Fundamentals of Business Mathematics and Statistics · Statistical Representation of Data

Ogives and Locating Median and Quartiles Graphically

Q: What is the difference between a less-than and a more-than ogive?

A less-than ogive plots upper class limits against the number of items below them, so it rises. A more-than ogive plots lower class limits against the number of items at or above them, so it falls.

Q: How do you find the median from an ogive?

Draw both ogives on the same axes. The x-value of the point where they intersect is the median. You can also mark N/2 on the y-axis, go across to the less-than curve and read down to the x-axis.

Q: How do you find quartiles graphically?

Use the less-than ogive. Mark N/4 on the y-axis, move across to the curve and read down for Q1. Repeat at 3N/4 for Q3. The median is Q2 at N/2.

Q: Do I need to draw the graph in the CMA Foundation exam?

No. The paper is fully objective, so you answer MCQs. You need to know how an ogive is built and how to get values from the table by interpolation. Interpolation gives an approximation of the ogive reading, and it equals the formula-based median.

Updated 10 October 2026 · Fact-checked

An ogive is a graph of cumulative frequency. A less-than ogive plots upper class limits against cumulative frequency and rises. A more-than ogive plots lower limits and falls. The x-value where the two curves cross is the median. Quartiles are read at heights N/4 and 3N/4 on the less-than ogive.

Understand Ogives and Locating Median and Quartiles Graphically

A frequency table tells you how many items fall in each class. An ogive tells you how many items fall below or above a given value. It does this by plotting cumulative frequency, which is a running total of frequencies.

A less-than ogive uses the upper limit of each class. At each upper limit you plot the number of items less than that value. The curve starts at the lower limit of the first class with frequency 0 and rises to the total N. A more-than ogive uses the lower limit of each class. At each lower limit you plot the number of items that are that value or more. It starts at N and falls to 0 at the upper limit of the last class.

Both curves are drawn by joining the plotted points with a smooth freehand curve. Both must use the same scale. The two curves cross at one point. The x-value of that point is the median, because there the number of items below equals the number of items above. The height at the crossing is N/2.

Quartiles work the same way. Mark the height N/4 on the vertical axis, move across to the less-than ogive, then drop down to the x-axis. That value is Q1. Do the same at 3N/4 for Q3. The median is the same as Q2, read at N/2. Classes must be continuous (exclusive type) before you start.

Key formulas to remember

Less-than ogive points: Plot (upper limit, cumulative frequency of classes up to that limit); Add a starting point (lower limit of first class, 0).
More-than ogive points: Plot (lower limit, frequency of that class and all later classes); The last point is (upper limit of last class, 0).
Median from ogive: x-value where the two ogives intersect; also x at height N/2; N is the total frequency.
First quartile Q1: x-value at height N/4 on the less-than ogive; One quarter of items lie below it.
Third quartile Q3: x-value at height 3N/4 on the less-than ogive; Three quarters of items lie below it.

How to solve Ogives and Locating Median and Quartiles Graphically questions

Use this method for any question on drawing ogives or reading the median and quartiles.

1Check that classes are continuous. If they are inclusive like 10-19, 20-29, convert them to 9.5-19.5, 19.5-29.5 by adjusting the limits.
2Find N, the total frequency, and compute N/4, N/2 and 3N/4.
3Build the less-than cumulative table: each upper limit with its running total. Add the first lower limit with 0.
4Build the more-than table: each lower limit with the total of that class and all later classes. Add the last upper limit with 0.
5Plot the points on the same axes with classes on the x-axis and cumulative frequency on the y-axis. Join them with smooth curves.
6For the median, read the x-value of the intersection of the two curves. It should sit at height N/2.
7For Q1 and Q3, go across from N/4 and 3N/4 to the less-than curve, then down to the x-axis.
8In an MCQ, check that your answer lies inside the correct class by looking at the cumulative frequencies.

