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Statistics - Cumulative Frequency and Box Plots

Grade 9IGCSE

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

🔑Concepts

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Cumulative Frequency: A running total of frequencies, calculated by adding each frequency to the sum of all previous frequencies.

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Cumulative Frequency Graph: A smooth 'S-shaped' curve (ogive) plotted using the upper class boundaries on the x-axis and cumulative frequency on the y-axis.

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Median (Q2): The middle value of the data set, found at the 50% mark of the total frequency on the cumulative frequency curve.

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Quartiles: The Lower Quartile (Q1) is at 25% of the total frequency, and the Upper Quartile (Q3) is at 75% of the total frequency.

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Interquartile Range (IQR): A measure of spread representing the range of the middle 50% of the data (Q3−Q1Q3 - Q1).

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Box Plot (Box-and-Whisker): A visual summary of five key values: Minimum, Lower Quartile, Median, Upper Quartile, and Maximum.

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Comparing Distributions: Use the Median to compare 'average' performance and the IQR to compare 'consistency' (a smaller IQR means more consistent data).

📐Formulae

Cumulative Frequency Position (Median)=n2\text{Cumulative Frequency Position (Median)} = \frac{n}{2}

Lower Quartile (Q1) Position=n4\text{Lower Quartile (Q1) Position} = \frac{n}{4}

Upper Quartile (Q3) Position=3n4\text{Upper Quartile (Q3) Position} = \frac{3n}{4}

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

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

💡Examples

Problem 1:

A group of 80 students took a math test. The results are: 0<x≤200 < x \le 20 (freq: 10), 20<x≤4020 < x \le 40 (freq: 20), 40<x≤6040 < x \le 60 (freq: 35), 60<x≤8060 < x \le 80 (freq: 15). Calculate the cumulative frequencies and identify the position of the median.

Solution:

  1. CF for x≤20x \le 20 is 10.
  2. CF for x≤40x \le 40 is 10+20=3010 + 20 = 30.
  3. CF for x≤60x \le 60 is 30+35=6530 + 35 = 65.
  4. CF for x≤80x \le 80 is 65+15=8065 + 15 = 80. Median Position: 802=40th\frac{80}{2} = 40^{th} value.

Explanation:

To find cumulative frequency, we keep a running total. The median position in a continuous data set of nn items is found at n/2n/2. To find the actual median score, you would locate 40 on the y-axis of a CF graph and read the corresponding x-value.

Problem 2:

From a cumulative frequency graph, the following values were found: Min = 12, Q1=25Q1 = 25, Median = 34, Q3=42Q3 = 42, Max = 58. Construct the description of the box plot.

Solution:

The box starts at 25 and ends at 42. A vertical line is drawn inside the box at 34. Whiskers extend from the box left to 12 and right to 58.

Explanation:

A box plot visually represents the five-number summary. The 'box' covers the IQR (Q1Q1 to Q3Q3), and the 'whiskers' cover the full range of the data.

Problem 3:

Compare two sets of data: Class A has a Median of 65 and IQR of 10. Class B has a Median of 60 and IQR of 20. Which class performed better and which was more consistent?

Solution:

Class A performed better on average (higher Median: 65 > 60). Class A was also more consistent (lower IQR: 10 < 20).

Explanation:

In IGCSE statistics, 'better performance' is indicated by a higher median, while 'consistency' or 'reliability' is indicated by a smaller Interquartile Range (less spread in the middle 50% of data).