Question 1
Find mean deviation about the mean for 4,7,8,9,10,12,13,17.
Steps only
Sum=80, so mean=80/8=10.
Sum |x−10|=24.
Mean deviation=24/8=3.
Answer
Mean=10; mean deviation=3.
CLASS 11 · MATHEMATICS · CHAPTER 13
Statistics
Measure spread with mean deviation, variance and standard deviation. Short calculations come first, with clear explanations for grouped and ungrouped data.
29 worked answersAll 2 exercise setsAll 6 end-of-chapter questions
What you will learn
- Calculate mean deviation about the mean or median.
- Use frequency tables and class marks for grouped data.
- Find variance and standard deviation.
- Understand how data transformations change these measures.
29 worked answers · short steps first, explanations when you need them
Exercise 13.1
Textbook pages 270–271
Question 2
Find mean deviation about the mean for 38,70,48,40,42,55,63,46,54,44.
Steps only
Sum=500, mean=50.
Sum of absolute deviations from 50=84.
84/10=8.4.
Answer
Mean=50; mean deviation=8.4.
Question 3
Find mean deviation about the median for 13,17,16,14,11,13,10,16,11,18,12,17.
Steps only
Ordered data: 10,11,11,12,13,13,14,16,16,17,17,18.
Median=(13+14)/2=13.5.
Sum of distances from 13.5=28; divide by 12.
Answer
Median=13.5; mean deviation=7/3≈2.33.
Question 4
Find mean deviation about the median for 36,72,46,42,60,45,53,46,51,49.
Steps only
Ordered data: 36,42,45,46,46,49,51,53,60,72.
Median=(46+49)/2=47.5.
Sum of absolute deviations=70; 70/10=7.
Answer
Median=47.5; mean deviation=7.
Question 5
Find mean deviation about the mean for xi=5,10,15,20,25 with frequencies 7,4,6,3,5.
Steps only
N=25 and Σfixi=350, so mean=14.
Σfi|xi−14|=158.
158/25=6.32.
Answer
Mean=14; mean deviation=6.32.
Question 6
Find mean deviation about the mean for xi=10,30,50,70,90 with frequencies 4,24,28,16,8.
Steps only
N=80 and weighted mean=50.
Σfi|xi−50|=1280.
1280/80=16.
Answer
Mean=50; mean deviation=16.
Question 7
Find mean deviation about the median for xi=5,7,9,10,12,15 with frequencies 8,6,2,2,2,6.
Steps only
N=26; observations 13 and 14 both equal 7.
Σfi|xi−7|=84.
84/26≈3.2308.
Answer
Median=7; mean deviation=3.2308 (about 3.23).
Question 8
Find mean deviation about the median for xi=15,21,27,30,35 with frequencies 3,5,6,7,8.
Steps only
N=29; the 15th observation is 30.
Σfi|xi−30|=148.
148/29≈5.1034.
Answer
Median=30; mean deviation≈5.1034.
Question 9
Find mean deviation about the mean for daily incomes in classes 0–100,…,700–800 with frequencies 4,8,9,10,7,5,4,3.
Steps only
Class marks: 50,150,…,750; N=50.
Weighted mean=358.
Σfi|xi−358|=7,896; divide by 50.
Answer
Mean=₹358; mean deviation=₹157.92.
Question 10
Find mean deviation about the mean for heights 95–105,…,145–155 cm with frequencies 9,13,26,30,12,10.
Steps only
Class marks: 100,110,120,130,140,150; N=100.
Weighted mean=125.3.
Σfi|xi−125.3|=1128.8; divide by 100.
Answer
Mean=125.3 cm; mean deviation=11.288 cm.
Question 11
Find mean deviation about the median for marks classes 0–10,…,50–60 with frequencies 6,8,14,16,4,2.
Steps only
N=50; median class is 20–30.
M=20+[(25−14)/14]×10=27.857.
Using class marks 5,15,…,55 gives Σfi|xi−M|/50≈10.343.
Answer
Median≈27.86; mean deviation≈10.34 marks.
Question 12
Find mean deviation about median age for 100 persons in age groups 16–20,…,51–55 with frequencies 5,6,12,14,26,12,16,9.
Steps only
Convert to continuous intervals; median class is 35.5–40.5.
M=35.5+[(50−37)/26]×5=38.
Using class marks gives weighted absolute-deviation sum=735; divide by 100.
Answer
Median=38 years; mean deviation=7.35 years.
Exercise 13.2
Textbook pages 281–282
Question 1
Find mean and variance of 6,7,10,12,13,4,8,12.
Steps only
Σx=72, n=8, so mean=9.
Σ(x−9)²=74.
Variance=74/8=9.25.
Answer
Mean=9; variance=9.25.
Question 2
Find mean and variance of the first n natural numbers.
Steps only
Sum= n(n+1)/2, so mean=(n+1)/2.
Use Σk²=n(n+1)(2n+1)/6.
Substitute in variance=Σk²/n−mean² and simplify.
Answer
Mean=(n+1)/2; variance=(n²−1)/12.
Question 3
Find mean and variance of the first 10 multiples of 3.
Steps only
Values are 3,6,…,30, which are 3 times 1,…,10.
