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Class 11 Maths NCERT Solutions

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Statistics Class 11 Maths NCERT Solutions

The complete NCERT exercise solutions for Chapter 13, Statistics — 22 questions from Ex 13.1 to Ex 13.2, each worked through step by step in the CBSE marking pattern. Mean, variance and standard deviation, with the computational shortcut for grouped and ungrouped data.

Class:11Subject:MathsChapter:13
4 Key Formulas
DWritten byDeep Narayan
Updated
Key Concept Summary

How many questions are in NCERT Class 11 Maths Chapter 13?

Chapter 13 carries 2 exercise questions, numbered Ex 13.1 to Ex 13.2. All of them are solved step by step on this page, along with the chapter's key formulas and exam pointers.

01

Chapter Overview

Statistics measures how spread out a data set is. Three measures appear in this chapter: mean deviation (average absolute distance from a centre), variance (average squared distance from the mean) and standard deviation (the square root of variance). Grouped data uses midpoints of classes, and the shortcut method shifts the working to smaller numbers using a provisional mean A with step size h.

Board pattern

Mean deviation = Σf|xᵢ − mean or median| / Σf. Variance = Σfx²/Σf − (mean)². Shortcut: mean = A + h·Σfd/Σf and variance = h²[Σfd²/Σf − (Σfd/Σf)²], where d = (x − A)/h. For median-based deviations, first locate the median class where the cumulative frequency crosses N/2, then use the formula, or read positions for ungrouped data.
02

Exercise 13.1 — Mean Deviation

12Exercise questions

Step-by-step solution

  1. 1Mean = (4+7+8+9+10+12+13+17)/8 = 80/8 = 10.
  2. 2Deviations |x − 10|: 6, 3, 2, 1, 0, 2, 3, 7; sum = 24.
  3. 3MD = 24/8 = 3.

Final answer

MD = 3.

Step-by-step solution

  1. 1Sum = 500, n = 10 → mean = 50.
  2. 2Deviations: 12, 20, 2, 10, 8, 5, 13, 4, 4, 6; sum = 84.
  3. 3MD = 84/10 = 8.4.

Final answer

MD = 8.4.

Step-by-step solution

  1. 1Sorted: 10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18.
  2. 2n = 12 → median = (13 + 14)/2 = 13.5.
  3. 3Deviations sum = 3.5 + 5 + 1.5 + 1 + 0.5 + 5 + 7 + 4.5 = 28.
  4. 4MD = 28/12 = 7/3 ≈ 2.33.

Final answer

MD = 7/3 ≈ 2.33.

Step-by-step solution

  1. 1Sorted: 36, 42, 45, 46, 46, 49, 51, 53, 60, 72.
  2. 2n = 10 → median = (46 + 49)/2 = 47.5.
  3. 3Deviations: 11.5, 5.5, 2.5, 1.5+1.5, 1.5, 3.5, 5.5, 12.5, 24.5; sum = 70.
  4. 4MD = 70/10 = 7.

Final answer

MD = 7.

Step-by-step solution

  1. 1N = 25; Σfx = 35 + 40 + 90 + 60 + 125 = 350 → mean = 14.
  2. 2Deviations: 9, 4, 1, 6, 11.
  3. 3Σf|x − mean| = 9·7 + 4·4 + 1·6 + 6·3 + 11·5 = 158.
  4. 4MD = 158/25 = 6.32.

Final answer

MD = 6.32.

Step-by-step solution

  1. 1N = 80; Σfx = 40 + 720 + 1400 + 1120 + 720 = 4000 → mean = 50.
  2. 2Deviations: 40, 20, 0, 20, 40.
  3. 3Σf·dev = 4·40 + 24·20 + 28·0 + 16·20 + 8·40 = 1280.
  4. 4MD = 1280/80 = 16.

Final answer

MD = 16.

Step-by-step solution

  1. 1N = 26, N/2 = 13. Cumulative: 8, 14, 16, 18, 20, 26 → the 13th value is 7, so median = 7.
  2. 2Deviations: 2, 0, 2, 3, 5, 8.
  3. 3Σf·dev = 8·2 + 6·0 + 2·2 + 2·3 + 2·5 + 6·8 = 84.
  4. 4MD = 84/26 = 42/13 ≈ 3.23.

Final answer

MD = 42/13 ≈ 3.23.

