Statistics and Probability

Overview

Statistics and Probability forms a crucial quantitative reasoning component in JTET Paper II Mathematics. This topic tests your ability to summarize data using central tendency measures and to calculate basic chances of events occurring. Questions typically involve numerical datasets requiring calculation of mean, median, or mode, and simple probability scenarios involving coins, dice, cards, or everyday situations.

For upper-primary teaching, understanding these concepts is essential because they connect mathematics to real-world data interpretation—skills students need for informed decision-making. Expect 2–4 questions from this topic, often presented as word problems or data tables. Mastery requires both computational accuracy and conceptual clarity about when to use each measure.

The JTET syllabus specifically mentions "mean, median, mode and elementary probability," so focus on ungrouped data calculations and single-event probability problems rather than advanced statistical techniques.

Key Concepts

• **Mean (Arithmetic Average)**: The sum of all observations divided by the number of observations. Best used when data has no extreme outliers.

• **Median**: The middle value when data is arranged in ascending or descending order. Preferred when data contains outliers or is skewed.

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

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

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

• **Sample Space**: The set of all possible outcomes in a probability experiment.

• **Favourable Outcomes**: Outcomes that satisfy the condition of the event we're calculating probability for.

• **Complementary Events**: If P(E) is 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

**Median (Ungrouped Data)**

• Arrange data in ascending order
• 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** Mode = Value with highest frequency (count which number appears most often)

**Range** Range = Maximum value − Minimum value

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

**Probability Rules**

• 0 ≤ P(E) ≤ 1 for any event E
• P(certain event) = 1
• P(impossible event) = 0
• P(E) + P(not E) = 1

**Standard Sample Spaces**

• Coin toss: 2 outcomes (Head, Tail)
• Dice roll: 6 outcomes (1, 2, 3, 4, 5, 6)
• Two dice: 36 outcomes
• Standard deck of cards: 52 cards (4 suits × 13 cards each)

Worked Examples

**Example 1: Finding Mean, Median, and Mode**

*The marks obtained by 7 students are: 45, 55, 60, 55, 70, 55, 80. Find the mean, median, and mode.*

**Solution:**

Mean = (45 + 55 + 60 + 55 + 70 + 55 + 80) ÷ 7 Mean = 420 ÷ 7 = **60**

For Median, arrange in order: 45, 55, 55, 55, 60, 70, 80 n = 7 (odd), so median position = (7+1)/2 = 4th value Median = **55**

Mode = value appearing most frequently = **55** (appears 3 times)

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**Example 2: Median with Even Number of Observations**

*Find the median of: 12, 18, 15, 22, 19, 25*

**Solution:**

Arrange in order: 12, 15, 18, 19, 22, 25 n = 6 (even) Median = average of 3rd and 4th values Median = (18 + 19) ÷ 2 = 37 ÷ 2 = **18.5**

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**Example 3: Probability with Dice**

*A fair dice is thrown once. What is the probability of getting (a) an even number, (b) a number greater than 4?*

**Solution:**

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

(a) Even numbers = {2, 4, 6} = 3 favourable outcomes P(even) = 3/6 = **1/2**

(b) Numbers greater than 4 = {5, 6} = 2 favourable outcomes P(greater than 4) = 2/6 = **1/3**

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**Example 4: Probability with Cards**

*A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of getting a king.*

**Solution:**

Total cards = 52 Number of kings = 4 (one in each suit) P(king) = 4/52 = **1/13**

Common Mistakes

• **Forgetting to arrange data before finding median** → Always sort the data in ascending order first; the middle position in unsorted data gives wrong answer.

• **Confusing mean and median for skewed data** → Mean is affected by extreme values; if asked "which measure is better for salary data with one very high earner," answer is median, not mean.

• **Saying "no mode" when all values appear once** → This is actually correct! A dataset where every value appears equally often has no mode. Don't force an answer.

• **Writing probability greater than 1** → If your answer exceeds 1, recheck. Probability is always between 0 and 1. Common error: writing 3/2 when favourable outcomes exceed total (impossible—recount).

• **Miscounting sample space for two dice** → Two dice have 36 outcomes (6 × 6), not 12. Each die is independent, so multiply possibilities.

• **Forgetting to reduce fractions** → Probability 4/52 should be written as 1/13 in final answer. Always simplify.

Quick Reference

• Mean = Total sum ÷ Number of items

• Median needs sorted data; for n items, middle is at position (n+1)/2

• Mode = most frequent value (can be none or multiple)

• Probability = Favourable outcomes ÷ Total outcomes

• Probability always lies between 0 and 1 inclusive

• Coin: 2 outcomes | Dice: 6 outcomes | Cards: 52 total (4 kings, 4 queens, 13 of each suit)