Statistics | Maths Grade 10 CBSE

Play with it

Build a distribution

Drag each class's frequency. The mean, mode and median of the grouped data recompute live, and the modal class (tallest bar) is highlighted.

marks (each class width 10)drag the frequencies

0–10 310–20 7

20–30 1230–40 9

40–50 4

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The three ideas in this chapter

When data is grouped into classes, we use the class mark x (the midpoint) to stand for each class. The mean is then:

Mean = Σ(f · x) / Σf

Worked example · direct method

Classes 0–10, 10–20, 20–30 with frequencies 2, 3, 5.

Class marks x = 5, 15, 25.
Σf·x = 2·5 + 3·15 + 5·25 = 10 + 45 + 125 = 180. Σf = 10.
Mean = 180/10 = 18.

Shortcuts

The assumed-mean and step-deviation methods just make this arithmetic lighter — they give the same mean.

The modal class is the class with the highest frequency. The mode inside it is:

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

where L = lower boundary of the modal class, f₁ = its frequency, f₀ = the class before, f₂ = the class after, and h = class width.

Common mistake: using the highest frequency itself as the mode. The mode is a value inside the modal class, found with the formula — not the frequency.

The median is the middle value. For grouped data, find the median class (where the cumulative frequency first reaches n/2), then:

Median = L + [ (n/2 − cf) / f ] × h

L = lower boundary of the median class, cf = cumulative frequency before it, f = its frequency, h = class width, n = Σf.

The empirical relation

The three averages are linked: Mode = 3 Median − 2 Mean. If you know two, you can estimate the third.

Worked example · empirical relation

If mean = 20 and median = 22, estimate the mode.

Mode = 3·Median − 2·Mean = 3(22) − 2(20).
= 66 − 40 = 26.

Why this matters

Where you'll actually use this

"What's typical?" runs the modern world — salaries, exam results, cricket averages, app usage. But mean, median and mode can disagree, and knowing which to trust is a real-life superpower.

Why "average salary" can mislead

A few huge salaries pull the mean far above what most people earn, while the median (middle person) stays realistic. That's why economists report median income, not mean — the same maths you're learning decides which number is honest.

Mean vs median

Stocking & planning by the mode

A shop stocks more of the most common shoe size — that's the mode. Factories, restaurants and streaming services all plan around the most frequent choice. The mean wouldn't help; you can't sell a "size 8.37".

Mode = most frequent

🏏 Sports averages

A batting average is a mean; selectors also look at medians to judge consistency.

🏥 Public health

Median age, average BMI and modal blood group guide hospitals and policy.

📱 Product analytics

Apps track median session time and the modal feature used to decide what to build.

🔒 More real-world applications

Sports, public health, analytics and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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