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Statistics - Cumulative Frequency and Quartiles

Grade 12IGCSE

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

🔑Concepts

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

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Cumulative Frequency Curve (Ogive): A graph where the cumulative frequency is plotted on the y-axis against the upper class boundaries on the x-axis.

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Median (Q2): The middle value of the data set, located at the 50th percentile (total frequency / 2).

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Lower Quartile (Q1): The value at the 25th percentile (total frequency / 4).

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Upper Quartile (Q3): The value at the 75th percentile (3 * total frequency / 4).

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

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Percentiles: Values that divide the data into 100 equal parts (e.g., the 90th percentile is at 0.9 * total frequency).

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Box-and-Whisker Plot: A visual summary of data using the minimum, Q1, Median, Q3, and maximum values.

📐Formulae

Cumulative Frequency (CF)=∑f\text{Cumulative Frequency (CF)} = \sum f

Position of Median (Q2)=n2\text{Position of Median } (Q_2) = \frac{n}{2}

Position of Lower Quartile (Q1)=n4\text{Position of Lower Quartile } (Q_1) = \frac{n}{4}

Position of Upper Quartile (Q3)=3n4\text{Position of Upper Quartile } (Q_3) = \frac{3n}{4}

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

Position of pth Percentile=p100×n\text{Position of } p^{th} \text{ Percentile} = \frac{p}{100} \times n

💡Examples

Problem 1:

A group of 80 students took a math test. The cumulative frequency curve shows that the 20th student (Q1) scored 45 marks, the 40th student (Median) scored 62 marks, and the 60th student (Q3) scored 78 marks. Calculate the Interquartile Range (IQR).

Solution:

IQR=Q3−Q1=78−45=33IQR = Q_3 - Q_1 = 78 - 45 = 33

Explanation:

To find the IQR, subtract the value of the Lower Quartile (25th percentile) from the Upper Quartile (75th percentile) as read from the x-axis (marks).

Problem 2:

The total frequency of a data set is 200. At what cumulative frequency value should you look on the y-axis to find the 85th percentile?

Solution:

Value=85100×200=170\text{Value} = \frac{85}{100} \times 200 = 170

Explanation:

To find a specific percentile, multiply the percentage (as a decimal) by the total frequency (nn). You then locate 170 on the y-axis and read across to the curve and down to the x-axis to find the corresponding data value.

Problem 3:

Given the class intervals for mass (mm): 0<m≤100 < m \leq 10 (freq: 5), 10<m≤2010 < m \leq 20 (freq: 12), 20<m≤3020 < m \leq 30 (freq: 8). State the coordinates to be plotted for the second class on a cumulative frequency graph.

Solution:

(20,17)(20, 17)

Explanation:

On a cumulative frequency graph, we plot the upper class boundary on the x-axis and the cumulative frequency on the y-axis. For the second class, the upper boundary is 20. The cumulative frequency is 5+12=175 + 12 = 17.