Quartiles & Percentiles

Quartiles and percentiles are measures used to understand the position of data within a data set. Key points to remember:

  • Quartiles divide the data into four equal parts, while percentiles divide the data into one hundred equal parts.
  • The horizontal axis represents the data values or measurements.
  • The vertical axis represents the frequency, or the number of data points.
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How Cumulative Freq Graph Is Plotted

In this video, we will learn how to plot a cumulative frequency graph from a table of data. When drawing the graph, keep these key points in mind:

  • Arrange the data values in ascending order.
  • Create a new column called cumulative frequency by adding up the frequencies step by step.
  • Label the horizontal axis with the data values.
  • Label the vertical axis with the cumulative frequency.
  • Plot each cumulative frequency against its corresponding value.
  • Join the points with a smooth curve.

Mean & IQR

In this video, we will learn how to use a cumulative frequency graph to find the median and interquartile range (IQR) & how the stem-and-leaf diagram is derived. Keep these key points in mind:

  • Divide the cumulative frequency scale on the vertical axis into four equal parts: 25%, 50%, 75%, and 100%.
  • The value at 50% gives the median.
  • The values at 25% and 75% give the lower quartile and upper quartile. The interquartile range (IQR) is found by subtracting the lower quartile from the upper quartile.
  • The range is the difference of the maximum and minimum value
  • Always read the corresponding values from the horizontal axis when finding the median and IQR.

Reading Values In Cumulative Freq Graph

When reading values from a cumulative frequency graph, remember that it is always a less than graph:

  • If the question asks for the number of values less than a given value, you can read the answer directly from the graph.
  • If the question asks for the number of values more than a given value, first find the cumulative frequency up to that value, then subtract it from the total frequency.

This is because a cumulative frequency graph shows the total number of data points that are less than or equal to each value.

Example 1 - Continuous Data

The cumulative frequency curve below illustrates the marks obtained, out of 60, by 300 students in a Mathematics Examination.

  1. Use the graph to find (i) the median mark (ii) the upper quartile (iii) the interquartile range (iv) the 35th percentile mark.
  2. Given that 72% of the students passed the paper, use the graph to find the pass mark.

Example 2 - Discrete Data

The marks scored by 10 students in a Math test are shown below.

44 , 28 , 32 , 83 , 45 , 55 , 54 , 58 , 60 , 84

Find the

  1. range
  2. median
  3. upper and lower quartiles
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