Ogives
Ogives
Cumulative histograms, also known as ogives, are graphs that can be used to determine how many data values lie above or below a particular value in a data set. The cumulative frequency is calculated from a frequency table, by adding each frequency to the total of the frequencies of all data values before it in the data set. The last value for the cumulative frequency will always be equal to the total number of data values, since all frequencies will already have been added to the previous total.
An ogive is drawn by
- plotting the beginning of the first interval at a \(y\)-value of zero;
- plotting the end of every interval at the \(y\)-value equal to the cumulative count for that interval; and
- connecting the points on the plot with straight lines.
In this way, the end of the final interval will always be at the total number of data since we will have added up across all intervals.
Example
Question
Determine the cumulative frequencies of the following grouped data and complete the table below. Use the table to draw an ogive of the data.
| Interval | Frequency | Cumulative frequency |
| \(10 < n \le 20\) | \(\text{5}\) | |
| \(20 < n \le 30\) | \(\text{7}\) | |
| \(30 < n \le 40\) | \(\text{12}\) | |
| \(40 < n \le 50\) | \(\text{10}\) | |
| \(50 < n \le 60\) | \(\text{6}\) |
Compute cumulative frequencies
To determine the cumulative frequency, we add up the frequencies going down the table. The first cumulative frequency is just the same as the frequency, because we are adding it to zero. The final cumulative frequency is always equal to the sum of all the frequencies. This gives the following table:
| Interval | Frequency | Cumulative frequency |
| \(10 < n \le 20\) | \(\text{5}\) | \(\text{5}\) |
| \(20 < n \le 30\) | \(\text{7}\) | \(\text{12}\) |
| \(30 < n \le 40\) | \(\text{12}\) | \(\text{24}\) |
| \(40 < n \le 50\) | \(\text{10}\) | \(\text{34}\) |
| \(50 < n \le 60\) | \(\text{6}\) | \(\text{40}\) |
Plot the ogive
The first coordinate in the plot always starts at a \(y\)-value of \(\text{0}\) because we always start from a count of zero. So, the first coordinate is at \((10;0)\) — at the beginning of the first interval. The second coordinate is at the end of the first interval (which is also the beginning of the second interval) and at the first cumulative count, so \((20;5)\). The third coordinate is at the end of the second interval and at the second cumulative count, namely \((30;12)\), and so on.
Computing all the coordinates and connecting them with straight lines gives the following ogive.
Ogives do look similar to frequency polygons, which we saw earlier. The most important difference between them is that an ogive is a plot of cumulative values, whereas a frequency polygon is a plot of the values themselves. So, to get from a frequency polygon to an ogive, we would add up the counts as we move from left to right in the graph.
Ogives are useful for determining the median, percentiles and five number summary of data. Remember that the median is simply the value in the middle when we order the data. A quartile is simply a quarter of the way from the beginning or the end of an ordered data set. With an ogive we already know how many data values are above or below a certain point, so it is easy to find the middle or a quarter of the data set.
Example
Question
Use the following ogive to compute the five number summary of the data. Remember that the five number summary consists of the minimum, all the quartiles (including the median) and the maximum.
Find the minimum and maximum
The minimum value in the data set is \(\text{1}\) since this is where the ogive starts on the horizontal axis. The maximum value in the data set is \(\text{10}\) since this is where the ogive stops on the horizontal axis.
Find the quartiles
The quartiles are the values that are \(\cfrac{1}{4}\), \(\cfrac{1}{2}\) and \(\cfrac{3}{4}\) of the way into the ordered data set. Here the counts go up to \(\text{40}\), so we can find the quartiles by looking at the values corresponding to counts of \(\text{10}\), \(\text{20}\) and \(\text{30}\). On the ogive a count of
- \(\text{10}\) corresponds to a value of \(\text{3}\) (first quartile);
- \(\text{20}\) corresponds to a value of \(\text{7}\) (second quartile); and
- \(\text{30}\) corresponds to a value of \(\text{8}\) (third quartile).
Write down the five number summary
The five number summary is \((1; 3; 7; 8; 10)\). The box-and-whisker plot of this data set is given below.
This lesson is part of:
Statistics and Probability