A cumulative frequency graph (also called an ogive) shows a running total of how many data points fall at or below each value in a data set. The y-axis shows the cumulative count, the x-axis shows the variable being measured, and the S-shaped curve lets you read off key statistics like the median, quartiles, and percentiles without any calculation. Once you understand the basic technique of drawing horizontal and vertical lines to and from the curve, you can extract almost any summary statistic at a glance.
What the Axes Tell You
The vertical (y) axis always represents the cumulative frequency: the running total of observations counted so far. The horizontal (x) axis represents whatever variable is being measured, such as test scores, heights, incomes, or temperatures. Each point on the curve tells you how many observations fall at or below the corresponding x-value. The final point on the curve always equals the total number of observations in the data set, because by the last class every data point has been counted.
For example, if the curve passes through the point (50, 120), that means 120 observations in the data set have a value of 50 or less.
The Core Technique: Drawing Lines
Almost everything you do with a cumulative frequency graph follows the same two-step pattern. You start on one axis, draw a straight line to the curve, then draw a second line down (or across) to the other axis. The direction depends on what you’re looking for.
If you want to know how many observations fall below a specific value, start on the x-axis at that value, draw a vertical line up to the curve, then draw a horizontal line across to the y-axis. The number you land on is the cumulative frequency.
If you want to find the value that corresponds to a specific count or position (like the median), you do the reverse. Start on the y-axis at your target count, draw a horizontal line across to the curve, then drop a vertical line down to the x-axis. The x-value you land on is your answer.
Finding the Median
The median is the middle value of your data set. To find it, divide the total frequency by 2. If your data set contains 80 observations, the median position is the 40th value. Locate 40 on the y-axis, draw a horizontal line to the right until it hits the curve, then drop straight down to the x-axis. The value you read off is the median.
This works because the curve is a running total. The point where the curve reaches half the total count is, by definition, the value that splits the data into two equal halves.
Finding Quartiles and the Interquartile Range
Quartiles divide your data into four equal parts, and you find them the same way you find the median.
- Lower quartile (Q1): Calculate one quarter of the total frequency. For 40 observations, that’s the 10th value. Read across from 10 on the y-axis to the curve, then down to the x-axis.
- Upper quartile (Q3): Calculate three quarters of the total frequency. For 40 observations, that’s the 30th value. Same process: across to the curve, then down.
The interquartile range (IQR) is simply Q3 minus Q1. It tells you how spread out the middle 50% of your data is. If Q1 is 38 and Q3 is 47, the IQR is 9. A small IQR means the central data points are tightly clustered; a large one means they’re spread out.
Finding Any Percentile
Percentiles work on the same principle, just with different fractions. To find the 90th percentile, calculate 90% of the total frequency and locate that value on the y-axis. Draw across to the curve, then down to the x-axis. The number you land on is the value below which 90% of observations fall. For the 10th percentile, use 10% of the total. You can find any percentile this way.
This is especially useful for grading or ranking. If you know that the 80th percentile of exam scores is 76.7 and the 90th percentile is 78, you can assign letter grades to exact score ranges just by reading lines off the graph.
What the Shape of the Curve Tells You
The steepness of the curve at any point reveals where your data is concentrated. A steep section means many observations fall within that narrow range of x-values, because the cumulative count is climbing quickly. A flat or gently sloping section means few observations fall in that range, so the running total barely increases.
Most cumulative frequency curves have an S-shape. The curve starts shallow (few observations at the low end), becomes steep in the middle (where most data clusters), and flattens again at the top (few observations at the high end). If the steep section is shifted to the left, the data is skewed toward lower values. If it’s shifted to the right, most observations are at the higher end.
Why Points Are Plotted at Upper Class Boundaries
One detail that trips people up: when data is grouped into ranges (like 10–20, 20–30, 30–40), each point on a cumulative frequency graph is plotted at the upper boundary of the range, not the midpoint. This is because the cumulative frequency for the class “10–20” tells you how many observations are at or below 20, not at or below 15. Plotting at the midpoint would shift the entire curve to the left and give you inaccurate readings.
The very first point on the graph sits at the lower boundary of the first class with a cumulative frequency of zero, since no data points fall below the start of the range. From there, each successive upper boundary gets a point at the running total up to that boundary.
Putting It All Together
Suppose you have a cumulative frequency graph of 200 students’ exam scores. The x-axis runs from 0 to 100 (scores), and the y-axis runs from 0 to 200 (cumulative number of students). To extract useful information, you would:
- Find the median: Locate 100 on the y-axis (half of 200), draw across to the curve, then down. If you land on 62, the median score is 62.
- Find Q1: Locate 50 on the y-axis (one quarter of 200), draw across and down. If you land on 48, Q1 is 48.
- Find Q3: Locate 150 on the y-axis (three quarters of 200), draw across and down. If you land on 74, Q3 is 74.
- Calculate the IQR: 74 minus 48 equals 26.
- Find how many students scored below 55: Start at 55 on the x-axis, draw up to the curve, then across to the y-axis. If you land on 72, then 72 students scored below 55.
- Find how many scored above 55: Subtract from the total. 200 minus 72 equals 128 students.
That last point is worth emphasizing. Cumulative frequency graphs directly show “how many below” a value, but you can always find “how many above” by subtracting from the total. This makes the graph useful for answering questions in both directions.