Ogives and quartiles

Plot a running total of frequencies and something remarkable happens: the whole shape of a grouped distribution collapses into a single smooth curve, and every summary statistic you need — median, quartiles, inter-quartile range — can be read straight off it with a ruler and a pencil, no algebra required. This curve is the ogive, and in the ICSE Class 10 syllabus it means specifically the "less than" ogive, built from cumulative frequencies plotted against the upper boundary of each class. The idea is simple: cumulative frequency grows from $0$ up to $N$ (the total number of observations) as you sweep across the classes, so a graph of cumulative frequency against class boundary is automatically a rising curve. Marking $N/2$ on the vertical axis and reading across to the curve gives the median; $N/4$ gives the lower quartile $Q_1$ ; $3N/4$ gives the upper quartile $Q_3$ . This article builds the ogive from a frequency table, uses it to read off all three statistics, and checks the results against the same 60-student data set used in the shorter version of this article.

The core insight

A "less than" ogive plots cumulative frequency against the upper class boundary, so the median, $Q_1$ and $Q_3$ can all be read off graphically by locating $N/2$ , $N/4$ and $3N/4$ on the vertical axis.

The core idea

Step 1 — build the cumulative frequency table. For each class, the cumulative frequency is the count of all observations less than that class's upper boundary — that is, the running sum of the class frequency plus every frequency before it. This table is the raw material for the ogive; nothing is plotted yet.

Step 2 — plot and join. Each cumulative frequency is plotted at the upper boundary of its class, giving points such as $(10, 5), (20, 13), (30, 25), \dots$ . A point at the lower boundary of the very first class with cumulative frequency $0$ is included so the curve genuinely starts from zero. The points are joined with a smooth freehand curve, not straight polyline segments — this smooth rising curve is the ogive itself.

Step 3 — locate statistics using $N$ . If $N = \Sigma f$ is the total frequency, the median, lower quartile and upper quartile correspond to these cumulative-frequency levels:

\text{Median} \leftrightarrow \frac{N}{2}, \qquad Q_1 \leftrightarrow \frac{N}{4}, \qquad Q_3 \leftrightarrow \frac{3N}{4}

where: $N$ is the total number of observations, $Q_1$ is the value below which one-quarter of the data lies, and $Q_3$ is the value below which three-quarters of the data lies. To read any of these off the graph: mark the target cumulative frequency on the vertical axis, draw a horizontal line to meet the curve, then drop a perpendicular to the horizontal axis — the reading there is the statistic.

Step 4 — combine into a spread measure. Once $Q_1$ and $Q_3$ are known, the inter-quartile range summarises how spread out the middle half of the data is:

\text{IQR} = Q_3 - Q_1

where: IQR is the inter-quartile range, in the same units as the data. Because it ignores the extreme quarter at each end, the IQR is far less distorted by outliers than the full range (maximum minus minimum).

Figure

A rising smooth ogive curve with horizontal and vertical guide lines showing how the median is read off at cumulative frequency thirty

Derivation ALocating the median, quartiles and IQR from cumulative frequency

The goal is to show precisely how the three horizontal levels $N/2$ , $N/4$ and $3N/4$ on the ogive's vertical axis correspond to the median, lower quartile and upper quartile, and how subtracting the two quartile readings produces the inter-quartile range, using the 60-student marks data from the worked example below as the running check.

Step 1. State what the ogive represents. Every point on the curve is a pair (mark value $x$ , number of students scoring less than $x$ ). Since the curve rises from $0$ to $N$ , it is a graphical version of "what fraction of the data lies below this value" — exactly the information needed to locate any percentile of the data, including the median and quartiles.

Step 2. Fix the median level. The median is the value below which exactly half the data lies, so its cumulative frequency is $N/2$ . For $N = 60$ , that level is $60/2 = 30$ . Locating $30$ on the vertical axis and reading across to the curve, then down to the horizontal axis, gives the mark value at which exactly half the class scored less — the median.

Step 3. Fix the quartile levels. $Q_1$ is the value below which one quarter of the data lies, so its cumulative frequency level is $N/4 = 60/4 = 15$ . $Q_3$ is the value below which three quarters of the data lies, so its level is $3N/4 = 3 \times 60/4 = 45$ . Both are located on the ogive exactly as the median was, by drawing a horizontal line across to the curve and dropping a perpendicular.

Step 4. Combine into the spread. Once the two readings $Q_1$ and $Q_3$ are taken from the graph, subtracting gives $\text{IQR} = Q_3 - Q_1$ , the width of the mark range containing the middle $50\%$ of the class. This single subtraction is why the IQR is preferred over the full range when a few very high or very low scores would otherwise distort the picture of "typical" spread. Check against the worked example: with $N=60$ , the levels $30$ , $15$ and $45$ read off the ogive as median $\approx 33$ , $Q_1 \approx 21.7$ , $Q_3 \approx 40.9$ , giving $\text{IQR} \approx 19.2$ — consistent across every reading.

