Measure of Central Tendency

Chapter in a nutshell: A measure of central tendency is a single value representing a data set — the mean, median and mode. You compute these for raw, discrete (frequency) and grouped (class-interval) data, using different formulas for each.

1. Key Ideas

• Mean = arithmetic average. Median = the middle value (when ordered). Mode = the most frequent value.
• Data may be raw, discrete with frequencies, or grouped into class intervals.
• Class mark (mid-value) $=\dfrac{\text{lower limit}+\text{upper limit}}{2}$.

2. Formulas (current chapter — all of them)

• Raw data: $\bar x=\dfrac{\sum x}{n}$
• Discrete/grouped (direct): $\bar x=\dfrac{\sum f x}{\sum f}$
• Assumed-mean method: $\bar x=A+\dfrac{\sum f d}{\sum f}$, where $d=x-A$
• Step-deviation method: $\bar x=A+\dfrac{\sum f u}{\sum f}\times h$, where $u=\dfrac{x-A}{h}$

Median

• Raw (n odd): the $\left(\frac{n+1}{2}\right)$th value; (n even): mean of the $\frac n2$th and $(\frac n2+1)$th values.
• Grouped: $\text{Median}=l+\dfrac{\frac{n}{2}-cf}{f}\times h$ (l = lower boundary of median class, cf = cumulative freq before it, f = its freq, h = class width).

Mode

• Raw/discrete: the value with the highest frequency.
• Grouped: $\text{Mode}=l+\dfrac{f_1-f_0}{2f_1-f_0-f_2}\times h$ ($f_1$ = modal-class freq, $f_0$ before, $f_2$ after).

Empirical relation: $\text{Mode}=3\,\text{Median}-2\,\text{Mean}$.

3. Prerequisite Formulas / Ideas (in full)

• Frequency distribution & cumulative frequency (cf).
• Class mark $=\frac{l+u}{2}$; class size $h=u-l$.
• $\sum f=n$ (total frequency).
• Converting inclusive classes to exclusive (boundaries) by adjusting ±0.5.

4. Worked Example (one, for the method)

5. Common Mistakes to Avoid

• Dividing $\sum fx$ by $n$ wrongly — divide by $\sum f$.
• Using frequency instead of cumulative frequency in the median formula.
• Wrong modal class (highest frequency, not highest class mark).
• Forgetting to convert inclusive classes to boundaries before using h.
• Mixing up $f_0$ (before) and $f_2$ (after) the modal class.

6. Likely Exam Questions (with crisp answers)

1. Name the three measures of central tendency. → Mean, median, mode.
2. Formula for the mean of grouped data (direct)? → $\frac{\sum fx}{\sum f}$.
3. State the step-deviation formula. → $\bar x=A+\frac{\sum fu}{\sum f}\times h$.
4. Median formula for grouped data? → $l+\frac{n/2-cf}{f}\times h$.
5. Mode formula for grouped data? → $l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$.
6. Empirical relation between the three? → Mode = 3 Median − 2 Mean.
7. What is a class mark? → The mid-value of a class interval.
8. Median of 3, 5, 7, 9 (even count)? → (5+7)/2 = 6.
9. Mode of 2,2,3,4,4,4,5? → 4.
10. What does cf stand for? → Cumulative frequency.

Extended & Prerequisite Formulas (full)

• Mean: $\bar x=\tfrac{\sum x}{n}=\tfrac{\sum fx}{\sum f}$; assumed-mean $A+\tfrac{\sum fd}{\sum f}$; step-deviation $A+\tfrac{\sum fu}{\sum f}h$ (u=(x−A)/h).
• Median (grouped) $l+\tfrac{n/2-cf}{f}h$; Mode (grouped) $l+\tfrac{f_1-f_0}{2f_1-f_0-f_2}h$; Mode=3Median−2Mean.
• Prereq: class mark (l+u)/2; class size h; cumulative frequency; Σf=n; inclusive→exclusive (±0.5).