# Averages, Range & Measures of Spread

> Mathematics · CIE IGCSE 0580 (2025-2027)
> Source: https://www.owlsprep.com/study/cie-0580-u9-averages-range-measures-of-spread/

This guide covers Core and Extended content for CIE IGCSE Maths 0580 averages and spread, including ungrouped data calculations, frequency table problems, grouped mean, quartiles and IQR for Extended candidates.

**Prerequisites:** Ability to perform basic arithmetic operations (addition, division, ordering values); Understanding of discrete and grouped frequency tables

## Learning objectives

- Calculate mode, median, mean and range for ungrouped discrete data (Core)
- Compute averages and range from ungrouped frequency tables (Core)
- Find quartiles and interquartile range for ungrouped data (Extended)
- Estimate mean for grouped data and identify modal class (Extended)

## Core: Averages & Range for Ungrouped Discrete Data

**Core Measures for Ungrouped Data** — Mode = most frequent value; Median = middle value when data is ordered ascending; Mean = sum of all values divided by number of values; Range = largest value minus smallest value.

**Worked example:** Find the mode, median, mean and range of the following ungrouped data set: 3, 7, 2, 3, 9, 5, 3

1. 1. Order the data ascending: 2, 3, 3, 3, 5, 7, 9
2. 2. Mode is the most frequent value: 3 (appears 3 times)
3. 3. Median is the 4th value (middle of 7 values): 3
4. $$Mean = \frac{2+3+3+3+5+7+9}{7} = \frac{32}{7} \thickapprox 4.57$$
5. 5. Range = 9 - 2 = 7

> **Exam Tip**
>
> Always order data first before calculating the median: this is the single most common error for this type of question.

## Core: Calculations from Ungrouped Frequency Tables

**Worked example:** The frequency table below shows scores on a 5-mark quiz. Calculate the mode, median, mean and range. 

| Score | 1 | 2 | 3 | 4 | 5 |
|-------|---|---|---|---|---|
| Frequency | 2 | 5 | 7 | 3 | 3 |

1. 1. Mode is the score with the highest frequency: 3
2. 2. Total frequency n = 2+5+7+3+3 = 20. Median is the average of the 10th and 11th values. Cumulative frequency up to score 3 is 14, so both values are 3: median = 3
3. $$Mean = \frac{(1\times2)+(2\times5)+(3\times7)+(4\times3)+(5\times3)}{20} = \frac{60}{20} = 3$$
4. 4. Range = 5 - 1 = 4

> **warning**
>
> Never ignore frequency values when calculating mean from a table: you must multiply each score by its frequency first.

## Extended Adds: Quartiles & Interquartile Range

**Quartiles and IQR** — Q1 (lower quartile) = value at the 25th percentile of ordered data; Q3 (upper quartile) = value at the 75th percentile; IQR = Q3 - Q1. IQR measures spread of the middle 50% of data, so it is not affected by outliers.

**Worked example:** For the ordered data set: 2, 3, 3, 3, 5, 7, 9, 10, 12, 14, 15, calculate Q1, Q3 and IQR.

1. 1. Number of values n = 11
2. 2. Q1 position = (n+1)/4 = 3rd value: Q1 = 3
3. 3. Q3 position = 3*(n+1)/4 = 9th value: Q3 = 12
4. 4. IQR = Q3 - Q1 = 12 - 3 = 9

## Extended Adds: Grouped Data Calculations

**Grouped Data Measures** — Modal class = class interval with the highest frequency; Estimated mean = sum of (mid-interval value × frequency) divided by total frequency. Mid-interval values are used because exact individual data points are not available in grouped data.

**Worked example:** The grouped frequency table shows heights of students in cm. Find the modal class and estimated mean height. 

| Height (h) | 120≤h<130 | 130≤h<140 | 140≤h<150 | 150≤h<160 |
|------------|-----------|-----------|-----------|-----------|
| Frequency | 4 | 9 | 12 | 5 |

1. 1. Modal class is the interval with highest frequency: 140≤h<150
2. 2. Calculate mid-interval values: 125, 135, 145, 155
3. $$Sum(mid-interval value × frequency) = (125×4)+(135×9)+(145×12)+(155×5) = 4230$$
4. 3. Total frequency = 4+9+12+5 = 30
5. $$Estimated mean = \frac{4230}{30} = 141 cm$$

> **tip**
>
> Always label your answer as 'estimated mean' for grouped data: exam markers award marks for explicitly stating it is an approximation.

## Common pitfalls

- **Wrong:** Forgetting to order data before calculating median or quartiles
  - Why it fails: Median and quartiles are calculated from ordered data, so unordered data will give an incorrect middle value
  - Correct: Sort all data points in ascending order first before calculating median or quartiles
- **Wrong:** Calculating mean from a frequency table by summing scores only, ignoring frequencies
  - Why it fails: Frequencies represent how many times each score occurs, so they must be included in the sum
  - Correct: Multiply each score by its frequency, sum these products, then divide by total frequency
- **Wrong:** Using exact values instead of mid-interval values for grouped mean calculations
  - Why it fails: Exact individual values are not available in grouped data, so mid-interval values are the only valid estimate
  - Correct: Calculate the midpoint of each class interval first before multiplying by frequency for grouped mean
- **Wrong:** Calculating range of grouped data as difference between midpoints of highest and lowest classes
  - Why it fails: Range measures the full spread of possible values, not just midpoints
  - Correct: Range of grouped data = upper bound of highest class minus lower bound of lowest class
- **Wrong:** Confusing IQR with range when a question asks for a spread measure unaffected by outliers
  - Why it fails: Range uses all data points including extreme outliers, while IQR only uses the middle 50% of data
  - Correct: Use IQR instead of range when you need to eliminate the effect of extreme values, as specified in the question

## Cheatsheet

| Measure | Core/Extended | Ungrouped Data | Ungrouped Frequency Table | Grouped Data |
| --- | --- | --- | --- | --- |
| Mode | Core | Most frequent value | Score with highest frequency | Modal class (Ext) |
| Median | Core | Middle ordered value | (n+1)/2 th value in cumulative freq | Estimate via cumulative frequency curve (see CF & Histograms page, E9.6) (Ext) |
| Mean | Core/Ext | Sum of values / n | Sum(score×freq) / n | Sum(mid×freq) / n (Ext) |
| Range | Core | Max - Min | Max score - Min score | Upper highest - Lower lowest class (Ext) |
| Q1 / Q3 | Ext | 25th / 75th percentile of ordered data | Same as ungrouped using cumulative freq | Estimate via cumulative frequency curve (see CF & Histograms page, E9.6) (Ext) |
| IQR | Ext | Q3 - Q1 | Q3 - Q1 | Q3 - Q1 |

## What's next

Now that you have mastered averages and measures of spread for CIE IGCSE Maths 0580, you can apply these skills to solve combined statistics problems involving data representation, including cumulative frequency graphs and histograms. These topics are frequently paired with averages in 4-6 mark exam questions, so practicing mixed statistics problems will help you secure full marks in the statistics section of your paper. Core candidates can move on to probability basics, while Extended candidates should focus on cumulative frequency calculations next to build on your knowledge of quartiles and spread.

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