A cumulative frequency graph, also called an ogive, shows how many data values are less than or equal to each value on the horizontal axis. It is useful because it turns a frequency table into a visual summary of the whole distribution. From one curve, you can estimate the median, quartiles, percentiles, and spread.
This makes it easier to compare groups, such as test scores from two classes or heights from two samples.
To draw an ogive, plot cumulative frequency against the upper class boundary or data value, then join the points with a smooth increasing curve or straight line segments. To read a percentile, move horizontally from the chosen cumulative frequency to the curve, then move down to the value axis. The median is found at 50% of the total frequency, while the lower and upper quartiles are found at 25% and 75%.
Comparing two ogives on the same axes shows which distribution tends to have larger values and which has more variability.
Understanding Statistics: Cumulative Frequency Graphs
For grouped measurements, class boundaries matter. Imagine a height class written as 150 to 159 centimetres. If heights were measured to the nearest centimetre, its true boundaries run from 149.5 up to 159.5 centimetres.
Plotting at the boundary puts each class in the correct place on a continuous scale. The graph normally begins at the lowest boundary with a cumulative frequency of zero.
This starting point shows that no observations lie below the first possible value. Using class midpoints would be wrong because a midpoint describes the centre of a class, not the point by which every value in that class has been counted.
Values read from a graph are usually estimates, especially when the original data were grouped. A graph cannot reveal exactly how values were arranged inside a class interval. It assumes a reasonable spread between the plotted points.
This is called interpolation. A careful reader uses the scale on both axes, draws neat horizontal and vertical guide lines, and reports only sensible precision.
If the scale allows readings only to the nearest two units, giving a result to three decimal places suggests accuracy that the graph does not have. A percentile found on a steep part of the curve may change by only a small value, while the same vertical movement on a flat part can produce a much larger change in the data value.
The shape of the curve gives clues about the distribution. A steep section means many observations occur within a short range of values. A nearly flat section means few observations occur there.
When class widths are unequal, pay attention to the horizontal width as well as the vertical rise. A rise of ten across a narrow interval shows a stronger concentration than a rise of ten across a wide interval. When comparing two groups, use the same axes and check that the totals are comparable.
One curve lying further right often indicates larger values at many percentile levels. Curves can cross, however, so neither group is larger throughout the whole distribution. The distance between the first and third quartiles describes how spread out the central half is.
Ogives appear in real situations where a decision depends on a cutoff. A school might find the score reached by the top ten percent of a year group. A delivery company might estimate the time by which ninety percent of parcels arrive.
A health study might compare the middle range of resting heart rates between age groups. Before trusting a reading, check the total number of observations, the units, and whether every class has been included. Missing data changes every later cumulative total.
Cumulative graphs work best for numerical data with a meaningful order, such as time, distance, mass, or marks. They are less useful for categories such as eye colour because the order of those categories has no numerical meaning.
Key Facts
- Cumulative frequency is the running total of frequencies up to and including a value or class.
- For total frequency n, median position = n/2 on the cumulative frequency axis.
- Lower quartile position = n/4 and upper quartile position = 3n/4.
- Percentile position = (p/100)n, where p is the percentile number.
- Interquartile range = Q3 - Q1, measuring the spread of the middle 50% of the data.
- An ogive should never decrease because cumulative frequency can only stay the same or increase.
Vocabulary
- Cumulative frequency
- The total number of data values up to and including a particular value or class.
- Ogive
- A graph of cumulative frequency plotted against data values or class boundaries.
- Median
- The middle value of a data set, found at 50% of the total cumulative frequency.
- Quartile
- One of the values that divides an ordered data set into four equal parts.
- Percentile
- A value below which a given percentage of the data lies.
Common Mistakes to Avoid
- Plotting ordinary frequency instead of cumulative frequency is wrong because an ogive uses running totals, not the height of each class alone.
- Using class midpoints when the table gives grouped intervals can be wrong because ogives are usually plotted against upper class boundaries.
- Reading the median directly from the horizontal axis first is wrong because you must start at n/2 on the cumulative frequency axis, go across to the curve, then down.
- Assuming the curve gives exact raw data values is wrong because an ogive from grouped data gives estimates based on interpolation between plotted points.
Practice Questions
- 1 A data set has total frequency 80. On a cumulative frequency graph, what cumulative frequency levels should you use to estimate Q1, the median, and Q3?
- 2 An ogive for 120 students shows that the 30th percentile corresponds to a score of 46. How many students scored about 46 or below?
- 3 Two cumulative frequency graphs are drawn on the same axes. Graph A is mostly to the left of Graph B, and Graph B has a wider distance between Q1 and Q3. Explain what this suggests about the typical values and spread of the two distributions.