Quickest way: Interpolate instead of drawing

When to use it: Use this when the MCQ gives a frequency table and asks for the median or a quartile read from an ogive. The exam has no graph paper. Linear interpolation gives an approximation of the ogive reading, since a hand-drawn smooth curve may differ slightly. The interpolated value is the same as the formula-based median.

Compute N/2 (or N/4, 3N/4) and find the first class whose cumulative frequency reaches it.
Read the answer as a straight-line position inside that class: lower limit + (target − cumulative frequency before) ÷ class frequency × class width.
Check that the answer lies between the class limits.
Eliminate options outside that class at once. Usually only one option is left.

Common mistakes in Ogives and Locating Median and Quartiles Graphically

Plotting cumulative frequencies against class midpoints.
Students mix up the ogive with the frequency polygon.
Fix: For a less-than ogive use upper limits. For a more-than ogive use lower limits.
Using the wrong limit for the more-than ogive.
Students copy the upper limits used for the less-than curve.
Fix: A more-than curve means 'this value or more', so use lower limits.
Reading the median at height N instead of N/2.
Students forget the median splits the data into two halves.
Fix: Compute N/2 first and mark it on the y-axis before reading across.
Skipping the starting point with frequency 0.
The table has no row for it.
Fix: Add (first lower limit, 0) for the less-than curve and (last upper limit, 0) for the more-than curve.
Drawing the curves with straight segments and different scales.
Rushing or using two separate graphs.
Fix: Draw both on one graph, one scale, and join points with a smooth curve.
Not converting inclusive classes to exclusive classes.
Class limits like 10-19 look usable as they are.
Fix: Subtract 0.5 from lower limits and add 0.5 to upper limits before plotting.

Worked examples

Example 1

Marks of 40 students: 0-10: 4, 10-20: 6, 20-30: 12, 30-40: 10, 40-50: 8. Find the median graphically, using the interpolated ogive reading.

Show the solution

N = 4 + 6 + 12 + 10 + 8 = 40, so N/2 = 20.
Less-than cumulative frequencies: below 10 = 4, below 20 = 10, below 30 = 22, below 40 = 32, below 50 = 40.
More-than cumulative frequencies: 0 or more = 40, 10 or more = 36, 20 or more = 30, 30 or more = 18, 40 or more = 8.
Height 20 first falls in the class 20-30, since 10 < 20 ≤ 22.
On the less-than ogive the curve rises from (20, 10) to (30, 22). Take it as a straight line inside the class.
Median = 20 + (20 − 10) ÷ 12 × 10 = 20 + 8.33 = 28.33 approximately.
Check with the more-than ogive, taking it as a straight line from (20, 30) to (30, 18): at x = 28.33 the value is 30 − (28.33 − 20) ÷ 10 × 12 = 30 − 10 = 20. This matches the height N/2 = 20, so the curves cross at x = 28.33.

Answer: Median ≈ 28.33 marks, where the two ogives cross at height 20.

Example 2

Using the same data (N = 40), find Q1 and Q3 from the less-than ogive.

Show the solution

Q1 is at height N/4 = 10. Q3 is at height 3N/4 = 30.
Less-than values: below 10 = 4, below 20 = 10, below 30 = 22, below 40 = 32, below 50 = 40.
For Q1, the height 10 is exactly at the point (20, 10), so Q1 = 20.
For Q3, the height 30 lies between (30, 22) and (40, 32), so the class is 30-40.
Q3 = 30 + (30 − 22) ÷ 10 × 10 = 30 + 8 = 38.

Answer: Q1 = 20 marks and Q3 = 38 marks.

Exam tips

Convert inclusive classes to exclusive ones before doing anything else, and note which curve starts or ends at zero.
Remember the pairing: less-than uses upper limits, more-than uses lower limits. Many MCQs test only this.
The median is the x-value at the crossing of the two ogives, and the height there is N/2. Use this to eliminate options.
Use interpolation inside the correct class to get a number fast, then pick the closest option. It approximates the ogive reading, so a nearby option may be the intended one.

Practice questions from Statistical Representation of Data