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Statistics and Probability - Mean, Median, Mode, and Range

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Mean: The arithmetic average calculated by dividing the sum of values by the count of values.

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Median: The middle value in a data set when arranged in numerical order. If there is an even number of values, it is the average of the two middle values.

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Mode: The value that occurs most frequently in a data set. A set can be bimodal (two modes) or have no mode.

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Range: A measure of spread calculated by subtracting the smallest value from the largest value.

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Frequency Tables: Data organized by how often each value occurs, often used to find the 'Estimated Mean' or 'Modal Class'.

📐Formulae

Mean(xˉ)=∑xn\text{Mean} (\bar{x}) = \frac{\sum x}{n}

Median Position=n+12\text{Median Position} = \frac{n + 1}{2}

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

Mean from Frequency Table=∑(f×x)∑f\text{Mean from Frequency Table} = \frac{\sum (f \times x)}{\sum f}

💡Examples

Problem 1:

Find the Mean, Median, Mode, and Range of the following data set: 12, 15, 12, 18, 20, 15, 12.

Solution:

Mean = 14.86, Median = 15, Mode = 12, Range = 8.

Explanation:

  1. Arrange data: 12, 12, 12, 15, 15, 18, 20. 2. Mean: (12+12+12+15+15+18+20)/7=104/7≈14.86(12+12+12+15+15+18+20) / 7 = 104 / 7 \approx 14.86. 3. Median: Middle value (4th position) is 15. 4. Mode: 12 appears most frequently (3 times). 5. Range: 20−12=820 - 12 = 8.

Problem 2:

Calculate the mean from this frequency table: Score (x): 1, 2, 3; Frequency (f): 4, 5, 1.

Solution:

1.7

Explanation:

First, find ∑f=4+5+1=10\sum f = 4 + 5 + 1 = 10. Then find ∑(f×x)=(1×4)+(2×5)+(3×1)=4+10+3=17\sum (f \times x) = (1 \times 4) + (2 \times 5) + (3 \times 1) = 4 + 10 + 3 = 17. Mean = 17/10=1.717 / 10 = 1.7.

Problem 3:

The mean of five numbers is 8. Four of the numbers are 6, 10, 7, and 9. Find the fifth number.

Solution:

8

Explanation:

Total sum = Mean×n=8×5=40\text{Mean} \times n = 8 \times 5 = 40. Sum of known numbers = 6+10+7+9=326 + 10 + 7 + 9 = 32. Fifth number = 40−32=840 - 32 = 8.