Statistics

Why This Matters

Imagine your school collects the marks of 200 students, or a city records the daily temperature for a year. That’s a flood of numbers. Nobody learns anything by staring at 200 figures — what we want is one number that stands in for the whole pile: a typical mark, a typical temperature, the size that shows up most often.

In Class 9 you found the mean, median and mode of a short list of numbers you could read one by one. But real data is usually too big to list individually, so it gets squeezed into a table of class intervals — “10–25 marks: 2 students, 25–40 marks: 3 students, …”. Once the exact values are hidden inside intervals, the old formulas don’t apply as-is.

This chapter teaches you how to pull the three measures of central tendency — mean, mode and median — out of grouped data. You’ll learn three ways to find the mean (and why, magically, they all give the same answer), a formula that finds the mode inside the busiest interval, and a formula that finds the exact middle value. These are the tools that turn a wall of numbers into a single, meaningful summary.

The Big Idea

Grouped data hides the exact values inside class intervals, so we work with each interval’s mid-point (class mark) as its stand-in. The mean is the average of those mid-points weighted by how many values fall in each class — found by the direct, assumed-mean or step-deviation method, which only differ in how tidy the arithmetic is, never in the answer. The mode lives inside the class with the highest frequency (the modal class), and the median lives inside the class where the running total of frequencies first reaches halfway (the median class). Each has a formula that pinpoints the value within that class.

Let’s Break It Down

A quick word on class marks, because all of statistics rests on them. For grouped data we assume every value in a class sits at its centre. The mid-point (class mark) is just the average of the two limits:

class mark = (lower limit + upper limit) / 2

So for the class 10–25 the mark is (10 + 25)/2 = 17.5. This single number represents the whole class.

Mean — the direct method

The mean of grouped data is the sum of (each class mark × its frequency) divided by the total frequency:

x̄ = Σfᵢxᵢ / Σfᵢ

where xᵢ is the class mark and fᵢ the frequency of the i-th class. You build one extra column for the products fᵢxᵢ, add it up, and divide by the total number of observations.

Consider the marks of 30 students:

Class interval	Frequency (fᵢ)	Class mark (xᵢ)	fᵢxᵢ
10–25	2	17.5	35.0
25–40	3	32.5	97.5
40–55	7	47.5	332.5
55–70	6	62.5	375.0
70–85	6	77.5	465.0
85–100	6	92.5	555.0
Total	Σfᵢ = 30		Σfᵢxᵢ = 1860

Mean — the assumed-mean method

When the class marks are big (say 200, 300, 400…), multiplying each by its frequency is tedious. The trick: guess a mean called the assumed mean a (pick a class mark near the middle), and work with how far each class mark sits from it. That gap is the deviation dᵢ = xᵢ − a. Because we subtracted a from every value, we just add it back at the end:

x̄ = a + (Σfᵢdᵢ / Σfᵢ)

It gives the exact same mean because subtracting a constant from every value lowers the mean by that constant — adding it back restores it. Take a = 47.5 for the marks data:

Class interval	fᵢ	xᵢ	dᵢ = xᵢ − 47.5	fᵢdᵢ
10–25	2	17.5	−30	−60
25–40	3	32.5	−15	−45
40–55	7	47.5	0	0
55–70	6	62.5	15	90
70–85	6	77.5	30	180
85–100	6	92.5	45	270
Total	30			Σfᵢdᵢ = 435

Mean — the step-deviation method

Look at the deviations −30, −15, 0, 15, 30, 45: they’re all multiples of 15, the class size. So divide each by h (the class size) to get even smaller whole numbers uᵢ = (xᵢ − a) / h. Since we shrank every deviation by a factor of h, we scale back up by multiplying by h at the end:

x̄ = a + h × (Σfᵢuᵢ / Σfᵢ)

Class interval	fᵢ	xᵢ	uᵢ = (xᵢ − 47.5)/15	fᵢuᵢ
10–25	2	17.5	−2	−4
25–40	3	32.5	−1	−3
40–55	7	47.5	0	0
55–70	6	62.5	1	6
70–85	6	77.5	2	12
85–100	6	92.5	3	18
Total	30			Σfᵢuᵢ = 29

