Complete Math Magic

All 15 Problems Solved with Wizardry!

✨ The Range Enchantment ✨

1. Find the range and coefficient of range of the following data:

(i) 63, 89, 98, 125, 79, 108, 117, 68

(ii) 43.5, 13.6, 18.9, 38.4, 61.4, 29.8

Part (i) Solution
Range = Largest value - Smallest value
Coefficient of Range = (Largest - Smallest) ÷ (Largest + Smallest)

Given data: 63, 89, 98, 125, 79, 108, 117, 68

Sorted data: 63, 68, 79, 89, 98, 108, 117, 125

Largest value (L) = 125

Smallest value (S) = 63

Range = L - S = 125 - 63 = 62

Coefficient of Range = (125 - 63) ÷ (125 + 63) = 62 ÷ 188 ≈ 0.3298

Part (ii) Solution

Given data: 43.5, 13.6, 18.9, 38.4, 61.4, 29.8

Sorted data: 13.6, 18.9, 29.8, 38.4, 43.5, 61.4

Largest value (L) = 61.4

Smallest value (S) = 13.6

Range = L - S = 61.4 - 13.6 = 47.8

Coefficient of Range = (61.4 - 13.6) ÷ (61.4 + 13.6) = 47.8 ÷ 75 ≈ 0.6373

2. If the range and the smallest value of a set of data are 36.8 and 13.4 respectively, then find the largest value.

Solution
Range = Largest value - Smallest value
∴ Largest = Range + Smallest

Given:

Range = 36.8

Smallest value = 13.4

Let Largest value be L

36.8 = L - 13.4

L = 36.8 + 13.4 = 50.2

The largest value is 50.2

3. Calculate the range of the following data.

Income 400-450 450-500 500-550 550-600 600-650
Number of workers 8 12 30 21 6
Solution
For grouped data:
Range = Upper limit of highest class - Lower limit of lowest class

Highest class: 600-650

Upper limit of highest class = 650

Lowest class: 400-450

Lower limit of lowest class = 400

Range = 650 - 400 = 250

4. A teacher asked the students to complete 60 pages of a record note book. Eight students have completed only 32, 35, 37, 30, 33, 36, 35 and 37 pages. Find the standard deviation of the pages completed by them.

Step 1: Calculate Mean

Data: 32, 35, 37, 30, 33, 36, 35, 37

Mean (μ) = (32 + 35 + 37 + 30 + 33 + 36 + 35 + 37) ÷ 8

μ = 275 ÷ 8 = 34.375

Step 2: Calculate Squared Differences
Pages (xᵢ) xᵢ - μ (xᵢ - μ)²
32-2.3755.6406
350.6250.3906
372.6256.8906
30-4.37519.1406
33-1.3751.8906
361.6252.6406
350.6250.3906
372.6256.8906
Total45.875
Step 3: Calculate Variance
Variance = Σ(xᵢ - μ)² ÷ N

Variance = 45.875 ÷ 8 = 5.7344

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √5.7344 ≈ 2.395

The standard deviation is approximately 2.395 pages

5. Find the variance and standard deviation of the wages of 9 workers given below:

310, 290, 320, 280, 300, 290, 320, 310, 280

Step 1: Calculate Mean

Data: 310, 290, 320, 280, 300, 290, 320, 310, 280

Mean (μ) = (310 + 290 + 320 + 280 + 300 + 290 + 320 + 310 + 280) ÷ 9

μ = 2700 ÷ 9 = 300

Step 2: Calculate Squared Differences
Wage (xᵢ) xᵢ - μ (xᵢ - μ)²
31010100
290-10100
32020400
280-20400
30000
290-10100
32020400
31010100
280-20400
Total2000
Step 3: Calculate Variance
Variance = Σ(xᵢ - μ)² ÷ N

Variance = 2000 ÷ 9 ≈ 222.22

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √222.22 ≈ 14.91

The standard deviation is approximately 14.91

🔮 Magical Standard Deviations 🔮

6. A wall clock strikes the bell once at 1 o'clock, 2 times at 2 o'clock, 3 times at 3 o'clock and so on. How many times will it strike in a particular day? Find the standard deviation of the number of strikes the bell makes in a day.

