Statistics Class 10 — Mean, Median, Mode Shortcuts & Examples

Q: What are the three methods to find the mean of grouped data?

The three methods are the direct method (Mean = Σfixi/Σfi), the assumed mean method (Mean = a + Σfidi/Σfi, where di = xi−a), and the step deviation method (Mean = a + (Σfiui/Σfi) × h, where ui = (xi−a)/h). All three give the same result; the assumed mean and step deviation methods are just faster for large or awkward numbers.

Q: 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 (the class with the highest frequency), f1 is its frequency, f0 is the frequency of the class before it, f2 is the frequency of the class after it, and h is the class size.

Q: 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, n is the total frequency, cf is the cumulative frequency of the class before the median class, f is the frequency of the median class, and h is the class size.

Q: What is the empirical relationship between mean, median and mode?

The empirical relationship is Mode = 3 × Median − 2 × Mean. It is an approximate relationship, useful for estimating one measure when the other two are known, or for sanity-checking calculated values — it does not hold exactly for every data set.

Q: What are the most common mistakes in Class 10 Statistics?

Common mistakes include using the wrong class boundary for l in the mode or median formula, misidentifying which frequency is f0 versus f2 in the mode formula, forgetting to find the correct median class using cumulative frequency and n/2, and forgetting to multiply back by the class size h in the step deviation method.

Statistics Class 10 – Mean Median Mode formulas and shortcuts

Statistics Class 10: Mean, Median, Mode Shortcuts & Examples (2026 Guide)

📅 Updated Sep 16, 2026 ⏱ 13 min read 📘 CBSE / NCERT Class 10, Statistics ✍️ MyTestSeries Expert Team

Statistics Class 10 — mean median mode formulas for grouped data explained by MyTestSeries

Class 10 Statistics takes something you already know — mean, median, and mode — and applies it to grouped data, where individual values are bundled into class intervals instead of listed one by one. The formulas look longer than what you learned in earlier grades, but each one is really just the old idea adapted to work with frequency tables instead of raw numbers.

This guide walks through all three methods for finding the mean, the mode and median formulas for grouped data, and the empirical relationship that connects all three — each with a worked example — followed by the specific mistakes that quietly cost marks even when a student clearly understands the underlying idea.

3Mean Methods

6Solved Examples

7Common Mistakes

1Quick Formula Table

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1. Mean of Grouped Data — 3 Methods

Direct method: Mean = Σfᵢxᵢ / Σfᵢ
Assumed mean method: Mean = a + (Σfᵢdᵢ/Σfᵢ), where dᵢ = xᵢ − a
Step deviation method: Mean = a + (Σfᵢuᵢ/Σfᵢ) × h, where uᵢ = (xᵢ−a)/h xᵢ = class mark (midpoint) of each class · fᵢ = frequency · a = assumed mean (any class mark, usually the middle one) · h = class size. All three methods give exactly the same final answer.

💡 Quick check: Use the direct method for small, simple numbers. Switch to assumed mean or step deviation when the class marks or frequencies are large or awkward — step deviation is fastest when the class size h is constant, which it usually is.

2. Mode of Grouped Data

Mode = l + [ (f₁ − f₀) / (2f₁ − f₀ − f₂) ] × h l = lower limit of the modal class (the class with the highest frequency) · f₁ = frequency of the modal class · f₀ = frequency of the class before it · f₂ = frequency of the class after it · h = class size

3. Median of Grouped Data

Median = l + [ (n/2 − cf) / f ] × h l = lower limit of the median class · n = total of all frequencies · cf = cumulative frequency of the class just before the median class · f = frequency of the median class · h = class size

The median class is found by locating n/2 in the cumulative frequency column — it's the first class whose cumulative frequency is greater than or equal to n/2.

4. The Empirical Relationship

Mode = 3 × Median − 2 × Mean An approximate relationship — useful for a quick sanity check or for estimating one value when the other two are already known, but it won't match a directly calculated value exactly.

5. Six Fully Solved Examples

Example 1 — Mean by the direct method

Find the mean of the following grouped data:

Class	0–10	10–20	20–30	30–40	40–50
Frequency (f)	5	10	15	15	5

Class marks (x): 5, 15, 25, 35, 45. Σf = 50.
Σfx = (5×5)+(10×15)+(15×25)+(15×35)+(5×45) = 25+150+375+525+225 = 1300
Mean = 1300/50 = 26

Example 2 — Mean by the assumed mean method

Using the same data as Example 1, find the mean by the assumed mean method (take a = 25).

dᵢ = xᵢ−25: −20, −10, 0, 10, 20.
Σfᵢdᵢ = (5×−20)+(10×−10)+(15×0)+(15×10)+(5×20) = −100−100+0+150+100 = 50
Mean = 25 + (50/50) = 25 + 1 = 26 (matches Example 1 exactly, as expected)

Example 3 — Mean by the step deviation method

Using the same data again, find the mean by the step deviation method (a = 25, h = 10).

uᵢ = (xᵢ−25)/10: −2, −1, 0, 1, 2.
Σfᵢuᵢ = (5×−2)+(10×−1)+(15×0)+(15×1)+(5×2) = −10−10+0+15+10 = 5
Mean = 25 + (5/50)×10 = 25 + 1 = 26 (all three methods agree — this consistency is a useful way to check your own work)

Example 4 — Mode of grouped data

Find the mode of the following data:

Class	0–20	20–40	40–60	60–80	80–100
Frequency (f)	10	15	30	20	15

The modal class is 40–60 (highest frequency, 30). l=40, f₁=30, f₀=15, f₂=20, h=20.
Mode = 40 + [(30−15)/(2×30−15−20)] × 20 = 40 + [15/25] × 20 = 40 + 12 = 52

Example 5 — Median of grouped data

Find the median of the same data as Example 4.

