Measures of central tendency — mean (grouped - ungrouped), median (grouped), mode (grouped)

1. Mean (average)

1.1 Ungrouped mean — derive it

WHAT: We want the "fair share". If total is $S$ and we split it equally among $n$ people: $\bar{x} = \frac{S}{n} = \frac{\sum x_i}{n}$

WHY this is the balance point: The mean is the value $m$ that makes the deviations cancel: $\sum (x_i - m) = 0 \;\Rightarrow\; \sum x_i - nm = 0 \;\Rightarrow\; m = \frac{\sum x_i}{n}$ Why this step? Setting total deviation to zero is the definition of a balance point — pushes above equal pulls below.

1.2 Grouped mean (frequency data)

When data is in classes, each class has a midpoint $x_i$ (assume every value in that class ≈ its midpoint) and a frequency $f_i$.

2. Median (the middle)

2.1 Ungrouped median

Order the data. Then:

• If $n$ is odd: median = the $\left(\frac{n+1}{2}\right)^\text{th}$ value.
• If $n$ is even: median = mean of the $\frac{n}{2}^\text{th}$ and $\left(\frac{n}{2}+1\right)^\text{th}$ values.

2.2 Grouped median — derive it

We can't see individual values, so we find where the middle person lands using cumulative frequency (cf).

Derivation (linear interpolation), WHY every piece: We need the position of the $\frac{n}{2}$-th item. Before the median class, $cf$ items are already used up. We still need $\frac{n}{2} - cf$ more items. Assume the $f$ items in the median class are spread evenly across its width $h$. So each item occupies $\frac{h}{f}$ of width. To reach $\left(\frac{n}{2}-cf\right)$ items into the class we move: $\left(\frac{n}{2}-cf\right)\times\frac{h}{f}$ past the lower boundary $l$. Add to $l$ → done.

3. Mode (the peak)

3.1 Grouped mode — derive it

The modal class = class with highest frequency $f_1$. But the true peak may lie left or right of centre depending on the neighbouring classes.

Derivation intuition, WHY: Look at how much the peak rises above the class before it, $(f_1 - f_0)$, vs the total "excess" over both neighbours, $(f_1-f_0)+(f_1-f_2) = 2f_1-f_0-f_2$. The fraction tells us how far into the modal class the true peak sits. If the class before is heavier, the mode leans left; if the class after is heavier, it leans right. Multiply by width $h$, add to $l$.

4. Worked Examples

5. Common Mistakes

6. Flashcards

Connections

• Cumulative Frequency & Ogives — median read graphically as the $n/2$ ordinate.
• Histograms & Frequency Distributions — modal class is the tallest bar.
• Measures of Dispersion — Variance & Standard Deviation — spread around the mean.
• Skewness — why Mean, Median, Mode differ; the empirical relation.
• Weighted Mean — generalisation of the grouped mean idea.
• Probability Distributions — Expected Value — mean is the expectation $E[X]$.

Concept Map

Hinglish (regional understanding)