StudentAidUnit 11: Statistics — Exercise 11 2
10th Class Mathematics · Unit 11: Statistics
11.2.1.Find the range of the following data sets: (i) 63, 89, 98, 125, 79, 108, 117, 60 (ii) 43.5, 13.6, 18.9, 38.4, 61.4, 29.4.
(i) sorted
60,63,79,89,98,108,117,125
Result
(i) Range
Range = 125 - 60 = 65
(ii) sorted
13.6,18.9,29.4,38.4,43.5,61.4
(ii) Range
Range = 61.4 - 13.6 = 47.8
11.2.2.If the range and the lowest value of a set of data are 46.7 and 13.4 respectively, then find the highest value.
Formula
Range
Range = Highest - Lowest
Working
Substituting
46.7 = Highest - 13.4
Result
Highest value
Highest = 46.7 + 13.4 = 60.1
11.2.3.Calculate the range of the following data: Income (in Rs.) 4000-4500, 4500-5000, 5000-5500, 5500-6000, 6000-6500; No. of workers 8, 12, 30, 21, 6.
Formula
Range (grouped data)
Range = Upper limit of highest class - Lower limit of lowest class
Working
Substituting
= 6500 - 4000
Result
Range
Range = 2500
11.2.4.A group of 7 workers reported the number of items they assembled in a day as: 52, 55, 50, 53, 54, 56, 52. Find the standard deviation and variance of the items assembled.
Given
Data (n=7)
52,55,50,53,54,56,52
Mean
Sigma x = 372, bar{x} = 3727 = 53.14
Sum of squared deviations
Sigma (x-bar{x})2 = 24.8568
Formula
Variance
sigma2 = frac{Sigma(x-bar{x})2}{n} = 24.85687 = 3.5510
Result
Standard Deviation
sigma = sqrt{3.5510} = 1.885
11.2.5.A librarian recorded the number of visitors during 5 days of a week: 120, 135, 130, 125, 140. Calculate the variance and standard deviation of visitors.
Given
Data (n=5)
120,135,130,125,140
Mean
Sigma x = 650, bar{x} = 6505 = 130
Sum of squared deviations
Sigma (x-bar{x})2 = 250
Formula
Variance
sigma2 = 2505 = 50
Result
Standard Deviation
sigma = sqrt{50} = 7.071
11.2.6.Find the range, variance and standard deviation of first 23 odd numbers.
Given
First 23 odd numbers (n=23)
1,3,5,dots,43,45
Range
Range = 45 - 1 = 44
Formula
Mean and variance formulas for first n odd numbers
bar{x} = n,quad sigma2 = n2-13
Working
Substituting n=23
sigma2 = 232-13 = 5283 = 176
Result
Standard Deviation
sigma = sqrt{176} = 13.266
11.2.7.The rainfall recorded in various places of five districts in a week is given below. Find its variance and standard deviation. Rainfall (in mm) 42, 51, 54, 61, 63, 71; Number of places (f) 5, 13, 4, 9, 5, 4.
Given
Discrete frequency distribution
N = Sigma f = 40
Sums
Sigma fx = 2237, Sigma fx2 = 127795
Formula
Mean
bar{x} = Sigma fxN = 223740 = 55.925
Variance
sigma2 = Sigma fx2N - bar{x}2 = 3194.875 - 3127.605625 = 67.269375 ≈ 67.27
Result
Standard Deviation
sigma = sqrt{67.269375} = 8.2018 ≈ 8.20
11.2.8.Machine A Output (units): 98, 100, 102, 101, 99. Machine B Output (units): 95, 100, 105, 90, 110. (i) Which machine has better performance? (ii) Which machine is more consistent?
Machine A
n=5, Sigma x = 500, bar{x}A = 100, Sigma(x-bar{x})2 = 10
Result
Machine A
sigmaA2 = 105 = 2,quad sigmaA = sqrt{2} = 1.414
Machine B
n=5, Sigma x = 500, bar{x}B = 100, Sigma(x-bar{x})2 = 250
Machine B
sigmaB2 = 2505 = 50,quad sigmaB = sqrt{50} = 7.071
Conclusion
Same average output (100); Machine A is more consistent (1.414 < 7.071)
11.2.9.The monthly sales (rupees in lacs) for two salespersons over 6 months are: Person A: 5.5, 5.7, 5.4, 5.6, 5.8, 5.6. Person B: 5.5, 6.5, 4.5, 6.0, 5.0, 6.0. Compare their performance and consistency.
11.2.10.The table given below shows the daily wages of workers in a textile mill, grouped into six income brackets: Daily Wage (Rs) 800-1000, 1000-1200, 1200-1400, 1400-1600, 1600-1800, 1800-2000; Frequency (f) 2, 4, 6, 8, 2, 1. Calculate the mean, variance and standard deviation of the wages.
11.2.11.A company forecasts monthly sales (rupees in millions): 15, 18, 14, 20, 13. Find variability in sales predictions.
Given
Data (n=5)
15,18,14,20,13
Mean
Sigma x = 80, bar{x} = 16
Sum of squared deviations
Sigma(x-bar{x})2 = 34.20
The deviation table underneath uses 16.2 instead of the stated mean 16 (e.g. 15-16.2=-1.2 instead of 15-16=-1); with the correct mean 16, Σ(x-x̄)²=1+4+4+16+9=34, not 34.20 — very close, but not from the stated mean.
Formula
Variance
sigma2 = 34.205 = 6.84
Result
Standard Deviation
sigma = sqrt{6.84} = 2.615
11.2.12.Unemployment rates (%) in five provinces are 5.2, 6.0, 4.8, 5.5, 6.2. Calculate standard deviation and describe is there balanced unemployment rate?
Given
Data (n=5)
5.2,6.0,4.8,5.5,6.2
Mean
Sigma x = 27.7, bar{x} = 5.54
Sum of squared deviations
Sigma(x-bar{x})2 = 1.3300
The deviation table uses 5.48 instead of the stated mean 5.54 (e.g. 5.2-5.48=-0.28 instead of 5.2-5.54=-0.34).
Formula
Variance
sigma2 = 34.205 = 6.84
This line's numbers (34.20, 6.84) are leftover figures from problem 11.2.11, not this problem; using this problem's own Σ(x-x̄)²=1.33, variance should be 1.33/5=0.266.
Standard Deviation
sigma = sqrt{0.266} = 2.615
√0.266 ≈ 0.516, not 2.615 (2.615 is again a leftover figure from problem 11.2.11, where √6.84=2.615).
Result
Standard Deviation
0.516
Conclusion
Since SD is small (0.516), unemployment rates are fairly balanced across the provinces
11.2.13.Find variance and standard deviation: (i) Σx = 45, Σx² = 421, n = 5 (ii) Σfx = 210, Σfx² = 7560, Σf = 6 (iii) x̄ = 18, Σfx² = 1670, Σf = 5.
Formula
(i) Variance
sigma2 = Sigma x2n - left(Sigma xnright)2 = 4215 - left(455right)2 = 84.2 - 81 = 3.2
Result
(i) Standard Deviation
sigma = sqrt{3.2} = 1.789
Formula
(ii) Mean and Variance
bar{x} = Sigma fxSigma f = 2106 = 35;quad sigma2 = Sigma fx2Sigma f - bar{x}2 = 1260 - 1225 = 35
Result
(ii) Standard Deviation
sigma = sqrt{35} = 5.916
Formula
(iii) Variance
sigma2 = Sigma fx2Sigma f - bar{x}2 = 16705 - 182 = 334 - 324 = 10
Result
(iii) Standard Deviation
sigma = sqrt{10} = 3.162