Statistics and Probability forms a crucial scoring section in Assam TET Paper II Mathematics. This topic tests your ability to analyse data, calculate central tendencies, and understand basic chance events. Questions typically involve computing mean, median, and mode from given datasets, interpreting grouped frequency distributions, and solving simple probability problems.
For upper primary teaching (Classes VI–VIII), educators must help students transition from basic data handling to more formal statistical reasoning. The syllabus expects you to understand both the computational procedures and their real-world applications—such as interpreting rainfall data in Assam, analysing student performance, or understanding fair games and chance events.
Expect 2–4 questions from this topic, often combining calculation with conceptual understanding. Mastery here requires speed in arithmetic and clarity about when to use which measure of central tendency.
Key Concepts
**Mean (Arithmetic Average)**: The sum of all observations divided by the number of observations. Most affected by extreme values (outliers). Best used when data is evenly spread without extreme values.
**Median**: The middle value when data is arranged in ascending or descending order. For even number of observations, it is the average of two middle values. Robust against outliers—preferred for skewed data like income distributions.
**Mode**: The value that occurs most frequently. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal). Best for categorical data like favourite colour or most common shoe size.
**Range**: Difference between maximum and minimum values. Gives a rough measure of spread but ignores how data is distributed between extremes.
**Grouped Data**: When data is presented in class intervals (e.g., 10–20, 20–30), we use class marks (midpoints) to calculate mean and identify modal/median classes.
**Probability**: A measure of likelihood of an event, always between 0 and 1. P(Event) = Number of favourable outcomes ÷ Total number of possible outcomes.
**Complementary Events**: P(Event not happening) = 1 − P(Event happening). If probability of rain is 0.3, probability of no rain is 0.7.
**Equally Likely Outcomes**: When each outcome has the same chance of occurring—essential assumption for basic probability calculations (fair coin, fair dice).
Formulas / Key Facts
**Mean (Ungrouped Data)** Mean = Sum of all observations ÷ Number of observations Mean = Σx ÷ n
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**Mean (Grouped Data using Direct Method)** Mean = Σ(f × x) ÷ Σf where f = frequency, x = class mark (midpoint)
**Class Mark** Class Mark = (Lower limit + Upper limit) ÷ 2
**Median (Ungrouped Data)**
If n is odd: Median = ((n+1)/2)th observation
If n is even: Median = Average of (n/2)th and (n/2 + 1)th observations
**Median (Grouped Data)** Median = L + [(n/2 − CF) ÷ f] × h where L = lower limit of median class, CF = cumulative frequency before median class, f = frequency of median class, h = class width
**Mode (Ungrouped)**: Value with highest frequency
**Mode (Grouped Data)** Mode = L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h where L = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency of class before modal class, f₂ = frequency of class after modal class
**Probability** P(E) = Number of favourable outcomes ÷ Total outcomes P(E) + P(not E) = 1 0 ≤ P(E) ≤ 1
**Example 1: Finding Mean** The marks of 5 students are: 45, 52, 60, 48, 55. Find the mean.
Solution: Sum = 45 + 52 + 60 + 48 + 55 = 260 Number of students = 5 Mean = 260 ÷ 5 = **52 marks**
**Example 2: Finding Median** Find the median of: 12, 18, 10, 15, 20, 17, 14, 19
Solution: Step 1: Arrange in ascending order: 10, 12, 14, 15, 17, 18, 19, 20 Step 2: n = 8 (even), so median = average of 4th and 5th values Step 3: 4th value = 15, 5th value = 17 Median = (15 + 17) ÷ 2 = **16**
**Example 4: Basic Probability** A bag contains 4 red balls, 3 blue balls, and 5 green balls. If one ball is drawn at random, find the probability of getting a blue ball.
**Using wrong median position formula** → For ungrouped data with n observations, the position is (n+1)/2, not n/2. For n=7, median is 4th value, not 3.5th.
**Confusing class limits with class marks** → When calculating mean for grouped data, use class mark (midpoint), not upper or lower limit. Class 20–30 has class mark 25, not 20 or 30.
**Forgetting to arrange data before finding median** → Data must be in ascending or descending order first. Random arrangement gives wrong median.
**Adding probabilities incorrectly for overlapping events** → P(A or B) = P(A) + P(B) only if A and B are mutually exclusive. For overlapping events, subtract P(A and B).
**Assuming probability can exceed 1** → If your calculated probability is greater than 1 or negative, recheck your calculation—it must be between 0 and 1.
**Ignoring "at least" and "at most" wording** → "At least 2" means 2 or more. "At most 2" means 2 or fewer. These require adding multiple probabilities or using complement method.
Quick Reference
Mean = Σx ÷ n (affected by outliers)
Median = middle value when arranged in order (robust to outliers)
Mode = most frequent value (can be more than one)
P(E) = Favourable outcomes ÷ Total outcomes
P(not E) = 1 − P(E)
For fair dice: P(any single number) = 1/6; for fair coin: P(Head) = P(Tail) = 1/2