Q.Explain how the median of a continuous series can be located graphically using an ogive, and verify the graphic method against the algebraic median found in Worked Example 8.

Deciles divide a distribution into ten equal parts.

Partition values divide an ordered distribution into equal groups:

• Quartiles ($Q_1,Q_2,Q_3$) $\rightarrow$ 4 equal parts.
• Deciles ($D_1,D_2,\ldots,D_9$) $\rightarrow$ 10 equal parts.
• Percentiles ($P_1,\ldots,P_{99}$) $\rightarrow$ 100 equal parts.

The nine deciles therefore split the data into ten equal parts.

Quartile deviation is a dispersion measure, so it is the odd one out.

A measure of central value (central tendency / positional average) points to a representative or locating value of the data:

• Median — the middle value,
• Mode — the most frequent value,
• Quartile — a positional value dividing the data into four parts.

Quartile deviation, defined as $\dfrac{Q_3-Q_1}{2}$, instead measures the scatter of the middle 50% of the data — a measure of dispersion. Hence it is not a measure of central value.

Positional measures are located by rank (Median, Quartiles); Standard deviation is a computed average measure — the odd one out.

Positional measures are found by their place in the arranged data:

• Median — the middle positional value,
• Quartiles — positional values at the quarter points.

Standard deviation is obtained by a formula involving the squared deviations of all observations about the mean, $\sigma=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n}}$ — it is not defined by a position. Hence it is not a positional measure.

Three quartiles ⇒ four parts.

Quartiles are positional values that split the arranged data at the $25\%$, $50\%$ and $75\%$ points. These three cut-points $Q_1, Q_2, Q_3$ divide the whole distribution into $4$ equal parts (just as deciles give $10$ parts and percentiles give $100$).

Median $=P_{50}=Q_2=D_5$.

Percentiles divide an ordered distribution into 100 equal parts. The $P_{50}$ value has 50% of the data below it — which is precisely the definition of the median (and equally the second quartile $Q_2$ and fifth decile $D_5$).

Correct option: (b) Quartiles.

Positional values that divide an arranged data set into equal parts are called partition values. The ones that divide it into four equal parts are the quartiles $Q_1, Q_2, Q_3$. (Deciles divide into 10 parts, percentiles into 100.)