Statistics — Study Notes for TS TET
Overview
Statistics is a fundamental topic in the TS TET Mathematics section, appearing in both Paper I (Classes 1-5) and Paper II (Classes 6-8). At the primary and upper primary levels, statistics focuses on organising, representing and interpreting data using simple measures of central tendency (mean, median, mode) and spread (range).
This topic carries direct questions in the content section and also appears in pedagogy questions where you must suggest appropriate methods for teaching data handling. Examiners frequently test your ability to calculate averages from grouped/ungrouped data and interpret bar graphs, pictographs and pie charts. Mastery here requires both computational accuracy and conceptual clarity about when to use which measure.
For TET purposes, focus on quick mental calculation techniques, understanding the behaviour of mean/median/mode when data changes, and recognising common student misconceptions about averages.
Key Concepts
- Data is a collection of facts, figures or observations. It can be primary (collected first-hand) or secondary (obtained from existing sources).
- Mean (Arithmetic Average) is the sum of all observations divided by the number of observations. It uses every data point and is sensitive to extreme values (outliers).
- Median is 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. Median is resistant to outliers.
- Mode is the value that occurs most frequently. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal).
- Range measures spread: Range = Highest value − Lowest value. It indicates how scattered the data is but is affected by extreme values.
- Frequency is the number of times a particular observation occurs. A frequency distribution table organises data by grouping observations with their frequencies.
- Data Representation includes pictographs, bar graphs, double bar graphs, pie charts and line graphs. Each serves different purposes depending on the nature of data.
- The appropriate measure depends on context: mean for symmetric data without outliers, median for skewed data or when outliers exist, mode for categorical data or finding the most common item.
Formulas / Key Facts
| Measure | Formula / Rule |
|---|---|
| Mean (Ungrouped) | Mean = (Sum of all observations) ÷ (Number of observations) |
| Mean (Grouped) | Mean = Σ(f × x) ÷ Σf, where f = frequency, x = class mark |
| Class Mark | Class Mark = (Lower limit + Upper limit) ÷ 2 |
| Median (Odd n) | Middle value = value at position (n + 1) ÷ 2 |
| Median (Even n) | Average of values at positions n ÷ 2 and (n ÷ 2) + 1 |
| Mode | Value with highest frequency |
| Range | Highest observation − Lowest observation |
Key Facts:
- If all values are equal, Mean = Median = Mode.
- Adding a constant k to every observation increases mean, median and mode by k.
- Multiplying every observation by k multiplies mean, median and mode by k.
- Range remains unchanged when a constant is added but gets multiplied when observations are multiplied.
Worked Examples
Example 1: Finding Mean
The marks of 6 students are: 45, 52, 60, 58, 47, 54. Find the mean.
Solution: Sum = 45 + 52 + 60 + 58 + 47 + 54 = 316 Number of observations = 6 Mean = 316 ÷ 6 = 52.67 (approx.)
Example 2: Finding Median
Find the median of: 12, 18, 10, 15, 20, 17, 14
Solution: Step 1: Arrange in ascending order → 10, 12, 14, 15, 17, 18, 20 Step 2: Number of observations n = 7 (odd) Step 3: Median position = (7 + 1) ÷ 2 = 4th value Median = 15
Example 3: Finding Mode and Range
The shoe sizes of 10 students are: 6, 7, 6, 8, 7, 6, 9, 7, 6, 8. Find mode and range.
Solution: Frequency count: 6 appears 4 times, 7 appears 3 times, 8 appears 2 times, 9 appears 1 time Mode = 6 (highest frequency)
Range = 9 − 6 = 3
Example 4: Mean from Frequency Table
| Marks (x) | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| Frequency (f) | 4 | 6 | 5 | 5 |
Find the mean.
Solution: Σ(f × x) = (4 × 10) + (6 × 20) + (5 × 30) + (5 × 40) = 40 + 120 + 150 + 200 = 510 Σf = 4 + 6 + 5 + 5 = 20 Mean = 510 ÷ 20 = 25.5
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Forgetting to arrange data before finding median | Always sort data in ascending/descending order first, then locate the middle position |
| Using mean when outliers are present | Recognise that median is more appropriate for skewed data; mean gets distorted by extreme values |
| Confusing class mark with class interval | Class mark = (lower + upper) ÷ 2; class interval = upper − lower |
| Assuming every data set has exactly one mode | A data set can be mode-less (all values appear once) or multimodal (multiple values share highest frequency) |
| Calculating median position as n ÷ 2 for odd n | For odd n, position = (n + 1) ÷ 2; the n ÷ 2 rule applies only to even n for identifying the two middle values |
| Adding range when a constant is added to data | Range depends only on spread; adding the same constant to all values does not change the range |
Quick Reference
- Mean = Total sum ÷ Count — uses all values, affected by outliers.
- Median = Middle value after sorting — best for skewed data.
- Mode = Most frequent value — useful for categorical data.
- Range = Max − Min — simple measure of spread.
- For even n, median = average of the two central values.
- Grouped data mean: use class marks × frequencies, then divide by total frequency.