Step 1: Total Strikes per Day

Strikes per 12-hour cycle: 1+2+3+...+12 = 78

Total strikes per day (2 cycles): 78 × 2 = 156

Step 2: Standard Deviation of Strikes per Hour

We calculate SD for the numbers 1 through 12 (each occurring twice)

Mean (μ) = (1+2+...+12) ÷ 12 = 78 ÷ 12 = 6.5

Step 3: Calculate Sum of Squares

Sum of squares = 1² + 2² + ... + 12² = 650

Variance = (Sum of squares ÷ n) - μ² = (650 ÷ 12) - 6.5²

= 54.1667 - 42.25 = 11.9167

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √11.9167 ≈ 3.452

The standard deviation is approximately 3.452 strikes

7. Find the standard deviation of first 21 natural numbers.

Solution Using Formula
For first n natural numbers:
Standard Deviation = √[(n² - 1) ÷ 12]

For n = 21:

σ = √[(21² - 1) ÷ 12] = √[(441 - 1) ÷ 12] = √[440 ÷ 12]

= √36.6667 ≈ 6.055

Alternative Calculation

Mean (μ) = (1 + 21) ÷ 2 = 11

Sum of squares = n(n+1)(2n+1) ÷ 6 = 21×22×43 ÷ 6 = 3311

Variance = (Sum of squares ÷ n) - μ² = (3311 ÷ 21) - 121

= 157.6667 - 121 = 36.6667

σ = √36.6667 ≈ 6.055

8. If the standard deviation of a data is 4.5 and if each value of the data is decreased by 5, then find the new standard deviation.

Solution
Adding or subtracting a constant from all values:
• Changes the mean by that constant
• Does NOT change the standard deviation

Original standard deviation = 4.5

Subtracting 5 from each value doesn't affect spread

New standard deviation = 4.5 (unchanged)

9. If the standard deviation of a data is 3.6 and each value of the data is divided by 3, then find the new variance and new standard deviation.

Solution
Multiplying or dividing all values by a constant 'k':
• Multiplies the standard deviation by |k|
• Multiplies the variance by k²

Original standard deviation (σ) = 3.6

Dividing each value by 3:

New standard deviation = σ ÷ 3 = 3.6 ÷ 3 = 1.2

New variance = (σ ÷ 3)² = (1.2)² = 1.44

10. The rainfall recorded in various places of five districts in a week are given below. Find its standard deviation.

Rainfall (mm) 45 50 55 60 65 70
Number of places 5 13 4 9 5 4
Step 1: Calculate Mean
Rainfall (xᵢ) Frequency (fᵢ) fᵢxᵢ
455225
5013650
554220
609540
655325
704280
Total402240

Mean (μ) = Σfᵢxᵢ ÷ Σfᵢ = 2240 ÷ 40 = 56

Step 2: Calculate Squared Differences
xᵢ fᵢ xᵢ - μ (xᵢ - μ)² fᵢ(xᵢ - μ)²
455-11121605
5013-636468
554-114
609416144
655981405
70414196784
Total402410
Step 3: Calculate Variance
Variance = Σfᵢ(xᵢ - μ)² ÷ Σfᵢ

Variance = 2410 ÷ 40 = 60.25

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √60.25 = 7.762

The standard deviation is 7.762 mm

📊 Grouped Data Wizardry 📊

11. In a study about viral fever, the number of people affected in a town were noted as below. Find its standard deviation.

Age in years 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Number of people affected 3 5 16 18 12 7 4
Step 1: Find Midpoints and Calculate Mean
Class Midpoint (mᵢ) Frequency (fᵢ) fᵢmᵢ
0-105315
10-2015575
20-302516400
30-403518630
40-504512540
50-60557385
60-70654260
Total652305

Mean (μ) = Σfᵢmᵢ ÷ Σfᵢ = 2305 ÷ 65 ≈ 35.46

Step 2: Calculate Squared Differences
Class mᵢ fᵢ mᵢ - μ (mᵢ - μ)² fᵢ(mᵢ - μ)²
0-1053-30.46927.812783.43
10-20155-20.46418.612093.05
20-302516-10.46109.411750.56
30-403518-0.460.213.78
40-5045129.5491.011092.12
50-6055719.54381.812672.67
60-7065429.54872.613490.44
Total6513885.05
Step 3: Calculate Variance
Variance = Σfᵢ(mᵢ - μ)² ÷ Σfᵢ

Variance = 13885.05 ÷ 65 ≈ 213.62

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √213.62 ≈ 14.62

The standard deviation is approximately 14.62 years

12. The measurements of the diameters (in cms) of the plates prepared in a factory are given below. Find its standard deviation.

