What are the three methods for finding the mean of grouped data?

The direct method (x̄ = Σfᵢxᵢ/Σfᵢ), the assumed mean method (x̄ = a + Σfᵢdᵢ/Σfᵢ), and the step-deviation method (x̄ = a + h×Σfᵢuᵢ/Σfᵢ). All three always give exactly the same answer — choose whichever keeps the arithmetic simplest.

What is the formula for the mode of grouped data?

Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)]×h, where l is the lower limit of the modal class, f₁ its frequency, f₀ and f₂ the frequencies of the preceding and succeeding classes, and h the class size.

What is cumulative frequency?

Cumulative frequency is the running total of frequencies up to and including a given class — it answers 'how many observations fall at or below this point'.

What is the formula for the median of grouped data?

Median = l + [(n/2 − cf)/f]×h, where l is the lower limit of the median class (the first class whose cumulative frequency reaches or exceeds n/2), cf is the cumulative frequency before that class, f is its own frequency, and h is the class size.

How do I find class frequencies from a 'less than' cumulative frequency table?

Subtract each cumulative frequency value from the next one. The frequency of a class equals the cumulative frequency at its upper boundary minus the cumulative frequency at its lower boundary.

Chapter 13 · CBSE Class 10 Maths

Statistics

Two ways to learn — a crisp formula reference, or a story that makes mean/mode/median feel obvious.

Mean (3 methods)Mode of grouped dataCumulative frequencyMedian of grouped data

Also useful:Chapter 12 — Surface Areas and Volumes Chapter 14 — Probability NCERT Solutions — Class 10 Maths 1-on-1 CBSE Class 10 Tuition

CBSEClass 10MathematicsChapter 13⏱ 8 min crisp · 11 min story

Four scenes that build the ideas from scratch, step by step.

How this works

Four scenes, one thread: mean, mode, and median each answer a genuinely different question about the same data — "what's the average", "what's most common", and "what's the middle value" — and each has its own reliable formula for grouped data. Read the story, then tap "The Maths" for the formal result.

Three Roads, Same Destination

30 students' exam marks are grouped into classes. Each class is represented by its class mark (midpoint) — so the class 40-55 is represented by 47.5. Finding the mean means computing Σfᵢxᵢ/Σfᵢ — but with numbers like 47.5, 62.5, 77.5, the multiplication gets tedious fast.

Teacher

Instead of multiplying by the full class marks, subtract a reference point 'a' from each one first — say a = 47.5, the most central class mark. The differences become much smaller: −30, −15, 0, 15, 30, 45.

Direct method: x̄ = Σfᵢxᵢ/Σfᵢ = 1860/30 = 62

Assumed mean method: x̄ = a + Σfᵢdᵢ/Σfᵢ = 47.5 + 435/30 = 47.5 + 14.5 = 62

Same answer, less arithmetic. Now notice: all those deviations (−30, −15, 0, 15, 30, 45) are multiples of 15 — the class size. Dividing by 15 shrinks them to −2, −1, 0, 1, 2, 3 — tiny numbers.

Step-deviation method: x̄ = a + h×(Σfᵢuᵢ/Σfᵢ) = 47.5 + 15×(29/30) = 47.5 + 14.5 = 62

All three methods are algebraically the SAME calculation, just rearranged — they can never disagree. Pick whichever keeps the numbers smallest: direct for small data, assumed mean for large numbers, and step-deviation when the deviations share a common factor (the class size).

The Most Popular Slot

A survey of 20 households records family sizes in classes: 1-3, 3-5, 5-7, 7-9, 9-11, with frequencies 7, 8, 2, 2, 1. The class 3-5 has the highest frequency (8) — that's the modal class. But is the mode itself just "4" (the class mark)?

Student

Not quite — the mode is somewhere INSIDE that class, but exactly where depends on how the neighbouring classes compare too, not just the modal class alone.

l=3 (lower limit), h=2 (class size), f₁=8 (modal), f₀=7 (before), f₂=2 (after)

Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)]×h = 3 + [(8−7)/(16−7−2)]×2

= 3 + (1/7)×2 ≈ 3.286

Notice the formula leans on BOTH neighbours — if the class before (f₀) had been much smaller, the mode would sit further from the modal class's lower edge (closer to where the data is denser).

A classic comparison: for the earlier 30-student marks data, the mode works out to 52, while the mean was 62. These aren't errors — they answer different questions. The mean tells you the average score; the mode tells you the SINGLE most common score range. Use whichever the actual question is asking for.

Counting Up to the Middle

100 students' marks (ungrouped) need a median. Since n=100 is even, the median is the average of the 50th and 51st values once everything is sorted. Building a running total — a cumulative frequency — makes finding those exact positions easy.

Teacher

Add up frequencies as you go: "up to 25" is 26 students, "up to 28" is 50, "up to 29" is 78. Where do the 50th and 51st students fall?

Cumulative frequency reaches exactly 50 at marks=28, and 78 at marks=29

50th observation = 28, 51st observation = 29

Median = (28+29)/2 = 28.5

For GROUPED data, you can't pinpoint an exact "50th observation" the same way — the middle falls somewhere inside a class interval, not at a single class mark. So find n/2, locate the first class whose cumulative frequency reaches or passes it (the median class), then use a formula to pinpoint the exact value inside.

53 students, n/2 = 26.5. First class with cf ≥ 26.5 is 60-70 (cf=29) → median class

l=60, cf(before)=22, f=7, h=10

Median = 60 + [(26.5−22)/7]×10 = 60 + 45/7 ≈ 66.4

Reading a Real Survey

A real height survey of 51 girls is reported as "less than" cumulative data: Less than 140 → 4, Less than 145 → 11, Less than 150 → 29, and so on. There's no explicit class-by-class frequency table here at all — just running totals.

Student

I need the median, but the median formula needs the frequency AND cumulative frequency of each class separately. How do I get plain frequencies out of a "less than" table?

Frequency of a class = (cumulative frequency up to its end) − (cumulative frequency up to its start)

140-145: 11 − 4 = 7 girls; 145-150: 29 − 11 = 18 girls; and so on

Once you have both columns side by side (frequency AND cumulative frequency), the median class and formula work exactly as before — this reconstruction step is the only new skill.

The Big Picture

Every idea in this chapter branches from representing each class by its class mark, then asking one of three different questions: what's the average (mean), what's most common (mode), or what's in the middle (median) — each with its own dedicated formula.

Fastest mean method?

Depends on the numbers — all 3 always agree

Mode formula?

l + [(f₁−f₀)/(2f₁−f₀−f₂)]×h

Finding the median class?

First class where cf ≥ n/2

Median formula?

l + [(n/2−cf)/f]×h

'Less than' table → frequency?

Subtract consecutive cumulative values

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