Summary and Quick Revision

Chapter at a Glance

For grouped data you find three measures of central tendency. Build a clean table first (class, frequency, class mark or cumulative frequency), then apply the right formula. Identify what the question wants — average (mean), typical value (mode), or middle value (median).

Mean — Three Methods

• Direct: $\bar{x}=\dfrac{\sum f_ix_i}{\sum f_i}$, class mark $x_i=\dfrac{\text{lower}+\text{upper}}{2}$.
• Assumed-mean: $\bar{x}=a+\dfrac{\sum f_id_i}{\sum f_i}$, where $d_i=x_i-a$.
• Step-deviation: $\bar{x}=a+h\dfrac{\sum f_iu_i}{\sum f_i}$, where $u_i=\dfrac{x_i-a}{h}$.

All three give the same answer; shortcuts just reduce the arithmetic.

Mode, Median and Ogives

• Mode: modal class $=$ highest-frequency class; $\text{Mode}=l+\dfrac{f_1-f_0}{2f_1-f_0-f_2}\times h$ ($f_1$ modal, $f_0$ before, $f_2$ after).
• Median: median class $=$ where the cumulative frequency first crosses $\dfrac{n}{2}$; $\text{Median}=l+\dfrac{\frac{n}{2}-cf}{f}\times h$ ($cf$ = cf of the class before).
• Ogive: less-than curve uses upper limits (rising); more-than curve uses lower limits (falling). The two ogives cross at the median; equivalently read the median where the less-than ogive reaches height $\dfrac{n}{2}$.
• Empirical relation: $3\,\text{Median}=\text{Mode}+2\,\text{Mean}$ (approximate).

Common Traps to Avoid

• Class mark is the midpoint $\dfrac{\text{lower}+\text{upper}}{2}$, not the width.
• In the mode formula the denominator is $2f_1-f_0-f_2$; keep $f_0$ (before) and $f_2$ (after) in the right places; use 0 if a neighbouring class is missing.
• In the median formula, $cf$ is the cumulative frequency of the class before the median class, and $f$ is the frequency of the median class.
• Use $\dfrac{n}{2}$ (not $\dfrac{n+1}{2}$) for grouped data.
• Less-than ogive $\to$ upper limits; more-than ogive $\to$ lower limits.
• The empirical relation is approximate — only for estimating the third measure.

Final Tip: One neat table (class, $f$, $x$ or $cf$) usually answers all three questions — mean, median and mode.