Statistics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Statistics — Quick Facts

Key Definitions:

• Statistics: The science of collecting, organising, analysing, and interpreting data
• Population: The entire set of items under consideration
• Sample: A subset of the population used for analysis
• Variable: A characteristic that can take different values
• Frequency: The number of times a value occurs

Measures of Central Tendency:

Measure	Formula	Best Used When
Mean ($\bar{x}$)	$\bar{x} = \frac{\sum x_i}{n}$	Symmetric data without outliers
Median	Middle value when data is ordered	Skewed data, outliers present
Mode	Most frequently occurring value	Categorical data, peaks in data

⚡ Exam Tips for NDA:

• Remember: Mean is affected by extreme values, median is not
• For grouped data, Mode $= L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$ where $L$ = lower limit of modal class
• In NDA exam, direct formula application questions are common
• Always arrange data in ascending order before finding median

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Statistics — Study Guide

Measures of Dispersion:

Dispersion tells us how spread out the data is.

1. Range: $R = X_{max} - X_{min}$

   • Simplest measure but sensitive to outliers

2. Variance ($\sigma^2$):

   • Population variance: $\sigma^2 = \frac{\sum(x_i - \bar{x})^2}{N}$
   • Sample variance: $s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$

3. Standard Deviation ($\sigma$ or $s$):

   • $\sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2}{N}}$ (population)
   • $s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$ (sample)

4. Mean Deviation:

   • $M.D = \frac{\sum|x_i - \bar{x}|}{n}$ (from mean)
   • $M.D = \frac{\sum|x_i - M|}{n}$ (from median)

5. Quartile Deviation (Semi-Interquartile Range):

   • $Q.D = \frac{Q_3 - Q_1}{2}$

Coefficient of Variation: $$CV = \frac{\sigma}{\bar{x}} \times 100%$$

• Used to compare variability of two datasets with different units or scales

Skewness:

• Positive Skew: Tail extends to the right (mean > median > mode)
• Negative Skew: Tail extends to the left (mean < median < mode)
• Pearson’s Coefficient of Skewness: $$Sk_p = \frac{Mean - Mode}{SD} = \frac{3(Mean - Median)}{SD}$$

Correlation:

• Measures the strength and direction of relationship between two variables
• Pearson’s Correlation Coefficient ($r$): $$r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \times \sum(y_i - \bar{y})^2}}$$
• $-1 \leq r \leq +1$
• $r > 0$: Positive correlation
• $r < 0$: Negative correlation
• $r = 0$: No correlation

Common Student Mistakes:

• Confusing sample variance denominator ($n-1$) with population variance ($n$)
• Mixing up correlation with causation
• Forgetting to order data before finding median

---

🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough preparation.

Statistics — Comprehensive Notes

Detailed Formulae for Grouped Data:

Mean for Grouped Data: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Where $x_i$ is the class mark (midpoint) and $f_i$ is the frequency.

Step Deviation Method (for simplification): $$\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \times h$$ Where $A$ = assumed mean, $d_i = \frac{x_i - A}{h}$, $h$ = class width.

Median for Grouped Data: $$Median = L + \frac{(n/2) - cf}{f} \times h$$ Where:

• $L$ = lower limit of median class
• $n$ = total frequency
• $cf$ = cumulative frequency before median class
• $f$ = frequency of median class
• $h$ = class width

Mode for Grouped Data: $$Mode = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$ Where:

• $L$ = lower limit of modal class
• $f_1$ = frequency of modal class
• $f_0$ = frequency before modal class
• $f_2$ = frequency after modal class
• $h$ = class width

Quartiles for Grouped Data: $$Q_r = L + \frac{(rn/4) - cf}{f} \times h$$ Where $r$ = 1, 2, or 3 for $Q_1$, $Q_2$, $Q_3$

Variance and Standard Deviation for Grouped Data: $$\sigma^2 = \frac{\sum f_i(x_i - \bar{x})^2}{N} = \frac{\sum f_i x_i^2}{N} - \bar{x}^2$$ (Using the computational formula: variance = mean of squares − square of mean)

Probability:

Classical Definition: $$P(A) = \frac{n(A)}{n(S)}$$ Where $n(A)$ = favourable outcomes, $n(S)$ = total outcomes.

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

Multiplication Theorem (Independent Events): $$P(A \cap B) = P(A) \times P(B)$$

Conditional Probability: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Bayes’ Theorem: $$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Random Variables and Expectation:

• Discrete Random Variable: Takes finite or countable values
• Expectation (Mean): $E(X) = \sum x_i P(x_i)$
• Variance: $Var(X) = E(X^2) - [E(X)]^2$

Binomial Distribution: $$P(X = r) = \binom{n}{r} p^r q^{n-r}$$ Where $p$ = probability of success, $q = 1-p$, $n$ = number of trials.

• Mean $= np$
• Variance $= npq$

Normal Distribution: The standard normal variable $z = \frac{x - \mu}{\sigma}$

Properties:

• Bell-shaped, symmetric about mean
• Mean = Median = Mode = $\mu$
• Total area under curve = 1
• 68.27% within 1 SD, 95.45% within 2 SD, 99.73% within 3 SD

Regression Analysis:

Lines of Regression:

• $y$ on $x$: $y - \bar{y} = r\frac{\sigma_y}{\sigma_x}(x - \bar{x})$
• $x$ on $y$: $x - \bar{x} = r\frac{\sigma_x}{\sigma_y}(y - \bar{y})$

Regression Coefficients: $$b_{yx} = r\frac{\sigma_y}{\sigma_x}, \quad b_{xy} = r\frac{\sigma_x}{\sigma_y}$$ $$r = \sqrt{b_{yx} \cdot b_{xy}}$$

⚡ NDA High-Yield Patterns:

• Questions on mean deviation from mean/median are very common
• Always check whether the question asks for sample or population variance
• Coefficient of variation is frequently asked in comparative problems
• Normal distribution and z-score calculations appear in Paper II
• Remember the relationship: $3 Median = Mode + 2 Mean$ (for moderately skewed distributions)

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.