Mathematics 10 — Exercise 11.2 · Q4

Full question

The table gives the daily numbers of items assembled by seven workers. Calculate the variance and standard deviation of these counts.

Step-by-step solution

1. Multiply the value (or class midpoint) by its frequency.

   $$1\times 52=52$$

2. Square that value.

   $${52}^2=2704$$

3. Multiply the square by its frequency.

   $$1\times 2704=2704$$

4. Multiply the value (or class midpoint) by its frequency.

   $$1\times 55=55$$

5. Square that value.

   $${55}^2=3025$$

6. Multiply the square by its frequency.

   $$1\times 3025=3025$$

7. Multiply the value (or class midpoint) by its frequency.

   $$1\times 50=50$$

8. Square that value.

   $${50}^2=2500$$

9. Multiply the square by its frequency.

   $$1\times 2500=2500$$

10. Multiply the value (or class midpoint) by its frequency.

    $$1\times 53=53$$

11. Square that value.

    $${53}^2=2809$$

12. Multiply the square by its frequency.

    $$1\times 2809=2809$$

13. Multiply the value (or class midpoint) by its frequency.

    $$1\times 54=54$$

14. Square that value.

    $${54}^2=2916$$

15. Multiply the square by its frequency.

    $$1\times 2916=2916$$

16. Multiply the value (or class midpoint) by its frequency.

    $$1\times 56=56$$

17. Square that value.

    $${56}^2=3136$$

18. Multiply the square by its frequency.

    $$1\times 3136=3136$$

19. Multiply the value (or class midpoint) by its frequency.

    $$1\times 52=52$$

20. Square that value.

    $${52}^2=2704$$

21. Multiply the square by its frequency.

    $$1\times 2704=2704$$

22. Add the frequencies to find n.

    $$1+1+1+1+1+1+1=7$$

23. Add the f×x column.

    $$52+55+50+53+54+56+52=372$$

24. Add the f×x² column.

    $$2704+3025+2500+2809+2916+3136+2704=19794$$

25. Mean = Σfx/Σf. Divide the first total by n.

    $$\frac{372}{7}$$

26. Calculate this operation before combining the remaining terms.

    $$372\times 1=372$$

27. Calculate this operation before combining the remaining terms.

    $$1\times 7=7$$

28. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{372\times 1}{1\times 7}$$

29. Calculate these two products separately.

    $$\frac{372}{7}$$

30. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 7}$$

31. Calculate these two products separately.

    $$\frac{1}{7}$$

32. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{7}=\frac{1}{7}$$

33. Calculate this operation before combining the remaining terms.

    $$372\times \frac{1}{7}=\frac{372}{7}$$

34. Population variance = Σfx²/n − (mean)². Keep the exact mean before rounding.

    $$\frac{19794}{7}-{\left(\frac{372}{7}\right)}^2$$

35. Calculate this operation before combining the remaining terms.

    $$19794\times 1=19794$$

36. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{19794\times 1}{1\times 7}$$

37. Calculate these two products separately.

    $$\frac{19794}{7}$$

38. Calculate this operation before combining the remaining terms.

    $$19794\times \frac{1}{7}=\frac{19794}{7}$$

39. Calculate this operation before combining the remaining terms.

    $${\left(\frac{372}{7}\right)}^2=\frac{138384}{49}$$

40. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 138384=-138384$$

41. Calculate this operation before combining the remaining terms.

    $$1\times 49=49$$

42. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-138384\right)\times 1}{1\times 49}$$

43. Calculate this operation before combining the remaining terms.

    $$\left(-138384\right)\times \frac{1}{49}=-\frac{138384}{49}$$

44. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 138384}{1\times 49}$$

45. Calculate these two products separately.

    $$-\frac{138384}{49}$$

46. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{138384\times 1}{1\times 49}$$

47. Calculate these two products separately.

    $$\frac{138384}{49}$$

48. Calculate this operation before combining the remaining terms.

    $$138384\times \frac{1}{49}=\frac{138384}{49}$$

49. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{138384}{49}=-\frac{138384}{49}$$

50. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=49$$

51. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{19794}{7}=\frac{19794\times 7}{7\times 7}$$

52. Calculate the converted numerator; the denominator is now common.

    $$\frac{138558}{49}$$

53. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{138384}{49}=\frac{\left(-138384\right)\times 1}{49\times 1}$$

54. Calculate the converted numerator; the denominator is now common.

    $$-\frac{138384}{49}$$

55. Calculate this operation before combining the remaining terms.

    $$138558-138384=174$$

56. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{174\times 1}{1\times 49}$$

57. Calculate these two products separately.

    $$\frac{174}{49}$$

58. Calculate this operation before combining the remaining terms.

    $$174\times \frac{1}{49}=\frac{174}{49}$$

59. Keep the common denominator and add the signed numerators.

    $$\frac{138558-138384}{49}=\frac{174}{49}$$

60. Calculate this operation before combining the remaining terms.

    $$\frac{19794}{7}-\frac{138384}{49}=\frac{174}{49}$$

61. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{174}{49}}=\frac{1}{7}\times \sqrt{174}$$

62. Standard deviation is the POSITIVE square root of the variance.

    $$\sqrt{\frac{174}{49}}$$

63. Round the standard deviation only now.

    $$1.88442$$

64. Range is maximum minus minimum.

    $$56-50=6$$

Answer

mean=372/7; variance=174/49; standard deviation≈1.88442; range=6