Statistics and Probability
Overview
Statistics and Probability form a significant portion of the KAR TET Paper II Mathematics section. These topics test your ability to analyse data, calculate central tendencies, and understand the likelihood of events—skills essential for teaching upper-primary students how to interpret real-world information.
For the exam, you must be comfortable calculating mean, median, and mode from raw data, grouped frequency distributions, and simple datasets. Probability questions typically involve basic experiments like coin tosses, dice throws, and card draws. The pedagogy angle expects you to connect these concepts to everyday situations students encounter.
Mastery here requires both computational accuracy and conceptual clarity. Many questions are straightforward if you know the formulas, but careless errors in ordering data or misreading frequency tables are common traps.
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Key Concepts
- **Mean (Arithmetic Average)**: The sum of all observations divided by the number of observations. It uses every data point and is sensitive to extreme values (outliers).
- **Median**: The middle value when data is arranged in ascending or descending order. For an even number of observations, it is the average of the two middle values. It is not affected by extreme values.
- **Mode**: The value that occurs most frequently in a dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal).
- **Grouped Data**: When data is presented in class intervals, use class marks (mid-points) for calculations. Mean uses the assumed mean or direct method; median uses cumulative frequency; mode uses the modal class formula.
- **Probability**: A measure of how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%).
- **Sample Space**: The set of all possible outcomes of an experiment. For a die, sample space = {1, 2, 3, 4, 5, 6}.
- **Favourable Outcomes**: Outcomes that satisfy the condition of the event in question.
- **Complementary Events**: If P(E) is the probability of event E, then P(not E) = 1 − P(E).
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Formulas / Key Facts
**Measures of Central Tendency (Ungrouped Data)**
| Measure | Formula | |---------|---------| | Mean | x̄ = Σxᵢ / n | | Median (n odd) | Value at position (n+1)/2 | | Median (n even) | Average of values at positions n/2 and (n/2)+1 | | Mode | Most frequently occurring value |
**Measures of Central Tendency (Grouped Data)**
| Measure | Formula | |---------|---------| | Mean (Direct) | x̄ = Σfᵢxᵢ / Σfᵢ (where xᵢ = class mark, fᵢ = frequency) | | Median | Median = L + [(n/2 − cf) / f] × h | | Mode | Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h |
Where:
- L = lower boundary of median/modal class
- n = total frequency
- cf = cumulative frequency before median class
- f = frequency of median class
- f₁ = frequency of modal class
- f₀ = frequency of class before modal class
- f₂ = frequency of class after modal class
- h = class width
**Probability**
| Concept | Formula | |---------|---------| | Probability of event E | P(E) = Number of favourable outcomes / Total number of outcomes | | Range | 0 ≤ P(E) ≤ 1 | | Certain event | P(E) = 1 | | Impossible event | P(E) = 0 | | Complementary | P(E) + P(not E) = 1 |
**Standard Sample Spaces**
- Coin: 2 outcomes
- Die: 6 outcomes
- Two coins: 4 outcomes
- Two dice: 36 outcomes
- Deck of cards: 52 cards (4 suits × 13 ranks)
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Worked Examples
**Example 1: Mean and Median (Ungrouped)**
Find the mean and median of: 12, 15, 10, 18, 15, 20, 10
*Solution:*
- Arrange in order: 10, 10, 12, 15, 15, 18, 20
- Mean = (10 + 10 + 12 + 15 + 15 + 18 + 20) / 7 = 100 / 7 = 14.29 (approx)
- n = 7 (odd), so median = value at position (7+1)/2 = 4th position = **15**
- Mode = 10 and 15 both appear twice → **Bimodal (10 and 15)**
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**Example 2: Median from Grouped Data**
| Class Interval | Frequency | |----------------|-----------| | 0–10 | 5 | | 10–20 | 8 | | 20–30 | 12 | | 30–40 | 10 | | 40–50 | 5 |
Find the median.
*Solution:*
- Total frequency n = 40, so n/2 = 20
- Cumulative frequencies: 5, 13, 25, 35, 40
- Median class: The class where cumulative frequency first exceeds 20 → **20–30**
- L = 20, cf = 13, f = 12, h = 10
- Median = 20 + [(20 − 13) / 12] × 10 = 20 + (7/12) × 10 = 20 + 5.83 = **25.83**
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**Example 3: Basic Probability**
A bag contains 4 red, 3 blue, and 5 green balls. One ball is drawn at random. Find the probability of getting (a) a red ball, (b) not a green ball.
*Solution:*
- Total balls = 4 + 3 + 5 = 12
- (a) P(red) = 4/12 = **1/3**
- (b) P(not green) = 1 − P(green) = 1 − 5/12 = **7/12**
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Common Mistakes
- **Forgetting to arrange data before finding median** → Always sort data in ascending order first. The middle value of unsorted data is meaningless.
- **Confusing class limits with class boundaries** → For median/mode formulas in grouped data, use actual class boundaries (e.g., 10–20 means L = 10, not 9.5, unless specified as exclusive).
- **Using total observations instead of total frequency in grouped data** → In grouped data, n = Σfᵢ (sum of frequencies), not the number of class intervals.
- **Adding probabilities incorrectly for "or" events** → For mutually exclusive events, P(A or B) = P(A) + P(B). But if events can overlap, subtract P(A and B).
- **Calculating mode when no value repeats** → If all values occur once, there is **no mode**. Do not write the smallest or largest value as mode.
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Quick Reference
- Mean uses all values; affected by outliers. Median is the middle value; resistant to outliers.
- For ungrouped median: sort first, then locate position (n+1)/2 for odd n; average two middle values for even n.
- Modal class = class with highest frequency; use the mode formula with adjacent class frequencies.
- Probability = Favourable outcomes / Total outcomes; always between 0 and 1.
- P(not E) = 1 − P(E) is the fastest way to solve "not" type probability questions.
- Standard dice have 6 faces; standard cards have 52 cards (26 red, 26 black; 4 aces; 12 face cards).