Statistics

After eleven chapters of pure mathematics, the last chapter of Class 9 introduces data — how to collect it, organise it into tables and graphs, and summarise it with single numbers like the average.

1. Types of data

2. Presenting data

Frequency distribution table. Lists each value (or class interval) and how often it occurs.

Two conventions for class boundaries:

• Exclusive form (e.g. 0–10, 10–20): a value of 10 falls in 10–20, not 0–10.
• Inclusive form (e.g. 0–9, 10–19): boundaries don't overlap. Convert to exclusive by adding/subtracting 0.5 from limits before drawing histograms.

3. Graphical representations

• Bar graph: bars with gaps; for discrete or categorical data.
• Histogram: bars without gaps; for continuous (grouped) data. Bar height = frequency density (frequency / class width) — but when all classes are equal width, height = frequency directly.
• Frequency polygon: join the midpoints of the tops of histogram bars with straight lines. Useful for comparing two distributions.

4. Measures of central tendency (for raw / ungrouped data)

For $n$ observations $x_1,x_2,\dots ,x_n$:

Mean

$\bar{x}=\frac{x_1+x_2+\cdots +x_n}{n}=\frac{\sum x_i}{n}$

Median

Arrange the data in ascending order. Then:

• If $n$ is odd: median = middle value $=\left(\frac{n+1}{2}\right)$th observation.
• If $n$ is even: median = average of the two middle values $=\frac{1}{2}\left[\left(\frac{n}{2}\right)\text{th}+\left(\frac{n}{2}+1\right)\text{th}\right]$.

Mode

The value(s) that occur most often. A dataset can be:

• Unimodal (one mode), bimodal (two modes), multimodal, or have no mode (all values different).

5. Worked example

Find the mean, median and mode of: 4, 8, 7, 4, 6, 5, 4, 9, 7, 6.

Sorted: 4, 4, 4, 5, 6, 6, 7, 7, 8, 9 ($n=10$).

Mean = (4+4+4+5+6+6+7+7+8+9)/10 = 60/10 = 6.

Median ($n=10$, even): average of 5th and 6th values = (6 + 6)/2 = 6.

Mode = value occurring most often = 4 (occurs 3 times).

6. When does each measure win?

• Mean is sensitive to extreme values (a single outlier moves it a lot).
• Median is robust against outliers — preferred for income, house prices.
• Mode is the only measure that works for categorical data (e.g. most common favourite colour).

What's next

In Class 10 you'll meet the mean of grouped data, median formula ($l+\frac{n/2-cf}{f}\times h$) and mode formula for class intervals. The intuition you build here makes those formulas instantly clickable.