Averages, Range & Measures of Spread

Mathematics· 9.3· 35 min read

1. Core: Averages & Range for Ungrouped Discrete Data★★☆☆☆⏱ 10 min

📘 Definition

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
1. Mode is the most frequent value: 3 (appears 3 times)
3
1. 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
1. Range = 9 - 2 = 7

2. Core: Calculations from Ungrouped Frequency Tables★★☆☆☆⏱ 10 min

📐 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
1. 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
1. Range = 5 - 1 = 4

3. Extended Adds: Quartiles & Interquartile Range★★★☆☆Extended only⏱ 8 min

📘 Definition

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

4. Extended Adds: Grouped Data Calculations★★★☆☆Extended only⏱ 12 min

📘 Definition

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
1. Calculate mid-interval values: 125, 135, 145, 155
3
$Sum(mid-interval value × frequency) = (125×4)+(135×9)+(145×12)+(155×5) = 4230$
4
1. Total frequency = 4+9+12+5 = 30
5
$Estimated mean = \frac{4230}{30} = 141 cm$

5. Common Pitfalls

Wrong move:

Forgetting to order data before calculating median or quartiles

Why:

Median and quartiles are calculated from ordered data, so unordered data will give an incorrect middle value

Correct move:

Sort all data points in ascending order first before calculating median or quartiles

Wrong move:

Calculating mean from a frequency table by summing scores only, ignoring frequencies

Why:

Frequencies represent how many times each score occurs, so they must be included in the sum

Correct move:

Multiply each score by its frequency, sum these products, then divide by total frequency

Wrong move:

Using exact values instead of mid-interval values for grouped mean calculations

Why:

Exact individual values are not available in grouped data, so mid-interval values are the only valid estimate

Correct move:

Calculate the midpoint of each class interval first before multiplying by frequency for grouped mean

Wrong move:

Calculating range of grouped data as difference between midpoints of highest and lowest classes

Why:

Range measures the full spread of possible values, not just midpoints

Correct move:

Range of grouped data = upper bound of highest class minus lower bound of lowest class

Wrong move:

Confusing IQR with range when a question asks for a spread measure unaffected by outliers

Why:

Range uses all data points including extreme outliers, while IQR only uses the middle 50% of data

Correct move:

Use IQR instead of range when you need to eliminate the effect of extreme values, as specified in the question

6. Quick Reference 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.