Cumulative Frequency Graphs | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Cumulative Frequency

• Cumulative means "running total" or "adding up as you go along."
• In a table of grouped data, cumulative frequency means the total of all frequencies up to the end of that group.
• Cumulative frequencies can be presented in two ways:
  • A regular grouped data table with an extra column for cumulative frequencies.
    • Example: rows labeled 0 ≤ x < 20, 20 ≤ x < 40, 40 ≤ x < 60, etc.
  • A separate table where each group is relabeled to start at the beginning (often zero).
    • Example: rows labeled x < 20, x < 40, x < 60, etc.

Drawing Cumulative Frequency Diagrams

How do I draw a cumulative frequency graph?

• This is best explained with an example:
  • The times taken to complete a quiz by 50 students are shown in the table below:
    
• Cumulative frequency is the running total of the frequencies:
  • Time taken (s seconds) | Frequency | Cumulative Frequency
    
• To draw the cumulative frequency graph:
  • Plot cumulative frequency against the end (upper bound) of the class interval.
  • Example: first two points to plot would be (30, 3) and (35, 11).
  • Consider the second row (30 ≤ s < 35):
    • The 11 students in this group could have taken any time between 30 and 35 seconds.
    • They are all accounted for at 35 seconds.
• Plot all points from the table:
  • Add a starting point at the lowest time from the table (25 seconds) with a cumulative frequency of 0.
  • Plot the point (25, 0).
• Join the points with a smooth curve:
  • A cumulative frequency graph generally has a stretched-S-shape appearance.
  • The graph will never move back towards the x-axis.
• Here is the final cumulative frequency graph for the quiz times.

Interpreting Cumulative Frequency Diagrams

• A cumulative frequency graph is a useful tool for estimating important statistical values like the median, lower quartile, upper quartile, interquartile range, and percentiles.
• These values are approximations since the original raw data is not known.
  • Cumulative frequency graphs are typically employed when dealing with grouped or summarized data. Data points on the graph are connected by a smooth curve, indicating a smooth distribution over each interval.

How do I find the median, lower quartile and upper quartile from a cumulative frequency graph?

• Understanding the number of data values represented by a cumulative frequency graph is crucial.
• This may be explicitly stated in the question or determined by the highest value reached on the frequency (y-) axis of the graph.
• Typically, the highest value is located at the top right of the graph, given the stretched-S shape of cumulative frequency graphs.

Median:

• Step 1: Find the position of the median, which is n/2 for n data values.
  • No need to worry about whether nnn is odd or even, nor about using n+1n + 1n+1 as in other methods.
• Step 2: Draw a horizontal line from n/2 on the cumulative frequency axis until it intersects the curve.
• Step 3: Draw a vertical line from the curve down to the horizontal (x-) axis and take a reading. This value represents the median.

Lower Quartile:

• Step 1: Find the position of the lower quartile using n/4.
• Step 2: Draw a horizontal line from n/4 on the cumulative frequency axis until it intersects the curve.
• Step 3: Draw a vertical line from the curve to the horizontal (x-) axis and take a reading. This value represents the lower quartile.

Upper Quartile:

• Step 1: Find the position of the upper quartile using 3n/4.
• Step 2: Draw a horizontal line from 3n/4 on the cumulative frequency axis until it intersects the curve.
• Step 3: Draw a vertical line from the curve to the horizontal (x-) axis and take a reading. This value represents the upper quartile.

How do I find a percentile from a cumulative frequency graph?

• Percentiles divide the data into 100 parts, with each part representing a percentile.
• The 50th percentile is another term for the median.
• The 25th and 75th percentiles correspond to the lower and upper quartiles, respectively.

p^th Percentile:

• Step 1: Find the position of the percentile, which for nnn data values is .
• Step 2: Draw a horizontal line from on the cumulative frequency axis until it intersects the curve.
• Step 3: Draw a vertical line from the curve to the horizontal (x-) axis and take a reading. This value represents the p^th percentile.