I’m trying to get my head around cumulative frequency, and my graph keeps doing interpretive dance when I just want a nice smooth ogive. I have grouped data with class intervals, and I’m not sure where to plot the points for the cumulative totals. Do I put them at the midpoints of the classes, or at the upper class boundaries of each interval? I’ve seen both in different places, and my estimated median changes depending on what I pick.
Personal saga: on a test last term, I plotted everything at the midpoints and my median ended up way off. My teacher gave me that sympathetic “you stacked your blocks on the wrong shelf” look. Since then, I’ve been nervous about whether my curve should climb like stairs at the ends of classes or hover at the middle like a confused pigeon.
Analogy that may be completely wrong: is cumulative frequency like pouring sand into a series of buckets (so the total should be measured at the rim of each bucket), or like checking the weight at the balancing point (the midpoint)?
Could someone explain, plainly, where the cumulative points should go for a “less than” cumulative frequency graph? When, if ever, are midpoints okay? How do you handle open-ended classes or different class widths so that the median and quartiles you read from the curve actually make sense?
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3 Responses
Plot “less than” cumulative frequencies at the upper class boundaries (you’re topping up each bucket to its rim), though if all classes are equal width you can kinda get away with midpoints and the median usually only wobbles a smidge. Example: for 0–10:3, 10–20:5, 20–30:2, plot (10,3), (20,8), (30,10) and read the median where y = n/2 = 5 hits the curve; a friendly walkthrough is here: https://www.mathsisfun.com/data/cumulative-frequency.html
Go with the upper class boundaries for a “less than” cumulative frequency curve: each plotted point should be (upper boundary of the class, running total up to and including that class). That’s because F(x) literally means “how many values are ≤ x,” so the total only updates when you pass a class’s upper edge; plotting at midpoints shifts those updates left and can send your median wandering. Midpoints are great for frequency polygons and for estimating the mean from grouped data, but not for an ogive-unless the classes are tiny and you’re knowingly making a rough approximation. If your classes are written as 10–19, 20–29, etc., use the continuous boundaries (9.5, 19.5, 29.5, …) so the ogive aligns with the intended scale. Different class widths are no problem: just keep the x-values at the true boundaries and the curve will space itself correctly; you still read the median at N/2 on the y-axis (quartiles at N/4 and 3N/4), go across to the curve, then down to x. For open-ended classes, a top open class is fine if the median lies below it; if the median would fall inside that open class, you must assume a plausible upper boundary (e.g., by continuing the previous width) or switch tactics. For a bottom open class, it’s often cleaner to draw a “greater than” ogive (plot at lower boundaries with cumulative ≥ counts) or to assign a reasonable lower boundary so your curve has a proper start. Tiny analogy: think of filling buckets from left to right-the cumulative amount “clicks” up at each bucket’s rim (upper boundary), not at its balancing point (midpoint); midpoints tell you where the bucket would balance (useful for means), but rims tell you when the total has actually increased.
For a “less than” ogive, plot each cumulative total at the upper class boundary (begin at the lowest lower boundary with 0) and join them-midpoints are for frequency polygons/mean estimates, not cumulative graphs, which is why they shove the median by roughly half a class width. Unequal widths are fine because you’re just accumulating counts and then using linear interpolation within the median’s class; with open-ended classes you can plot up to the last finite boundary but can’t reliably interpolate a quantile that falls inside an open-ended class without an assumed boundary-something I learned the hard way on a quiz until my teacher said, “pour the sand to the rim, not the middle.”