Cumulative frequency is a way to show how many data values are at or below a given point. It is especially useful for grouped data, where individual values are sorted into class intervals. An ogive is the graph of cumulative frequency against class boundaries, and it gives a clear picture of how data builds up across a distribution.
These graphs help students estimate medians, quartiles, percentiles, and proportions without listing every data value.
Key Facts
- Cumulative frequency is found by adding each frequency to the total of all previous frequencies.
- For grouped data, plot cumulative frequency against the upper class boundary of each interval.
- Total frequency n is the final cumulative frequency on the ogive.
- Median position = n/2, lower quartile position = n/4, upper quartile position = 3n/4.
- Relative cumulative frequency = cumulative frequency / total frequency.
- Interquartile range = Q3 - Q1.
Vocabulary
- Cumulative frequency
- The running total of frequencies up to and including a given class or value.
- Ogive
- A graph that plots cumulative frequency against class boundaries to show how data accumulates.
- Class interval
- A range of values used to group continuous or discrete data in a frequency table.
- Quartile
- A value that divides an ordered data set into four equal parts.
- Relative cumulative frequency
- The fraction or percentage of the data at or below a certain value.
Common Mistakes to Avoid
- Plotting ordinary frequency instead of cumulative frequency is wrong because an ogive must always show running totals that do not decrease.
- Using class midpoints instead of upper class boundaries is wrong for a standard ogive because each plotted point represents all data up to the end of that interval.
- Forgetting to start near zero cumulative frequency is wrong because the graph should show that no data have accumulated before the first lower boundary.
- Reading quartiles from the x-axis first is wrong because you must locate n/4, n/2, or 3n/4 on the cumulative frequency axis, move across to the curve, then drop down to the data axis.
Practice Questions
- 1 A grouped frequency table has intervals 0 to 10, 10 to 20, 20 to 30, and 30 to 40 with frequencies 4, 9, 12, and 5. Find the cumulative frequencies and state the total frequency.
- 2 For a data set with total frequency n = 80, find the cumulative frequency positions for Q1, the median, and Q3. Explain how these positions would be used on an ogive.
- 3 Two ogives have the same total frequency, but Graph A rises steeply at low values while Graph B rises steeply at high values. Explain what this means about the distributions of the two data sets.