq$ frequency when class widths differ
  → Histograms, Frequency Polygons & Ogives
• All angles MUST sum to $360^\circ$
  → Pie Charts
• Cumulative frequency is RUNNING total, not class total
  → Reading Frequency Tables — Mode, Cumulative, Median
• Identify the median class FIRST, then plug into the formula
  → Reading Frequency Tables — Mode, Cumulative, Median