Step-by-step solution

  1. 1N = 29, N/2 = 14.5. Cumulative: 3, 8, 14, 21 → the 14.5th value lies in x = 30, so median = 30.
  2. 2Deviations: 15, 9, 3, 0, 5.
  3. 3Σf·dev = 3·15 + 5·9 + 6·3 + 7·0 + 8·5 = 148.
  4. 4MD = 148/29 ≈ 5.10.

Final answer

MD = 148/29 ≈ 5.10.

Step-by-step solution

  1. 1Midpoints: 50, 150, 250, 350, 450, 550, 650, 750; N = 50.
  2. 2Σfx = 200 + 1200 + 2250 + 3500 + 3150 + 2750 + 2600 + 2250 = 17900 → mean = 358.
  3. 3Deviations: 308, 208, 108, 8, 92, 192, 292, 392.
  4. 4Σf·dev = 1232 + 1664 + 972 + 80 + 644 + 960 + 1168 + 1176 = 7896.
  5. 5MD = 7896/50 = 157.92.

Final answer

MD = 157.92.

Step-by-step solution

  1. 1Midpoints: 100, 110, 120, 130, 140, 150; N = 100.
  2. 2Σfx = 900 + 1430 + 3120 + 3900 + 1680 + 1500 = 12530 → mean = 125.3.
  3. 3Deviations: 25.3, 15.3, 5.3, 4.7, 14.7, 24.7.
  4. 4Σf·dev = 227.7 + 198.9 + 137.8 + 141 + 176.4 + 247 = 1128.8.
  5. 5MD = 1128.8/100 = 11.288 ≈ 11.29.

Final answer

MD ≈ 11.29.

Step-by-step solution

  1. 1N = 50, N/2 = 25. Cumulative: 6, 14, 28 → median class is 20-30.
  2. 2Median = 20 + [(25 − 14)/14]·10 = 20 + 55/7 ≈ 27.86.
  3. 3Midpoints: 5, 15, 25, 35, 45, 55; deviations (×7): 160/7, 90/7, 20/7, 50/7, 120/7, 190/7.
  4. 4Σf·dev = (960 + 720 + 280 + 800 + 480 + 380)/7 = 3620/7.
  5. 5MD = (3620/7)/50 = 362/35 ≈ 10.34.

Final answer

MD = 362/35 ≈ 10.34.

Step-by-step solution

  1. 1Convert to 15.5-20.5, 20.5-25.5, …, 50.5-55.5 with the same frequencies.
  2. 2N = 100, N/2 = 50. Cumulative: 5, 11, 23, 37, 63 → median class 35.5-40.5.
  3. 3Median = 35.5 + [(50 − 37)/26]·5 = 35.5 + 2.5 = 38.
  4. 4Midpoints: 18, 23, 28, 33, 38, 43, 48, 53; deviations: 20, 15, 10, 5, 0, 5, 10, 15.
  5. 5Σf·dev = 100 + 90 + 120 + 70 + 0 + 60 + 160 + 135 = 735.
  6. 6MD = 735/100 = 7.35.

Final answer

MD = 7.35.

03

Exercise 13.2 — Variance and Standard Deviation

10Exercise questions

Step-by-step solution

  1. 1Sum = 72, n = 8 → mean = 9; Σx² = 36 + 49 + 100 + 144 + 169 + 16 + 64 + 144 = 722.
  2. 2Variance = 722/8 − 81 = 90.25 − 81 = 9.25.

Final answer

Mean = 9, variance = 9.25.

Step-by-step solution

  1. 1Mean = (1 + 2 + … + n)/n = (n + 1)/2.
  2. 2Σx² = n(n+1)(2n+1)/6 → first moment of squares = (n+1)(2n+1)/6.
  3. 3Variance = (n+1)(2n+1)/6 − (n+1)²/4 = (n² − 1)/12.

Final answer

Mean = (n + 1)/2, variance = (n² − 1)/12.

Step-by-step solution

  1. 1Data: 3, 6, …, 30; sum = 3·55 = 165 → mean = 16.5.
  2. 2Σx² = 9·(1² + … + 10²) = 9·385 = 3465.
  3. 3Variance = 3465/10 − (16.5)² = 346.5 − 272.25 = 74.25.

Final answer

Mean = 16.5, variance = 74.25.

Step-by-step solution

  1. 1N = 40; Σfx = 12 + 40 + 98 + 216 + 192 + 112 + 90 = 760 → mean = 19.
  2. 2Σfx² = 72 + 400 + 1372 + 3888 + 4608 + 3136 + 2700 = 16176.
  3. 3Variance = 16176/40 − 361 = 404.4 − 361 = 43.4.

