To determine the median using a cumulative frequency graph, also known as an ogive, you need to locate the 50th percentile on the cumulative frequency axis.

A cumulative frequency graph is an effective tool for visualizing the distribution of data. To start, you must identify the total number of data points, which is represented by the highest value on the cumulative frequency axis. The median is defined as the value that divides the data into two equal halves, so you will need to find the point corresponding to half of the total frequency.

For instance, if the total frequency is $100$, you would seek the $50$th percentile, which is $50$. From this point, draw a horizontal line from $50$ on the cumulative frequency axis until it intersects the ogive. Next, from the intersection point, draw a vertical line down to the horizontal axis (the x-axis). The point where this vertical line meets the x-axis is the median of your data set.

This method is effective because the cumulative frequency graph illustrates the running total of frequencies up to each data point. By locating the $50$th percentile, you are effectively identifying the middle value of the data set, which is the definition of the median. This graphical approach is particularly advantageous for large data sets, where manually calculating the median can be tedious.