Mathematics 10 — Exercise 11.2 · Q5

Full question

The table gives the numbers of visitors to a library on five days. Calculate the variance and standard deviation.

Step-by-step solution

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

   $$1\times 120=120$$

2. Square that value.

   $${120}^2=14400$$

3. Multiply the square by its frequency.

   $$1\times 14400=14400$$

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

   $$1\times 135=135$$

5. Square that value.

   $${135}^2=18225$$

6. Multiply the square by its frequency.

   $$1\times 18225=18225$$

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

   $$1\times 130=130$$

8. Square that value.

   $${130}^2=16900$$

9. Multiply the square by its frequency.

   $$1\times 16900=16900$$

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

    $$1\times 125=125$$

11. Square that value.

    $${125}^2=15625$$

12. Multiply the square by its frequency.

    $$1\times 15625=15625$$

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

    $$1\times 140=140$$

14. Square that value.

    $${140}^2=19600$$

15. Multiply the square by its frequency.

    $$1\times 19600=19600$$

16. Add the frequencies to find n.

    $$1+1+1+1+1=5$$

17. Add the f×x column.

    $$120+135+130+125+140=650$$

18. Add the f×x² column.

    $$14400+18225+16900+15625+19600=84750$$

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

    $$\frac{650}{5}$$

20. Calculate this operation before combining the remaining terms.

    $$650\times 1=650$$

21. Calculate this operation before combining the remaining terms.

    $$1\times 5=5$$

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

    $$\frac{650\times 1}{1\times 5}$$

23. Calculate these two products separately.

    $$\frac{650}{5}$$

24. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

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

    $$\frac{5\times 1}{1\times 5}$$

26. Calculate these two products separately.

    $$\frac{5}{5}$$

27. Show the two divisions explicitly.

    $$\frac{\frac{5}{5}}{\frac{5}{5}}$$

28. Write the reduced fraction.

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

29. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{5}=1$$

30. Calculate this operation before combining the remaining terms.

    $$\frac{1}{1}=1$$

31. Calculate this operation before combining the remaining terms.

    $$130\times 1=130$$

32. Show the two divisions explicitly.

    $$\frac{\frac{650}{5}}{\frac{5}{5}}$$

33. Evaluate the numerator division.

    $$\frac{650}{5}=130$$

34. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

35. Write the reduced fraction.

    $$130\times \frac{1}{1}$$

36. Calculate this operation before combining the remaining terms.

    $$650\times \frac{1}{5}=130$$

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

    $$\frac{84750}{5}-{130}^2$$

38. Calculate this operation before combining the remaining terms.

    $$84750\times 1=84750$$

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

    $$\frac{84750\times 1}{1\times 5}$$

40. Calculate these two products separately.

    $$\frac{84750}{5}$$

41. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

42. Calculate this operation before combining the remaining terms.

    $$16950\times 1=16950$$

43. Show the two divisions explicitly.

    $$\frac{\frac{84750}{5}}{\frac{5}{5}}$$

44. Evaluate the numerator division.

    $$\frac{84750}{5}=16950$$

45. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

46. Write the reduced fraction.

    $$16950\times \frac{1}{1}$$

47. Calculate this operation before combining the remaining terms.

    $$84750\times \frac{1}{5}=16950$$

48. Calculate this operation before combining the remaining terms.

    $${130}^2=16900$$

49. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 16900=-16900$$

50. Calculate this operation before combining the remaining terms.

    $$16950-16900=50$$

51. Extract a perfect-square factor from under the root.

    $$50=25\times 2$$

52. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{50}=5\times \sqrt{2}$$

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

    $$\sqrt{50}$$

54. Round the standard deviation only now.

    $$7.07107$$

55. Range is maximum minus minimum.

    $$140-120=20$$

Answer

mean=130; variance=50; standard deviation≈7.07107; range=20