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Statistics - Box-and-Whisker Plots

Grade 12IGCSE

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

🔑Concepts

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5-Number Summary: A box plot visually represents the Minimum value, Lower Quartile (Q1), Median (Q2), Upper Quartile (Q3), and Maximum value.

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Median (Q2): The middle value of the data set when arranged in order; represents the 50th percentile.

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Lower Quartile (Q1): The median of the lower half of the data; represents the 25th percentile.

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Upper Quartile (Q3): The median of the upper half of the data; represents the 75th percentile.

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Interquartile Range (IQR): The range of the middle 50% of the data, representing the spread and consistency.

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Outliers: Data points that fall significantly outside the range of the rest of the data, often defined by the 1.5 × IQR rule.

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Skewness: If the median is closer to Q1, the data is positively skewed; if closer to Q3, it is negatively skewed; if centered, it is symmetrical.

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Comparison: When comparing two datasets, use the Median to compare 'average' performance and the IQR to compare 'consistency' or 'variability'.

📐Formulae

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

Interquartile Range (IQR)=Q3−Q1\text{Interquartile Range (IQR)} = Q_3 - Q_1

Lower Boundary for Outliers=Q1−(1.5×IQR)\text{Lower Boundary for Outliers} = Q_1 - (1.5 \times \text{IQR})

Upper Boundary for Outliers=Q3+(1.5×IQR)\text{Upper Boundary for Outliers} = Q_3 + (1.5 \times \text{IQR})

💡Examples

Problem 1:

Given the dataset: 3, 7, 8, 5, 12, 14, 21, 15, 18. Calculate the 5-number summary and the IQR.

Solution:

  1. Order data: 3, 5, 7, 8, 12, 14, 15, 18, 21.
  2. Min = 3, Max = 21.
  3. Median (Q2) = 12 (the 5th value).
  4. Q1 = (5 + 7) / 2 = 6.
  5. Q3 = (15 + 18) / 2 = 16.5.
  6. IQR = 16.5 - 6 = 10.5.

Explanation:

To find quartiles, first find the median of the entire set. Then find the medians of the two halves created by the median. IQR is the difference between the upper and lower quartiles.

Problem 2:

A dataset has Q1 = 20 and Q3 = 35. Determine if a value of 60 is an outlier.

Solution:

  1. IQR = 35 - 20 = 15.
  2. Upper Boundary = Q3 + (1.5 * IQR) = 35 + (1.5 * 15) = 35 + 22.5 = 57.5.
  3. Since 60 > 57.5, the value 60 is an outlier.

Explanation:

Outliers are defined as values that exceed the upper boundary (Q3 + 1.5IQR) or fall below the lower boundary (Q1 - 1.5IQR).

Problem 3:

Compare two classes' test scores. Class A: Median 75, IQR 10. Class B: Median 70, IQR 25. Which class performed better and which was more consistent?

Solution:

  1. Class A performed better because its Median (75) is higher than Class B's (70).
  2. Class A was more consistent because its IQR (10) is smaller than Class B's (25).

Explanation:

In statistical comparison, a higher median indicates a higher average achievement, while a smaller IQR indicates less spread/variability, meaning the scores are more consistent.