Explore advanced tenets of statistics (measures of central tendency and dispersion: mean, variance and standard deviation by coding method, quartiles, and percentiles)

After obtaining $\sum f_i u_i$ we also need $\displaystyle\sum f_i u_i^2$.
The variance of grouped data is

and the standard deviation is $\sigma=\sqrt{\sigma^{2}}$.

Worked example (continuing the table above)

From the table we have $\sum f_i u_i^{2}= ( (-3)^2\! \times6)+\dots+(3^2\!\times3)=140$.

(If a smaller class width were used, the numbers would be smaller; the method stays the same.)

For grouped data the $i^{\text{th}}$ quartile is

where

• $L$ = lower real limit of the quartile class,
• $\sum f_b$ = cumulative frequency before the quartile class,
• $f_m$ = frequency of the quartile class,
• $c$ = class width.

• Interquartile range (IQR) = $Q_3-Q_1$.
• Semi‑interquartile range (SIR) = $\dfrac{Q_3-Q_1}{2}$.

Worked example (internet‑time data)

$N=60$.

• For $Q_1$: $\frac{N}{4}=15$ → the 15th observation lies in class 13‑15.
  $L=12.5,\;\sum f_b=4,\;f_m=13,\;c=3$.

• For $Q_3$: $\frac{3N}{4}=45$ → class 22‑24.
  $L=21.5,\;\sum f_b=36,\;f_m=24$.

The $p^{\text{th}}$ percentile follows the same pattern as quartiles:

The position $\frac{pN}{100}$ tells you which class contains the percentile.

Worked example (marks data)

$N=49$.

• $P_{50}$ (the median): position $=\frac{50\times49}{100}=24.5$.
  Class 15‑20 → $L=15,\;\sum f_b=11,\;f_m=15,\;c=5$.

• $P_{70}$: position $=\frac{70\times49}{100}=34.3$.
  Class 20‑25 → $L=20,\;\sum f_b=26,\;f_m=10$.

Thus 50 % of students scored below 19.5 marks and 70 % below 24.15 marks.

A market vendor in Dar es Salaam can use the coding method to compute the average daily sales (mean) and how much sales vary from day to day (standard deviation) from a table of weekly sales figures. Knowing the interquartile range helps the vendor identify the typical range of sales, while percentiles can be used, for example, to find the sales level that 90 % of days do not exceed—useful for planning stock and cash flow in a small‑scale shop.