Statistics and Probability

Overview

Statistics and Probability form an essential part of the CG TET Paper II Mathematics section. This topic tests your ability to analyse data, calculate central tendencies, and understand the likelihood of events—skills that every upper primary teacher must possess to make mathematics meaningful for students.

In CG TET, expect 2–4 questions from this topic. Questions typically involve calculating mean, median, or mode from a given data set, or finding the probability of simple events. The calculations are straightforward but require careful attention to data arrangement and formula application. Mastery here also helps in the pedagogy section, as you must know how to teach data handling concepts through real-life examples relevant to Chhattisgarh's context—rainfall data, agricultural yields, or population figures.

Key Concepts

• **Mean (Arithmetic Average)**: The sum of all observations divided by the total number of observations. It is affected by extreme values (outliers).

• **Median**: The middle value when data is arranged in ascending or descending order. For an odd number of observations, it is the central value; for even, it is the average of the two central values.

• **Mode**: The value that occurs most frequently in a data set. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal).

• **Range**: The difference between the highest and lowest values. It measures the spread of data.

• **Probability**: A measure of how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%).

• **Certain Event**: An event with probability = 1 (will definitely happen).

• **Impossible Event**: An event with probability = 0 (cannot happen).

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

Formulas / Key Facts

**Statistics Formulas:**

| Measure | Formula | |---------|---------| | Mean (ungrouped) | Mean = Sum of all observations ÷ Number of observations | | Mean (grouped) | Mean = Σ(f × x) ÷ Σf, where f = frequency, x = class mark | | Median (ungrouped, n odd) | Median = value at position (n + 1)/2 | | Median (ungrouped, n even) | Median = average of values at positions n/2 and (n/2 + 1) | | Mode | The most frequently occurring value | | Range | Range = Highest value − Lowest value |

**Probability Formulas:**

| Concept | Formula | |---------|---------| | Probability of event E | P(E) = Number of favourable outcomes ÷ Total number of outcomes | | Complementary probability | P(not E) = 1 − P(E) | | Range of probability | 0 ≤ P(E) ≤ 1 | | Sum of all probabilities | P(E) + P(not E) = 1 |

**Key Facts:**

• Class mark (mid-value) = (Lower limit + Upper limit) ÷ 2
• For a fair die: Total outcomes = 6
• For a fair coin: Total outcomes = 2
• For a deck of cards: Total cards = 52 (13 each of 4 suits)

Worked Examples

**Example 1: Finding Mean**

*The marks obtained by 5 students are: 45, 60, 55, 70, 50. Find the mean.*

Step 1: Add all observations = 45 + 60 + 55 + 70 + 50 = 280

Step 2: Count observations = 5

Step 3: Mean = 280 ÷ 5 = **56**

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**Example 2: Finding Median**

*Find the median of: 12, 7, 19, 5, 15, 8, 10*

Step 1: Arrange in ascending order: 5, 7, 8, 10, 12, 15, 19

Step 2: Number of observations (n) = 7 (odd)

Step 3: Median position = (7 + 1) ÷ 2 = 4th position

Step 4: Median = **10**

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**Example 3: Finding Median (Even observations)**

*Find the median of: 4, 8, 12, 16, 20, 24*

Step 1: Data is already arranged. n = 6 (even)

Step 2: Middle positions = 6/2 = 3rd and 4th positions

Step 3: Values at these positions = 12 and 16

Step 4: Median = (12 + 16) ÷ 2 = **14**

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**Example 4: Finding Mode**

*Find the mode of: 2, 3, 5, 3, 7, 3, 8, 5*

Step 1: Count frequency of each value

• 2 appears 1 time
• 3 appears 3 times
• 5 appears 2 times
• 7 appears 1 time
• 8 appears 1 time

Step 2: Highest frequency = 3 (for value 3)

Step 3: Mode = **3**

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**Example 5: Probability with a Die**

*A fair die is thrown once. Find the probability of getting a number greater than 4.*

Step 1: Total outcomes = {1, 2, 3, 4, 5, 6} = 6

Step 2: Favourable outcomes (greater than 4) = {5, 6} = 2

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

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**Example 6: Probability with Cards**

*A card is drawn from a well-shuffled deck. Find the probability of getting a red card.*

Step 1: Total cards = 52

Step 2: Red cards (hearts + diamonds) = 13 + 13 = 26

Step 3: P(red card) = 26/52 = **1/2**

Common Mistakes

| Wrong Thinking | Correct Approach | |----------------|------------------| | Calculating median without arranging data first | Always arrange data in ascending or descending order before finding median | | Using (n/2)th term directly for median when n is even | For even n, take the average of (n/2)th and (n/2 + 1)th terms | | Confusing mode with mean | Mode is the most frequent value, not the average | | Writing probability greater than 1 or less than 0 | Probability always lies between 0 and 1; if your answer is outside this range, recheck | | Forgetting to count all outcomes in probability | List all possible outcomes systematically before counting favourable ones | | Using class limits instead of class marks in grouped data mean | Always use class mark = (upper limit + lower limit) ÷ 2 |

Quick Reference

• **Mean** = Sum ÷ Count (affected by extreme values)

• **Median** = Middle value after arranging (not affected by extremes)

• **Mode** = Most frequent value (can be more than one)

• **Probability** = Favourable outcomes ÷ Total outcomes

• **P(E) + P(not E) = 1** (always)

• **0 ≤ P(E) ≤ 1** (probability can never be negative or exceed 1)