Worked example

The marks of 60 students are grouped as follows. Draw a "less than" ogive and use it to find the median, $Q_1$ , $Q_3$ and the inter-quartile range.

Marks	0–10	10–20	20–30	30–40	40–50	50–60
No. of students	5	8	12	18	10	7

Step 1. Build the cumulative frequency table by running totals: less than $10 \to 5$ ; less than $20 \to 5+8=13$ ; less than $30 \to 13+12=25$ ; less than $40 \to 25+18=43$ ; less than $50 \to 43+10=53$ ; less than $60 \to 53+7=60$ .

Step 2. Plot the points $(0,0), (10,5), (20,13), (30,25), (40,43), (50,53), (60,60)$ with marks on the horizontal axis and cumulative frequency on the vertical axis, and join them with a smooth curve — this is the ogive.

Step 3. Locate the median. $N = 60$ , so $N/2 = 30$ . A horizontal line from $30$ meets the curve between the plotted points $(30,25)$ and $(40,43)$ ; dropping a perpendicular reads approximately marks $= 33$ . The median is approximately 33 marks.

Step 4. Locate $Q_1$ and $Q_3$ . $N/4 = 15$ : the horizontal line from $15$ meets the curve between $(20,13)$ and $(30,25)$ , reading $Q_1 \approx 21.7$ . $3N/4 = 45$ : the horizontal line from $45$ meets the curve between $(40,43)$ and $(50,53)$ , reading $Q_3 \approx 40.9$ .

Step 5. Compute the inter-quartile range. $\text{IQR} = Q_3 - Q_1 \approx 40.9 - 21.7 = 19.2$ .

The median is approximately 33 marks, $Q_1 \approx 21.7$ , $Q_3 \approx 40.9$ , and the inter-quartile range is approximately 19.2 marks — the middle half of the class scored within a band about 19 marks wide.

Competitive stretch

If a question instead asks for the number of students scoring between $Q_1$ and $Q_3$ , the answer follows directly from the definitions: exactly half the class, since $Q_1$ marks the 25th percentile and $Q_3$ the 75th percentile, so $75\% - 25\% = 50\%$ of $60 = 30$ students lie between them — no fresh graph reading needed, just the definition of a quartile.

Common traps

Plotting cumulative frequency at the lower boundary instead of the upper. A "less than" ogive always uses the upper boundary of each class.
Skipping the starting point at zero. Without the point (lower boundary of the first class, 0), the ogive does not correctly begin from zero cumulative frequency, and readings near the start are unreliable.
Drawing a jagged polyline instead of a smooth curve. Examiners expect a careful freehand smooth curve through the points, not straight-line joins.
Mixing up $N/2$ , $N/4$ and $3N/4$ . The median uses $N/2$ , the lower quartile $Q_1$ uses $N/4$ , and the upper quartile $Q_3$ uses $3N/4$ — swapping these gives an entirely different statistic.

Ask the tutor: if your ogive readings don't line up closely with the algebraic median/quartile formulas, ask the AI tutor to check your graph scale and point plotting.

Why is only the "less than" ogive tested in ICSE Class 10, not the "more than" ogive?

The ICSE syllabus restricts Class-X statistics to the "less than" cumulative frequency curve, so that is the only version you need to construct and read from in exams. The "more than" ogive, built from cumulative frequencies counted downward from the highest class, is a valid idea used in some other courses, but unless a question explicitly asks you to construct one for comparison, stick to the "less than" curve — using the wrong one loses marks even if your reading is numerically sound.

My graphical median from the ogive does not exactly match the median formula for grouped data — is that a mistake?

Not necessarily. The ogive reading depends on how carefully you draw the smooth curve and how precisely you read the graph, so a small difference of half a mark or so from the algebraic grouped-median formula is normal and expected — both methods are estimates for grouped data, not exact values. If the two answers differ by several marks, re-check your cumulative frequency table and the class boundary you plotted each point against, since that is the most common source of a real error.

Do I have to draw the ogive to full graph-paper accuracy, or can I sketch it roughly for working?

In an exam you should use graph paper with a sensible, consistent scale on both axes, since the whole method depends on reading values accurately off the curve — a rough sketch will not give a reading close enough to the expected answer. Choose a scale where the curve occupies a good portion of the page, plot every cumulative frequency point precisely, and draw one smooth freehand curve through them before taking any readings.

Ogives and quartiles

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads