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Statistics - Cumulative Frequency and Box Plots

Grade 10IGCSE

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 (Ogive): A curve plotted using the upper class boundary on the x-axis and the cumulative frequency on the y-axis.

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Median (Q2Q_2): The middle value of the data set, found at the 50% mark of the total cumulative frequency.

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Lower Quartile (Q1Q_1): The value at 25% of the total cumulative frequency.

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Upper Quartile (Q3Q_3): The value at 75% of the total cumulative frequency.

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Interquartile Range (IQR): The difference between the upper and lower quartiles (Q3−Q1Q_3 - Q_1), representing the spread of the middle 50% of the data.

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Box Plot (Box-and-Whisker): A visual representation of the five-number summary: Minimum, Q1Q_1, Median, Q3Q_3, and Maximum.

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Comparing Distributions: When comparing two sets of data, compare a measure of central tendency (Median) and a measure of spread (IQR).

📐Formulae

Position of Median=n2\text{Position of Median} = \frac{n}{2}

Position of Q1=14n\text{Position of } Q_1 = \frac{1}{4}n

Position of Q3=34n\text{Position of } Q_3 = \frac{3}{4}n

IQR=Q3−Q1IQR = Q_3 - Q_1

Percentile Position=p100×n\text{Percentile Position} = \frac{p}{100} \times n

💡Examples

Problem 1:

A group of 80 students took a math test. The cumulative frequency graph shows that the Lower Quartile (Q1Q_1) is 42 marks and the Upper Quartile (Q3Q_3) is 68 marks. Calculate the Interquartile Range (IQR) and identify the number of students who scored above the Upper Quartile.

Solution:

  1. IQR=Q3−Q1=68−42=26IQR = Q_3 - Q_1 = 68 - 42 = 26 marks.
  2. Students above Q3Q_3: Since Q3Q_3 is at the 75% mark, 100100% - 75% = 25% of students scored above it.
  3. 25% of 80=0.25×80=2025\% \text{ of } 80 = 0.25 \times 80 = 20 students.

Explanation:

The IQR measures the range of the middle half of the scores. Because the Upper Quartile represents the 75th percentile, the remaining 25% of the total frequency (n=80) falls above this value.

Problem 2:

Given the following data from a cumulative frequency curve: Minimum = 10, Q1=25Q_1 = 25, Median = 35, Q3=45Q_3 = 45, and Maximum = 60. Describe how to construct the Box Plot.

Solution:

  1. Draw a horizontal scale from 10 to 60.
  2. Draw a rectangular box starting at 25 (Q1Q_1) and ending at 45 (Q3Q_3).
  3. Draw a vertical line inside the box at 35 (Median).
  4. Draw 'whiskers' extending from the box: one from 25 down to 10 (Min) and one from 45 up to 60 (Max).

Explanation:

A box plot summarizes the distribution. The 'box' covers the IQR, and the 'whiskers' show the full range of the data. This allows for a quick visual assessment of skewness and spread.

Problem 3:

In a dataset of 200 items, at what cumulative frequency value would you find the 60th percentile?

Solution:

Value=60100×200=120Value = \frac{60}{100} \times 200 = 120.

Explanation:

To find a specific percentile on a cumulative frequency graph, multiply the total frequency (nn) by the percentage (expressed as a decimal). You would then look for 120 on the y-axis to find the corresponding x-value on the curve.