Quartiles and the Graphic Location of the Median (the Ogive)

Quartiles divide an ordered dataset into four equal parts instead of the two the median divides it into. $Q_1$ (the lower or first quartile) is the value below which 25% of the observations lie; $Q_2$ is simply the median itself; $Q_3$ (the upper or third quartile) is the value below which 75% of the observations lie. For a continuous series, both are located by exactly the same interpolation logic as the median, just at $\dfrac{N}{4}$ and $\dfrac{3N}{4}$ instead of $\dfrac{N}{2}$:

with $L$, $cf$, $f$ and $h$ read off the quartile class, located exactly as the median class is located (Worked Example 8 computes both quartiles alongside the median on the same data).

Graphic location of the median — the ogive. A cumulative-frequency curve is called an ogive. Two ogives can be drawn from the same continuous series:

• The "less-than" ogive plots the upper class boundary of each class against its cumulative frequency, built from the lowest class upward — it is always rising.
• The "more-than" ogive plots the lower class boundary of each class against the cumulative frequency of "this value or more", built from the highest class downward — it is always falling.

<!-- FIGURE-NEEDED: ogive — less-than and more-than cumulative-frequency curves plotted on the same axes for the Worked Example 8/9 continuous series (class boundaries 0, 10, 20, 30, 40, 50 on the x-axis; less-than cumulative frequencies 0, 5, 13, 28, 44, 50; more-than cumulative frequencies 50, 45, 37, 22, 6, 0), the two curves intersecting at one point, with a perpendicular dropped from that intersection to the x-axis to read off the median (≈28) graphically --> …