Final answer

Mean = 19, variance = 43.4.

Step-by-step solution

  1. 1N = 22; Σfx = 276 + 186 + 291 + 196 + 612 + 312 + 327 = 2200 → mean = 100.
  2. 2Σfx² = 25392 + 17298 + 28227 + 19208 + 62424 + 32448 + 35643 = 220640.
  3. 3Variance = 220640/22 − 10000 = 10029.09 − 10000 = 320/11 ≈ 29.09.

Final answer

Mean = 100, variance = 320/11 ≈ 29.09.

Step-by-step solution

  1. 1Take A = 64, d = x − 64; N = 100.
  2. 2Σfd = −8 − 3 − 24 − 29 + 0 + 12 + 20 + 12 + 20 = 0 → mean = 64.
  3. 3Σfd² = 32 + 9 + 48 + 29 + 0 + 12 + 40 + 36 + 80 = 286.
  4. 4Variance = 286/100 − 0 = 2.86; SD = √2.86 ≈ 1.69.

Final answer

Mean = 64, SD ≈ 1.69.

Step-by-step solution

  1. 1Midpoints: 15, 45, 75, 105, 135, 165, 195; N = 30.
  2. 2Σfx = 30 + 135 + 375 + 1050 + 405 + 825 + 390 = 3210 → mean = 107.
  3. 3Σfx² = 450 + 6075 + 28125 + 110250 + 54675 + 136125 + 76050 = 411750.
  4. 4Variance = 411750/30 − 11449 = 13725 − 11449 = 2276.

Final answer

Mean = 107, variance = 2276.

Step-by-step solution

  1. 1Midpoints: 5, 15, 25, 35, 45; N = 50.
  2. 2Σfx = 25 + 120 + 375 + 560 + 270 = 1350 → mean = 27.
  3. 3Σfx² = 125 + 1800 + 9375 + 19600 + 12150 = 43050.
  4. 4Variance = 43050/50 − 729 = 861 − 729 = 132.

Final answer

Mean = 27, variance = 132.

Step-by-step solution

  1. 1Midpoints: 72.5, …, 112.5 (step h = 5). Take A = 92.5, d = (x − 92.5)/5: −4, −3, −2, −1, 0, 1, 2, 3, 4; N = 60.
  2. 2Σfd = −12 − 12 − 14 − 7 + 0 + 9 + 12 + 18 + 12 = 6.
  3. 3Mean = 92.5 + 5·(6/60) = 92.5 + 0.5 = 93.
  4. 4Σfd² = 48 + 36 + 28 + 7 + 0 + 9 + 24 + 54 + 48 = 254.
  5. 5Variance = 25·[254/60 − (6/60)²] = 25·4.2233 = 105.583; SD ≈ 10.28.

Final answer

Mean = 93, variance ≈ 105.58, SD ≈ 10.28.

Step-by-step solution

  1. 1Make continuous: 32.5-36.5, …, 48.5-52.5; midpoints 34.5, 38.5, 42.5, 46.5, 50.5; N = 100.
  2. 2Σfx = 517.5 + 654.5 + 892.5 + 1023 + 1262.5 = 4350 → mean = 43.5.
  3. 3Σfx² = 17853.75 + 25198.25 + 37931.25 + 47569.5 + 63756.25 = 192309.
  4. 4Variance = 192309/100 − 43.5² = 1923.09 − 1892.25 = 30.84; SD ≈ 5.55.

Final answer

Mean = 43.5 mm, SD ≈ 5.55 mm.

Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Arithmetic mean

Variance

Shortcut variance

Standard deviation

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

  • The shortcut variance is far less error-prone than the deviation form for raw data — use it whenever the values are large.
  • The two variance forms agree exactly; if your answers differ, the arithmetic has slipped rather than the method.

FAQ

Frequently asked questions

How many questions are in NCERT Class 11 Maths Chapter 13 (Statistics)?

There are 2 exercise questions in this chapter, numbered Ex 13.1 to Ex 13.2. Every one is solved step by step on this page in the official NCERT numbering.

Which formulas come up in Statistics Class 11 Maths?

The formulas this chapter's questions actually turn on are: Arithmetic mean, Variance, Shortcut variance, Standard deviation. They are listed with their expressions in the key formulas section below, and the solved questions show where each one is used.

Is Statistics important for JEE Main?

Moderate — mechanical and worth easy marks in boards, with the direct questions typical of the NEET syllabus.

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