Statistics — TN TET Paper II Study Notes

Overview

Statistics is a foundational topic in TN TET Paper II Mathematics, testing your ability to organise, summarise and interpret numerical data. Questions typically involve calculating measures of central tendency (mean, median, mode) and range from grouped or ungrouped data, along with reading bar graphs, pie charts and frequency tables.

This topic carries moderate weightage but offers easy marks if formulas and shortcuts are memorised. Exam questions often present data sets of 5–15 values or frequency distributions, asking you to compute one or more statistical measures. Mastery here also supports the pedagogy section—teachers must help students connect statistics to real-world contexts like census data, weather records and class test scores.

Focus on three skills: applying the correct formula, handling grouped vs ungrouped data differently, and interpreting graphical data accurately.

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Key Concepts

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

**Median**: The middle value when data is arranged in ascending or descending order. For even number of observations, it is the average of the two middle values. Unaffected by outliers.

**Mode**: The value that occurs most frequently. A data set can be unimodal (one mode), bimodal (two modes), multimodal or have no mode.

**Range**: The difference between the highest and lowest values. It measures spread but is sensitive to outliers.

**Ungrouped Data**: Raw data presented as individual values (e.g., 5, 7, 8, 10, 12).

**Grouped Data**: Data organised into class intervals with corresponding frequencies (e.g., 0–10: 5 students, 10–20: 8 students).

**Class Mark (Mid-value)**: For a class interval, class mark = (Lower limit + Upper limit) / 2. Used to compute mean of grouped data.

**Cumulative Frequency**: Running total of frequencies, essential for finding median in grouped data.

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Formulas / Key Facts

### 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 | | Range | Highest value − Lowest value |

### Grouped Data

| Measure | Formula | |---------|---------| | Mean (Direct) | x̄ = Σ(fᵢ × xᵢ) / Σfᵢ, where xᵢ = class mark | | Mean (Assumed Mean) | x̄ = A + Σ(fᵢ × dᵢ) / Σfᵢ, where dᵢ = xᵢ − A | | Median | Median = L + [(n/2 − cf) / f] × h | | Mode | Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h |

**Median formula terms**: L = lower limit of median class, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

**Mode formula terms**: 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, h = class width.

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Worked Examples

### Example 1: Mean and Median of Ungrouped Data

**Data**: 12, 15, 10, 18, 15, 20, 15

**Mean**:

Sum = 12 + 15 + 10 + 18 + 15 + 20 + 15 = 105
n = 7
Mean = 105 / 7 = **15**

**Median**:

Arrange in order: 10, 12, 15, 15, 15, 18, 20
n = 7 (odd), so median position = (7 + 1)/2 = 4th value
Median = **15**

**Mode**: 15 appears 3 times (most frequent). Mode = **15**

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### Example 2: Mean of Grouped Data (Direct Method)

| Class Interval | Frequency (f) | Class Mark (x) | f × x | |----------------|---------------|----------------|-------| | 0–10 | 4 | 5 | 20 | | 10–20 | 6 | 15 | 90 | | 20–30 | 10 | 25 | 250 | | 30–40 | 5 | 35 | 175 |

Σf = 25, Σ(f × x) = 535
Mean = 535 / 25 = **21.4**

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### Example 3: Median of Grouped Data

| Class | Frequency | Cumulative Frequency | |-------|-----------|---------------------| | 0–10 | 5 | 5 | | 10–20 | 8 | 13 | | 20–30 | 12 | 25 | | 30–40 | 7 | 32 |

n = 32, so n/2 = 16
Median class: where cf first exceeds 16 → class **20–30**
L = 20, cf = 13, f = 12, h = 10
Median = 20 + [(16 − 13) / 12] × 10 = 20 + 2.5 = **22.5**

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Common Mistakes

1. **Forgetting to arrange data before finding median** → Always sort data in ascending order first; picking the "middle" from unsorted data gives wrong answer.

2. **Using wrong median formula for even n** → When n is even, you must average the two middle values, not just pick one.

3. **Confusing class limits with class marks** → For grouped mean, multiply frequency by class mark (midpoint), not by class limits.

4. **Wrong identification of modal class** → Modal class has the highest frequency, not the highest class interval value.

5. **Calculation errors in cumulative frequency** → Build cf column carefully; one error cascades through median calculation.

6. **Ignoring "no mode" possibility** → If all values occur with equal frequency, state "no mode" rather than listing all values.

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Quick Reference

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

**Median** = Middle value after sorting (use for skewed data)

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

**Range** = Maximum − Minimum (simplest measure of spread)

**Grouped data median class**: First class where cumulative frequency ≥ n/2

**Empirical relationship**: Mode ≈ 3 × Median − 2 × Mean (approximate, for moderately skewed data)

Statistics