Diameter (cm) 21-24 25-28 29-32 33-36 37-40 41-44
Number of plates 15 18 20 16 8 7
Step 1: Using Assumed Mean Method

Let assumed mean A = 30.5 (midpoint of 29-32 class)

Class width (h) = 4

Class Midpoint (mᵢ) fᵢ dᵢ = (mᵢ - A)/h fᵢdᵢ fᵢdᵢ²
21-2422.515-2-3060
25-2826.518-1-1818
29-3230.520000
33-3634.51611616
37-4038.5821632
41-4442.5732163
Total845189
Step 2: Calculate Mean
Mean = A + (Σfᵢdᵢ ÷ Σfᵢ) × h

Mean = 30.5 + (5 ÷ 84) × 4 ≈ 30.74 cm

Step 3: Calculate Variance
Variance = [Σfᵢdᵢ² ÷ Σfᵢ - (Σfᵢdᵢ ÷ Σfᵢ)²] × h²

Variance = [189 ÷ 84 - (5 ÷ 84)²] × 16

= [2.25 - 0.0035] × 16 ≈ 35.94

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √35.94 ≈ 6.00 cm

13. The time taken by 50 students to complete a 100 meter race are given below. Find its standard deviation.

Time taken (sec) 8.5-9.5 9.5-10.5 10.5-11.5 11.5-12.5 12.5-13.5
Number of students 6 8 17 10 9
Step 1: Using Assumed Mean Method

Let assumed mean A = 11 (midpoint of 10.5-11.5 class)

Class width (h) = 1

Class Midpoint (mᵢ) fᵢ dᵢ = (mᵢ - A)/h fᵢdᵢ fᵢdᵢ²
8.5-9.596-2-1224
9.5-10.5108-1-88
10.5-11.51117000
11.5-12.5121011010
12.5-13.513921836
Total50878
Step 2: Calculate Mean
Mean = A + (Σfᵢdᵢ ÷ Σfᵢ) × h

Mean = 11 + (8 ÷ 50) × 1 = 11.16 seconds

Step 3: Calculate Variance
Variance = [Σfᵢdᵢ² ÷ Σfᵢ - (Σfᵢdᵢ ÷ Σfᵢ)²] × h²

Variance = [78 ÷ 50 - (8 ÷ 50)²] × 1

= [1.56 - 0.0256] = 1.5344

Step 4: Calculate Standard Deviation
Standard Deviation = √Variance

σ = √1.5344 ≈ 1.239 seconds

🎯 Advanced Statistical Magic 🎯

14. For a group of 100 candidates the mean and standard deviation of their marks were found to be 60 and 15 respectively. Later on it was found that the scores 45 and 72 were wrongly entered as 40 and 27. Find the correct mean and standard deviation.

Step 1: Calculate Correct Sum

Original sum = n × μ = 100 × 60 = 6000

Remove incorrect values: 6000 - 40 - 27 = 5933

Add correct values: 5933 + 45 + 72 = 6050

Correct mean = 6050 ÷ 100 = 60.5

Step 2: Calculate Correct Sum of Squares

Original sum of squares = n(σ² + μ²) = 100(225 + 3600) = 382,500

Remove incorrect squares: 382,500 - 40² - 27² = 382,500 - 1600 - 729 = 380,171

Add correct squares: 380,171 + 45² + 72² = 380,171 + 2025 + 5184 = 387,380

Step 3: Calculate Correct Variance
Correct Variance = (Correct Sum of Squares ÷ n) - (Correct Mean)²

Variance = (387,380 ÷ 100) - (60.5)² = 3873.8 - 3660.25 = 213.55

Step 4: Calculate Correct Standard Deviation
Standard Deviation = √Variance

σ = √213.55 ≈ 14.61

Correct values: Mean = 60.5, Standard Deviation ≈ 14.61

15. The mean and variance of seven observations are 8 and 16 respectively. If five of these are 2, 4, 10, 12 and 14, then find the remaining two observations.

Step 1: Find Sum of All Observations

Total sum = n × μ = 7 × 8 = 56

Sum of known five = 2 + 4 + 10 + 12 + 14 = 42

Let remaining two observations be a and b

a + b = 56 - 42 = 14 (Equation 1)

Step 2: Find Sum of Squares

Total sum of squares = n(σ² + μ²) = 7(16 + 64) = 560

Sum of squares of known five = 4 + 16 + 100 + 144 + 196 = 460

a² + b² = 560 - 460 = 100 (Equation 2)

Step 3: Solve the Equations

From Equation 1: b = 14 - a

Substitute into Equation 2: a² + (14 - a)² = 100

a² + 196 - 28a + a² = 100

2a² - 28a + 96 = 0

Divide by 2: a² - 14a + 48 = 0

Factorize: (a - 6)(a - 8) = 0

Solutions: a = 6 or a = 8

Corresponding b = 8 or b = 6

The remaining two observations are 6 and 8