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Complete IGCSE Edexcel guide to calculating averages and spread from raw data, frequency tables, and grouped data. Worked examples for grades 5-7.
The mean is the total of all values divided by the number of values. The median is the middle value in ordered data (or average of the two middle values for even-sized data sets). The mode is the most common value. The range is the difference between the largest and smallest values. For grouped data, you estimate the mean using midpoints, identify the modal class, and find the class containing the median.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The mean is found by adding all the values together and dividing by the number of values. For a frequency table, multiply each value by its frequency, add all the products (the sum of fx), then divide by the total frequency.
Mean = sum of all values / number of values
From a frequency table: Mean = sum of fx / sum of f
Arrange all values in order from smallest to largest. If there is an odd number of values, the median is the middle one. The position is (n + 1) / 2. If there is an even number of values, the median is the mean of the two middle values.
For a frequency table, find the cumulative frequency and identify which value corresponds to the (n + 1) / 2 position. For grouped data, you identify the median class (the class containing the median position) rather than finding an exact value.
The mode is the value that appears most often. A data set can have no mode (all values different), one mode (unimodal), or multiple modes (bimodal or multimodal).
For grouped data, you cannot find an exact mode. Instead, identify the modal class — the class interval with the highest frequency. This is usually the simplest question type in statistics and should never be missed.
The range is the difference between the largest and smallest values: range = maximum - minimum. It measures the spread of the data. A larger range means more spread; a smaller range means the data is more consistent.
For grouped data, you can only estimate the range using the class boundaries: upper boundary of the last class minus the lower boundary of the first class.
Grouped frequency tables require a modified approach because you do not have individual data values. Add a midpoint column (average of the lower and upper boundaries of each class) and an fx column (midpoint multiplied by frequency).
| Measure | From Grouped Data |
|---|---|
| Mean | Estimated mean = sum of fx / sum of f (using midpoints) |
| Median | Identify the median class (class containing the n/2 position) |
| Mode | Modal class (class with highest frequency) |
| Range | Estimated: highest upper boundary - lowest lower boundary |
Find the mean, median, mode, and range of: 3, 5, 7, 5, 8, 12, 5, 9, 6.
Mean: Sum = 3 + 5 + 7 + 5 + 8 + 12 + 5 + 9 + 6 = 60. Mean = 60 / 9 = 6.67 (3 sf).
Median: Ordered: 3, 5, 5, 5, 6, 7, 8, 9, 12. Position = (9 + 1) / 2 = 5th value = 6.
Mode: 5 appears three times (most frequent). Mode = 5.
Range: 12 - 3 = 9.
Answer: Mean = 6.67, Median = 6, Mode = 5, Range = 9
A grouped frequency table shows: 0-10 (freq 4), 10-20 (freq 8), 20-30 (freq 12), 30-40 (freq 6). Find the estimated mean and modal class.
Step 1: Find midpoints: 5, 15, 25, 35.
Step 2: Calculate fx: 5 x 4 = 20, 15 x 8 = 120, 25 x 12 = 300, 35 x 6 = 210.
Step 3: Sum of fx = 20 + 120 + 300 + 210 = 650. Sum of f = 30.
Step 4: Estimated mean = 650 / 30 = 21.7 (3 sf).
Step 5: Modal class = 20-30 (highest frequency = 12).
Answer: Estimated mean = 21.7, Modal class = 20-30
The mean of five numbers is 12. Four of the numbers are 8, 15, 10, and 14. Find the fifth number.
Step 1: Total of all five numbers = mean x count = 12 x 5 = 60.
Step 2: Sum of four known numbers = 8 + 15 + 10 + 14 = 47.
Step 3: Fifth number = 60 - 47 = 13.
Answer: The fifth number is 13
Using class boundaries instead of midpoints: The estimated mean formula uses midpoints (average of lower and upper bounds), not the boundaries themselves.
Dividing by the number of classes instead of total frequency: The denominator is the sum of all frequencies, not the number of rows in the table.
Not ordering data before finding the median: The median requires data in ascending order. Finding the "middle value" of unordered data gives the wrong answer.
Confusing modal class with the class containing the median: The modal class has the highest frequency. The median class contains the middle value. These are usually different classes.
Show the fx column: For estimated mean questions, always write out the midpoint and fx columns. This earns method marks even if the final answer has an arithmetic error.
Check your totals: After adding up the fx column, verify the total frequency matches the one given in the question. This catches addition errors.
Use (n+1)/2 for discrete data: For an ungrouped list of n values, the median position is (n + 1) / 2. For grouped data from a cumulative frequency curve, use n / 2.
Write "estimated" for grouped data: When the question involves grouped data, always say "estimated mean" since you used midpoints rather than actual values.
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