Mathematics 10 — Exercise 11.2 · Q13 (i)

Full question

Σx=45,Σx²=421,n=5. Find variance and standard deviation.

Step-by-step solution

1. Mean=sum/n.

   $$\frac{45}{5}$$

2. Calculate this operation before combining the remaining terms.

   $$45\times 1=45$$

3. Calculate this operation before combining the remaining terms.

   $$1\times 5=5$$

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

   $$\frac{45\times 1}{1\times 5}$$

5. Calculate these two products separately.

   $$\frac{45}{5}$$

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

   $$\mathrm{GCD}=5$$

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

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

8. Calculate these two products separately.

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

9. Show the two divisions explicitly.

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

10. Write the reduced fraction.

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

11. Calculate this operation before combining the remaining terms.

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

12. Calculate this operation before combining the remaining terms.

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

13. Calculate this operation before combining the remaining terms.

    $$9\times 1=9$$

14. Show the two divisions explicitly.

    $$\frac{\frac{45}{5}}{\frac{5}{5}}$$

15. Evaluate the numerator division.

    $$\frac{45}{5}=9$$

16. Evaluate the denominator division.

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

17. Write the reduced fraction.

    $$9\times \frac{1}{1}$$

18. Calculate this operation before combining the remaining terms.

    $$45\times \frac{1}{5}=9$$

19. Variance=sum of squares/n−mean².

    $$\frac{421}{5}-9^2$$

20. Calculate this operation before combining the remaining terms.

    $$421\times 1=421$$

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

    $$\frac{421\times 1}{1\times 5}$$

22. Calculate these two products separately.

    $$\frac{421}{5}$$

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

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

24. Calculate these two products separately.

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

25. Calculate this operation before combining the remaining terms.

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

26. Calculate this operation before combining the remaining terms.

    $$421\times \frac{1}{5}=\frac{421}{5}$$

27. Calculate this operation before combining the remaining terms.

    $$9^2=81$$

28. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 81=-81$$

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

    $$\mathrm{LCD}=5$$

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

    $$\frac{421}{5}=\frac{421\times 1}{5\times 1}$$

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

    $$\frac{421}{5}$$

32. Calculate this operation before combining the remaining terms.

    $$1\times 1=1$$

33. Calculate this operation before combining the remaining terms.

    $$\left(-81\right)\times 1=-81$$

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

    $$-81\times \frac{1}{1}=\frac{\left(-81\right)\times 5}{1\times 5}$$

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

    $$\frac{\left(-405\right)\times 1}{1\times 5}$$

36. Show the two divisions explicitly.

    $$\frac{-\frac{405}{5}}{\frac{5}{5}}$$

37. Evaluate the numerator division.

    $$-\frac{405}{5}=-81$$

38. Write the reduced fraction.

    $$-81\times \frac{1}{1}$$

39. Calculate this operation before combining the remaining terms.

    $$\left(-405\right)\times \frac{1}{5}=-81$$

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

    $$-\frac{405}{5}$$

41. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 405=-405$$

42. Calculate this operation before combining the remaining terms.

    $$421-405=16$$

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

    $$\frac{16\times 1}{1\times 5}$$

44. Calculate these two products separately.

    $$\frac{16}{5}$$

45. Calculate this operation before combining the remaining terms.

    $$16\times \frac{1}{5}=\frac{16}{5}$$

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

    $$\frac{421-405}{5}=\frac{16}{5}$$

47. Calculate this operation before combining the remaining terms.

    $$\frac{421}{5}-81=\frac{16}{5}$$

48. Standard deviation=positive square root of variance.

    $$\sqrt{\frac{16}{5}}=\frac{4}{5}\times \sqrt{5}$$

Answer

mean=9; variance=16/5; standard deviation=4·√(5)/5≈1.78885