Statistics and Probability

Overview

Statistics and Probability form a crucial scoring section in Bihar TET Paper II Mathematics. This topic tests your ability to analyse data, calculate central tendencies, and understand the basics of chance and likelihood. Questions typically involve calculating mean, median, and mode from given data sets, and solving elementary probability problems involving coins, dice, and cards.

For Bihar TET, expect 2-4 questions directly from this topic. The good news: once you master the formulas and practice standard problem types, these become quick marks. The key challenge is avoiding calculation errors and correctly identifying which measure of central tendency to use. Probability questions at this level remain straightforward—usually single-event scenarios without complex combinations.

Understanding these concepts also matters for your teaching career. You'll need to help upper-primary students interpret data from their environment and develop logical thinking about uncertain outcomes—skills central to the NCF vision of mathematics education.

Key Concepts

• **Mean (Arithmetic Average)**: The sum of all observations divided by the number of observations. Most affected by extreme values (outliers). Use when data is evenly distributed without extreme values.

• **Median**: The middle value when data is arranged in ascending or descending order. Not affected by outliers. Preferred when data has extreme values or is skewed.

• **Mode**: The value that occurs most frequently. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal). Best for categorical data or finding the most common item.

• **Probability**: A numerical measure of the likelihood of an event, always between 0 and 1 (or 0% to 100%). Probability of 0 means impossible; probability of 1 means certain.

• **Sample Space**: The set of all possible outcomes of an experiment. For a coin: {Head, Tail}. For a die: {1, 2, 3, 4, 5, 6}.

• **Favourable Outcomes**: Outcomes that satisfy the condition of the event we're interested in.

• **Complementary Events**: If P(E) is the probability of event E, then P(not E) = 1 - P(E).

Formulas / Key Facts

**Mean (Ungrouped Data)** Mean = Sum of all observations ÷ Number of observations Mean = Σx ÷ n

**Mean (Grouped Data - Direct Method)** Mean = Σ(f × x) ÷ Σf where f = frequency, x = class mark (mid-point)

**Class Mark** Class Mark = (Lower limit + Upper limit) ÷ 2

**Median (Ungrouped Data)**

• If n is odd: Median = value at position (n+1)/2
• If n is even: Median = average of values at positions n/2 and (n/2)+1

**Mode (Ungrouped Data)** Mode = Most frequently occurring value

**Probability of an Event** P(E) = Number of favourable outcomes ÷ Total number of outcomes

**Probability Range** 0 ≤ P(E) ≤ 1

**Complementary Probability** P(not E) = 1 - P(E)

**Standard Sample Spaces**

• Coin: 2 outcomes
• Die: 6 outcomes
• Deck of cards: 52 cards (26 red, 26 black; 13 each of hearts, diamonds, clubs, spades)

Worked Examples

**Example 1: Finding Mean** *The marks obtained by 7 students are: 45, 52, 60, 55, 48, 60, 50. Find the mean.*

Step 1: Add all observations Sum = 45 + 52 + 60 + 55 + 48 + 60 + 50 = 370

Step 2: Count observations n = 7

Step 3: Apply formula Mean = 370 ÷ 7 = 52.86

**Answer: Mean = 52.86 marks**

---

**Example 2: Finding Median** *Find the median of: 12, 18, 10, 15, 20, 17, 14*

Step 1: Arrange in ascending order 10, 12, 14, 15, 17, 18, 20

Step 2: Count terms n = 7 (odd)

Step 3: Find middle position Position = (7+1)/2 = 4th term

Step 4: Identify the value 4th term = 15

**Answer: Median = 15**

---

**Example 3: Probability with a Die** *A fair die is thrown once. What is the probability of getting a number greater than 4?*

Step 1: Identify sample space S = {1, 2, 3, 4, 5, 6} Total outcomes = 6

Step 2: Identify favourable outcomes Numbers greater than 4: {5, 6} Favourable outcomes = 2

Step 3: Apply formula P(number > 4) = 2/6 = 1/3

**Answer: Probability = 1/3**

---

**Example 4: Mode from Frequency Data** *Shoe sizes of 20 students: Size 5 (3 students), Size 6 (7 students), Size 7 (6 students), Size 8 (4 students). Find the mode.*

Step 1: Identify the highest frequency Size 6 has the maximum frequency (7 students)

**Answer: Mode = Size 6**

Common Mistakes

• **Forgetting to arrange data before finding median** → Always sort data in ascending order first. Without sorting, you'll pick the wrong "middle" value.

• **Confusing when to use mean vs median** → Mean is distorted by extreme values. If data has outliers (like incomes: 5000, 6000, 7000, 50000), use median instead.

• **Writing probability greater than 1** → If your answer exceeds 1, you've made an error. Probability is always between 0 and 1. Check if you've inverted the fraction.

• **Miscounting total outcomes in probability** → Be systematic. For a die, always 6 outcomes. For a coin tossed twice, 4 outcomes (HH, HT, TH, TT), not 3.

• **Taking class limits instead of class marks for grouped data** → For grouped frequency data, always calculate class mark (midpoint) first before computing mean.

Quick Reference

• Mean = Sum of observations ÷ Number of observations

• Median position (odd n) = (n+1)/2 th term

• Mode = Value with highest frequency

• P(E) = Favourable outcomes ÷ Total outcomes

• Probability is always between 0 and 1

• P(not E) = 1 - P(E)