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.
Understanding Statistics: Cumulative Frequency and Ogives
A cumulative frequency table must use the class boundaries that match the way measurements were recorded. Suppose heights are recorded to the nearest centimetre. A class labelled 150 to 159 really covers values from 149.5 up to 159.5 centimetres.
The boundary of 159.5 is used because a measured height of 159 centimetres could represent an actual height just below 159.5. This small adjustment prevents gaps between classes.
It matters most for continuous measurements such as height, mass, time, temperature, or distance. For whole-number counts, such as number of pets, grouping may need a different approach because the values cannot fall between integers.
Each point on an ogive represents a statement about the data. A point at a boundary of 159.5 with a cumulative frequency of 28 means that 28 observations are no greater than 159.5. The graph should start at the lower boundary of the first class with cumulative frequency zero.
Points are then joined with a smooth rising line or a sequence of straight sections, depending on the convention used in class. The graph can never move downward because totals do not decrease.
It must finish at the total number of observations. A flat section means no data values were added over that range.
Quartiles and percentiles are read by working from the vertical frequency scale first. For example, if there are 80 results, the lower quartile is the 20th value, the median is the 40th value, and the upper quartile is the 60th value. Mark the required position on the vertical axis, move horizontally to the curve, then move vertically down to the measurement axis.
The answer is an estimate, not an exact original value. Grouped data hides the individual values inside each interval. The reading assumes values are spread fairly evenly within the class, so a wide class interval can make estimates less precise.
Ogives are useful when a result must be described as a proportion below a limit. A school might examine the percentage of journey times below 30 minutes. A factory might check the proportion of bottle masses below a required amount.
In an exam, read axes carefully before making a conclusion. Do not confuse frequency with cumulative frequency. Check whether boundaries, class limits, or midpoints are shown.
Midpoints are used for some other graphs, but not for a standard cumulative frequency graph. When comparing two ogives with the same total, a curve farther left usually shows smaller values overall. A steeper part of a curve shows that many observations lie within a short interval.
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.