Chapter 14.5 — Exercise 14.4 — Ogive Curves

Two Kinds of Ogive

A cumulative frequency curve comes in two mirror-image forms, depending on which direction the running total accumulates:

• "Less than" ogive — plot each class's upper limit against its cumulative frequency (running total increasing from 0 up to n). The curve rises left to right, shaped like an elongated S.
• "More than" ogive — plot each class's lower limit against its cumulative frequency (running total decreasing from n down to 0). The curve falls left to right, an upside-down mirror of the "less than" version.

This crossing point is what makes plotting both curves together useful beyond just illustration: wherever the two ogives intersect, that intersection's x-coordinate is guaranteed to equal the dataset's median, without needing to read n/2 off either axis first. A single "less than" ogive on its own works too — locating the point whose y-coordinate is n/2 and reading off its x-coordinate reaches the same median a different way, the method the second problem below uses directly.

Daily Income of 50 Workers: A Less-Than Ogive

Fifty workers' daily incomes, grouped into five Rs. 50-wide classes from Rs. 250–500, are converted into a "less than" cumulative table before plotting — each class's upper limit paired with the running total up to that point.

Plotting (300,12), (350,26), (400,34), (450,40), and (500,50), then joining them with a smooth curve, produces the ogive itself — an x-axis scale of Rs. 50 per unit and a y-axis scale of 10 workers per unit keeps the graph a manageable size on paper. This problem asks only for the construction, not a median reading, making it the most direct illustration of the plotting mechanics alone, before the next problem adds a median on top.

Weights of 35 Students: Reading the Median Two Ways

Thirty-five students' weights are given directly as a "less than" cumulative table — no raw frequency column at all, only the running totals themselves.

Both routes land on exactly 46.5 kg — the graph isn't an approximation standing in for the "real" formula answer, it's a genuinely independent way of reaching the identical number. Reading a graph does introduce more room for small human error than the formula does (a slightly mis-drawn curve, an imprecise ruler reading), which is exactly why worked problems like this one that verify the two methods against each other are worth doing at least once before trusting a graph reading on its own.

Production Yield of 100 Farms: A More-Than Ogive

A hundred farms' wheat yield, grouped into six classes of size 5, are converted into a "more than" cumulative table this time — each class's lower limit paired with how many farms yielded at least that much, built from the top of the range downward.

Building this table works from the bottom class upward rather than the top class downward: the last class, 75–80, is where the running total starts (16, since all 16 of those farms yield more than 75), and each earlier "more than" total adds that class's own frequency on top — 16+38=54 farms yield more than 70, then 54+24=78 yield more than 65, and so on back to the full 100 farms yielding more than 50. Plotting (50,100), (55,98), (60,90), (65,78), (70,54), and (75,16) produces a falling curve, the mirror image of the rising "less than" shape seen in the two problems above.

This problem, like the daily-income problem earlier, asks only for the construction rather than a median reading — but the same n/2 technique used on the weights ogive would still apply here in principle, just read differently: since this curve falls rather than rises, the point where it crosses y=n/2=50 would sit between the 70–75 and 75–80 rows rather than climbing toward it, on the descending part of the curve instead of the ascending part. The reading technique itself doesn't change between a rising and a falling ogive, only which direction the eye follows the curve to find that horizontal line.

Choosing a Scale

All three graphs in this exercise chose axis scales deliberately rather than plotting every unit individually — Rs. 50 per x-axis unit and 10 workers per y-axis unit for the daily-income ogive, similarly rounded choices for the other two. A scale that's too fine wastes space and makes the curve's overall shape harder to see at a glance; a scale that's too coarse compresses the interesting detail near the median into a handful of pixels where a ruler reading becomes unreliable. Matching the scale to the actual spread of the data — roughly one axis unit per class width, and a y-axis that comfortably fits the full range from 0 to n — is what keeps an ogive both accurate to read and practical to draw by hand.

The Chapter, Complete

Statistics opened with a single number, the mean, in the chapter introduction, then built out the mode in Exercise 14.2 and the median in Exercise 14.3 as two genuinely different ways of summarising the same kind of data, before this exercise turned the median into something readable straight off a graph. All four measures — mean, mode, median, and the ogive's graphical median — answer some version of the same question: given a large, messy collection of numbers, what's the single best way to describe it. Revisit Exercise 14.1 for the three mean methods this entire chapter kept coming back to for comparison.

This closes out Class 10 Mathematics in full — from real numbers and polynomials at the start of the course through to statistics and probability here at the end. Every method across all fourteen chapters shares the same underlying spirit: turn a real, sometimes messy situation into numbers, apply a clearly defined formula, and check that the answer actually makes sense against the original question.