Mean=3(5.5)=16.5.
Variance=9×(10²−1)/12=74.25.
Answer
Mean=16.5; variance=74.25.
Question 4
Find mean and variance for xi=6,10,14,18,24,28,30 with frequencies 2,4,7,12,8,4,3.
Steps only
N=40, Σfixi=760; mean=19.
Σfixi²=16,176.
Variance=16,176/40−19²=43.4.
Answer
Mean=19; variance=43.4.
Question 5
Find mean and variance for xi=92,93,97,98,102,104,109 with frequencies 3,2,3,2,6,3,3.
Steps only
N=22 and Σfixi=2200, so mean=100.
Σfi(xi−100)²=640.
Variance=640/22=320/11≈29.09.
Answer
Mean=100; variance=29.09 (approximately).
Question 6
Use the short-cut method to find mean and standard deviation for xi=60,…,68 with frequencies 2,1,12,29,25,12,10,4,5.
Steps only
N=100; take assumed mean A=64 and d=xi−A.
Σfid=0; Σfid²=286.
Variance=286/100=2.86; σ=√2.86.
Answer
Mean=64; variance=2.86; standard deviation≈1.691.
Question 7
Find mean and variance for classes 0–30,…,180–210 with frequencies 2,3,5,10,3,5,2.
Steps only
Class marks: 15,45,75,105,135,165,195; N=30.
Σfixi=3210, mean=107.
Σfi(xi−107)²=68,280; divide by 30.
Answer
Mean=107; variance=2,276.
Question 8
Find mean and variance for classes 0–10,10–20,20–30,30–40,40–50 with frequencies 5,8,15,16,6.
Steps only
Class marks: 5,15,25,35,45; N=50.
Weighted mean=27.
Σfi(xi−27)²=6600; divide by 50.
Answer
Mean=27; variance=132.
Question 9
Find mean, variance and standard deviation for heights 70–75,…,110–115 cm with frequencies 3,4,7,7,15,9,6,6,3.
Steps only
Class marks 72.5,77.5,…,112.5; N=60.
Weighted mean=93.
Variance=6335/60≈105.583; σ≈10.275.
Answer
Mean=93 cm; variance≈105.58 cm²; standard deviation≈10.28 cm.
Question 10
Find mean diameter and standard deviation for classes 33–36,…,49–52 with frequencies 15,17,21,22,25.
Steps only
Use continuous class marks 34.5,38.5,42.5,46.5,50.5; N=100.
Mean=43.5 mm.
Variance=30.84; σ=√30.84≈5.553 mm.
Answer
Mean=43.5 mm; variance=30.84 mm²; standard deviation≈5.55 mm.
Miscellaneous Exercise on Chapter 13
Textbook pages 282–283
Question 1
Eight observations have mean 9 and variance 9.25. Six are 6,7,10,12,12,13. Find the other two.
Steps only
Total sum=8×9=72; missing values sum to 12.
Total sum of squares=8(9.25+9²)=722.
Known squares total 642, so missing squares sum to 80; the pair is 4,8.
Answer
4 and 8.
Question 2
Seven observations have mean 8 and variance 16. Five are 2,4,10,12,14. Find the other two.
Steps only
Missing values sum to 7×8−42=14.
Total squares=7(16+64)=560; known squares=460.
Missing squares sum to 100; the pair is 6,8.
Answer
6 and 8.
Question 3
Six observations have mean 8 and standard deviation 4. If each is multiplied by 3, find the new mean and standard deviation.
Steps only
Scaling observations by 3 scales their mean by 3.
Standard deviation scales by |3|.
Answer
New mean=24; new standard deviation=12.
Question 4
If x̄ and σ² are the mean and variance of x₁,…,xₙ, prove the mean and variance of ax₁,…,axₙ are ax̄ and a²σ².
Steps only
New mean=(1/n)Σaxi=a x̄.
New variance=(1/n)Σ(axᵢ−a x̄)².
Factor a² to get a²σ².
Answer
The result is proved for a≠0.
Question 5(i)
The mean and standard deviation of 20 values were 10 and 2. If the incorrect value 8 is omitted, find the corrected mean and standard deviation.
Steps only
Original sum=200; original sum of squares=20(2²+10²)=2080.
After removing 8: n=19, sum=192, sumsq=2016.
Variance=2016/19−(192/19)²≈3.989; take square root.
Answer
Mean=192/19≈10.105; standard deviation≈1.997.
Question 5(ii)
Instead replace the incorrect value 8 by 12. Find corrected mean and standard deviation.
Steps only
New sum=200−8+12=204.
New sum of squares=2080−64+144=2160.
Variance=2160/20−10.2²=3.96; σ=√3.96.
Answer
Mean=10.2; standard deviation≈1.990.
Question 6
For 100 observations, mean=20 and standard deviation=3. Three incorrect values 21,21,18 are omitted. Find corrected mean and standard deviation.
Steps only
Original sum=2000; sumsq=100(20²+3²)=40,900.
After omission: n=97, sum=1940, sumsq=39,694.
Variance=39,694/97−20²=894/97; σ≈3.036.
Answer
Mean=20; standard deviation≈3.036.