Histograms and Ogives — chapter notes

A histogram shows a grouped frequency distribution of a continuous variable. Each class is a bar standing on its class boundaries; the bars touch, and for classes of equal width the height of a bar is the frequency of its class.

Example: the typing speeds of $40$ trainees on classes of width $10$ from $20$ have heights $4, 6, 12, 9, 6, 3$; the heights add up to $40$ and the tallest bar is $40$–$50$.

If there is a gap between classes ($10$–$19$, $20$–$29$, …), subtract $0.5$ from each lower limit and add $0.5$ to each upper limit: $9.5$–$19.5$, $19.5$–$29.5$, …

Steps: write continuous classes; choose a scale and label both axes; mark the boundaries on the horizontal axis; draw touching bars; check the heights add up to $n$.

The tallest bar is the modal bar. Join the top right corner of the bar before it to the top right corner of the modal bar, and the top left corner of the modal bar to the top left corner of the bar after it. The lines cross inside the modal class; the horizontal value below the crossing point is the estimated mode. It leans towards the taller of the two neighbouring bars.

Example: bar heights $4, 8, 20, 12, 6$ on classes of width $5$ from $10$: the lines meet at $(23, 15.2)$, so the mode is about $23$.

The less-than cumulative frequency of a class is the number of observations below its upper boundary: add the frequencies from the first class. The last cumulative frequency equals $n$.

Example: frequencies $6, 10, 8, 8, 4, 4$ give $6, 16, 24, 32, 36, 40$.

Plot each (upper boundary, cumulative frequency), add the starting point (lower boundary of the first class, $0$), and join the points with a smooth rising curve (straight segments are used for readings). The ogive never falls; it is steepest over the class with the greatest frequency (for equal widths).

Not in the ICSE 2027 syllabus: the more-than ogive; only the less-than ogive is used.

Read the ogive at these heights:

• Median: $\dfrac{n}{2}$; lower quartile $Q_1$: $\dfrac{n}{4}$; upper quartile $Q_3$: $\dfrac{3n}{4}$.
• Inter-quartile range $= Q_3 - Q_1$, the spread of the middle half of the data.

On the segment containing the height, use proportional parts. Example ($n = 40$): $Q_1 = 14$, median $25$, $Q_3 = 37.5$, so the inter-quartile range is $23.5$ minutes.

Going from a value to a count: the height of the ogive at $x$ is the number of observations below $x$; the number above $x$ is $n$ minus that height; between two marks it is the difference of the two heights.

Example ($n = 60$ marks, height $49$ at $65$): $49$ below, $60 - 49 = 11$ above. Going the other way, the cut-off for the best $20$ of $60$ is read at height $60 - 20 = 40$.

Readings from a graph are accepted within a tolerance (for example $\pm 1$ on the horizontal scale). Give the value found by proportional parts on the right segment, write the coordinates of each point read, and use a ruler and a sensible scale. Check against the table: a reading must lie in the right class.