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Statistics - Mean, Median, Mode, and Range

Grade 12IGCSE

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

🔑Concepts

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Measures of Central Tendency: Statistical constants that describe the center of a data distribution (Mean, Median, Mode).

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Measures of Dispersion: Values that describe the spread or variability of the data (Range).

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Discrete vs. Grouped Data: For discrete data, exact values are used; for grouped data, midpoints are used to calculate estimates.

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Sensitivity to Outliers: The Mean is highly affected by extreme values, whereas the Median remains relatively stable.

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Modal Class: In grouped frequency distributions, the class interval with the highest frequency.

📐Formulae

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

Estimated Mean=∑(f×x)∑f\text{Estimated Mean} = \frac{\sum (f \times x)}{\sum f} (where xx is the midpoint of a class interval)

Median Position=n+12-th value\text{Median Position} = \frac{n + 1}{2}\text{-th value} (for a sorted list of nn items)

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

💡Examples

Problem 1:

A set of test scores is: 12, 15, 15, 17, 20, 22, 25. Find the Mean, Median, Mode, and Range.

Solution:

Mean = 18; Median = 17; Mode = 15; Range = 13.

Explanation:

  1. Mean: (12+15+15+17+20+22+25)/7=126/7=18(12+15+15+17+20+22+25) / 7 = 126 / 7 = 18. 2. Median: The data is already sorted; the 4th value is 17. 3. Mode: 15 appears twice, more than any other number. 4. Range: 25−12=1325 - 12 = 13.

Problem 2:

Estimate the mean for the following grouped data: Height (140 < h ≤ 150): Freq 4; Height (150 < h ≤ 160): Freq 11; Height (160 < h ≤ 170): Freq 5.

Solution:

Estimated Mean = 155.5 cm

Explanation:

  1. Find midpoints (xx): 145, 155, 165. 2. Multiply by frequency (f×xf \times x): (4×145)=580(4 \times 145) = 580, (11×155)=1705(11 \times 155) = 1705, (5×165)=825(5 \times 165) = 825. 3. Sum of fx=3110fx = 3110. 4. Total frequency ∑f=20\sum f = 20. 5. Mean =3110/20=155.5= 3110 / 20 = 155.5.