Cumulative frequencies: 10, 25, 55, 75, 90. n = 90, so n/2 = 45.
The median class is 40–60 (its cf of 55 is the first to reach or exceed 45; the previous cf is 25). l=40, cf=25, f=30, h=20.
Median = 40 + [(45−25)/30] × 20 = 40 + [20/30]×20 = 40 + 13.33 ≈ 53.33

Example 6 — Checking the empirical relationship

For the data in Examples 4 and 5, the mean (by the direct method) works out to 53.33, the median is 53.33, and the calculated mode is 52. Check how closely the empirical relationship holds.

Empirical estimate: Mode ≈ 3×Median − 2×Mean = 3(53.33) − 2(53.33) = 160 − 106.67 = 53.33
The directly calculated mode is 52 — close to the empirical estimate of 53.33, but not identical. This is expected: the relationship is approximate, not an exact identity, so small differences like this are normal, not a sign of a calculation error.

6. Common Mistakes Students Make

⚠️ Mistake 1 — Using the wrong class boundary for l. l is always the lower limit of the modal or median class — using the upper limit or the class mark instead gives a completely wrong answer.

⚠️ Mistake 2 — Mixing up f₀ and f₂ in the mode formula. f₀ is the frequency of the class immediately before the modal class; f₂ is the frequency of the class immediately after it — swapping them is a very common slip.

⚠️ Mistake 3 — Finding the wrong median class. The median class is the first class whose cumulative frequency reaches or exceeds n/2 — not the class containing the highest frequency, and not simply the "middle" row of the table.

⚠️ Mistake 4 — Forgetting to multiply back by h in the step deviation method. After computing Σfᵢuᵢ/Σfᵢ, that value must still be multiplied by h before adding it to a — skipping this step is a frequent, easy-to-miss error.

⚠️ Mistake 5 — Sign errors in dᵢ or uᵢ. Class marks below the assumed mean give negative dᵢ (and uᵢ) values — dropping the negative sign anywhere in the sum throws off the entire mean calculation.

⚠️ Mistake 6 — Treating the empirical relationship as exact. Mode = 3Median−2Mean is a useful approximation, not a strict equality — don't "correct" a properly calculated mode just to force an exact match.

⚠️ Mistake 7 — Rounding intermediate values too early. Carrying a rounded class mark or cumulative frequency into further calculations compounds small errors — keep full precision until the final answer.

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7. Quick Formula Recap Table

Concept	Formula
Mean — direct method	Σfᵢxᵢ / Σfᵢ
Mean — assumed mean method	a + Σfᵢdᵢ/Σfᵢ
Mean — step deviation method	a + (Σfᵢuᵢ/Σfᵢ) × h
Mode	l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h
Median	l + [(n/2−cf)/f] × h
Empirical relationship	Mode = 3×Median − 2×Mean

8. How to Revise This Chapter Effectively

📖 Always build the full table first — class marks, dᵢ or uᵢ, and cumulative frequency — before writing any formula. Most Statistics mistakes come from misreading a value mid-calculation, not from the arithmetic itself.

✍️ Pick your assumed mean (a) as one of the middle class marks. This keeps the dᵢ and uᵢ values small and symmetric, reducing arithmetic errors.

🧮 Double-check the modal and median class before substituting into any formula. A wrong class choice invalidates every number that follows it.

🎯 Use the three mean methods to cross-check each other when you have time. If direct and step deviation give different answers, you know exactly where to look for the error.

🔁 Revisit this page before your unit test. Bookmark it, and use the Quick Formula Recap Table (Section 7) as your final five-minute revision before the exam.

📚 Also Useful on MyTestSeries

🌐 Trusted External References

NCERT Official Website — download the official Class 10 Maths textbook
CBSE Academic — official syllabus & sample papers
Central Tendency — Wikipedia — broader statistical background
Grouped Data — Wikipedia — extended theory beyond the Class 10 syllabus

9. Frequently Asked Questions

What are the three methods to find the mean of grouped data?

The direct method, the assumed mean method, and the step deviation method. All three always give the same final answer — the later two are just faster for awkward numbers.

What is the formula for the mode of grouped data?

Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h, using the modal class (highest frequency) and its neighbouring classes.

What is the formula for the median of grouped data?

Median = l + [(n/2−cf)/f] × h, using the median class found via cumulative frequency.

What is the empirical relationship between mean, median and mode?

Mode = 3×Median − 2×Mean — an approximate relationship, useful for estimation or a rough sanity check, not an exact identity.

What are the most common mistakes in Class 10 Statistics?

Using the wrong class boundary for l, mixing up f₀ and f₂ in the mode formula, choosing the wrong median class, and forgetting to multiply back by h in the step deviation method.

Conclusion: Same Ideas, New Table Format

Mean, median, and mode haven't changed conceptually from earlier grades — what's new is reading them off a grouped frequency table instead of a plain list of numbers. Once you're comfortable building the class-mark, deviation, and cumulative-frequency columns correctly, every formula in this chapter becomes a direct substitution.

Rework the six solved examples above without checking the answers first, cross-check your mean using more than one method when time allows, and remember that the empirical relationship is an estimate, not a rule to force your answers into. That combination of habits is what turns this chapter into consistent exam marks.

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Tags: Statistics Class 10, Mean Median Mode, Assumed Mean Method, Step Deviation Method, Empirical Relationship, CBSE 2026, NCERT Class 10 Maths Statistics

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