Using linear interpolation and similar triangles to estimate percentiles from a cumulative frequency distribution

 On the screen you will see a graph with a short white line segment on it. This represents a part of a graph which is straight between two points. For instance, it could be part of a cumulative frequency graph (assuming it is straight between the two points). There are four red dots which can be clicked-and-dragged with your mouse. You will see that the line segment is redrawn with these end-points. The vertical axis on the left represents percentages. The horizontal axis represents the values corresponding to those percentages, the values are called percentiles. The blue dot on the left (between the two red ones) can also be moved with your mouse, and lines are drawn for you to see the equivalent on the horizontal axis. Calculating the percentile value for the percentage uses a technique called linear interpolation. The idea behind linear interpolation is simple, and uses the fact that the line is straight, and that the green lines mark out two similar triangles. Think about it, and particularly look at the ratio of the two vertical green lines. Then look at the working-out on the right of the screen. You'll soon get the idea! You can hide the working and the answer by removing the tick next to "Show working and answer" at the top. You might wish to test yourself in this way. For a more detailed look at this idea in practice, look at the Estimating Quantiles applet.