Why all three agree. They are the same calculation dressed differently. The assumed-mean method just shifts every value down by a, then adds a back. The step-deviation method does that shift and divides by h, then multiplies by h back. No information is lost — so the mean must be identical. Choose whichever keeps the arithmetic lightest:

Method	Formula	Best when	Extra columns
Direct	x̄ = Σfᵢxᵢ / Σfᵢ	xᵢ and fᵢ are small	fᵢxᵢ
Assumed mean	x̄ = a + Σfᵢdᵢ / Σfᵢ	xᵢ are large	dᵢ, fᵢdᵢ
Step-deviation	x̄ = a + h × (Σfᵢuᵢ / Σfᵢ)	deviations share a common factor h	dᵢ, uᵢ, fᵢuᵢ

Mode of grouped data

The mode is the value that occurs most often. In grouped data you can’t see individual values, but you can spot the class with the highest frequency — the modal class. The mode is some value inside that class, given by:

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

where l = lower limit of the modal class, f₁ = frequency of the modal class, f₀ = frequency of the class just before it, f₂ = frequency of the class just after it, and h = class size. The idea: the mode leans toward whichever neighbour is taller. If the class after the modal class is busier than the one before, the mode shifts that way.

Tap to zoom

Family-size survey of 20 households:

Family size	Number of families
1–3	7
3–5	8
5–7	2
7–9	2
9–11	1

Median of grouped data

The median is the middle value — half the data lies below it, half above. To find it in grouped data, first build the cumulative frequency (cf): a running total of frequencies down the table. The cumulative frequency of a class is the sum of its own frequency and all the frequencies before it, so it tells you “how many observations are at most this far down”.

Then locate the median class — the first class whose cumulative frequency reaches or passes n/2 (where n = total frequency). The median sits inside it:

Median = l + [(n/2 − cf) / f] × h

where l = lower limit of the median class, n = total frequency, cf = cumulative frequency of the class before the median class, f = frequency of the median class, h = class size. The bracket measures how far past the start of the median class we must walk to reach the (n/2)-th observation.

Marks of 53 students, with the cumulative-frequency column added:

Marks	Frequency (f)	Cumulative frequency (cf)
0–10	5	5
10–20	3	8
20–30	4	12
30–40	3	15
40–50	3	18
50–60	4	22
60–70	7	29
70–80	9	38
80–90	7	45
90–100	8	53

There’s also a handy empirical relationship linking the three: 3 Median = Mode + 2 Mean. It’s approximate, but useful for a quick sanity check or to find the third measure when you know the other two.

Common Mistakes

Quick Check

Practice Problems

Easy

Medium

Challenge

Summary

You should now be able to explain:

• For grouped data, each class is represented by its class mark = (lower + upper)/2.
• Mean — direct method: x̄ = Σfᵢxᵢ / Σfᵢ.
• Mean — assumed-mean method: x̄ = a + (Σfᵢdᵢ / Σfᵢ), where dᵢ = xᵢ − a; handy when class marks are large.
• Mean — step-deviation method: x̄ = a + h × (Σfᵢuᵢ / Σfᵢ), where uᵢ = (xᵢ − a)/h; handy when deviations share the factor h.
• All three mean methods give the same answer — they only differ in how light the arithmetic is.
• Mode: the modal class is the one with the highest frequency, and Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.
• Median: find n/2, locate the median class (first cumulative frequency ≥ n/2), then Median = l + [(n/2 − cf)/f] × h, with cf from the class before the median class.
• A quick link between them: 3 Median = Mode + 2 Mean.
• Mode and median formulas need continuous class intervals — convert inclusive classes first.

What’s Next

You’ve learnt to summarise data that has already happened — the typical mark, the most common family size, the middle weight. Next, in Probability, you flip to the future: instead of describing what did occur, you measure how likely something is to occur. From a single tossed coin to a deck of cards, you’ll put a number between 0 and 1 on uncertainty itself.