It is often used to define a geometric curve or to determine a limit of a series of values. Because the numbers are grouped into equal portions, the geometric shape of the curve can be determined by taking the difference between the first and second quartiles and dividing it by four.

An example of a log-log curve is the line connecting the points A and B in figure 5. The horizontal axis of this curve can be divided into thirds, starting at A. When drawn on graph paper, it would look something like a curved line, with A on the left and B on the right. The vertical axis of this curve shows the difference between A and B, starting from A.

The first quadrant of this curve separates the two points in which the two values are exactly the same. In this case, the mid-point, halfway between A and B, is B. This point is called the quartile or the middle value between the two. In the second quadrant, however, the mid-point is A.

The third quartile line marks the point where the difference between the mid-point and the fourth quartile, at the fourth position along the curve, is equal to the middle value. This point is called the end-point, or the fourth quartile, at the end of the line.

The fifth quartile, halfway between A and B, is used to separate the difference between two values, which are not exactly the same. The difference can range from one to three, but the third quartile in this case marks the mid-point between all but one of these differences. This fifth quartile can either be the mid-point between one and three, or it can be halfway between two and four.

Quartiles can be used to separate several values of a curve. By using three quartiles to separate the curves in figure 3, a quadrant can be drawn that separates the points where the third quartile lies between two of the previous quartiles. When drawn on graph paper, the graph will resemble a straight line, with each line joining the previous and subsequent quartiles, on either side of the curve.

Using quartiles to create curves is similar to using the Pythagorean triples, which allows a user to determine the distance between three points. In a linear equation, three straight lines will divide a series of points in half and connect the center of the points by the length of the lines. Quartiles can be used to break down a series of points in the same way, which makes it easier to calculate the value of a series, by grouping points together into equal quartiles, as seen in figure 4.

If four points are placed in a straight line, each point being separated by one or two quartiles, then the line will look something like an “L.” Each line can be drawn to a point within a quartile, and the distance between each point can be calculated. If the lines meet at one point, that point will be the middle of the quartile. If the lines meet at two points, each point will lie in half of the quartile.

A similar form of this pattern is often seen in linear equations, such as those used in calculus. The quadrants are drawn to meet at points within a series of points, such as those found in equation (2) above.

Quartiles are useful in a variety of applications. If a chart contains three or more data points, but only one of them is known, then by drawing a quartile line between the first two points and the middle point, it becomes easier to determine the distance between the unknown points. Using the quadrant to split up a series of points can also be useful in calculating the slope or the inclination of the series.

They can also be used to split a cubic polynomial curve into a series of equal portions. If a cubic polynomial curve consists of a series of polynomial terms, then using quartiles allows the user to draw a quadrant between each term, so that a new value of the curve is drawn by fitting a line between the original curve and the new point. This can also be useful in solving the cubic polynomial equation for a given parameter value, as is shown in equation (2) above.