Mathematics 10 — Exercise 11.2 · Q13 (iii)

Full question

Mean 18,Σfx²=1670,Σf=5. Find variance and standard deviation.

Step-by-step solution

1. Mean=sum/n.

   $$\frac{90}{5}$$

2. Calculate this operation before combining the remaining terms.

   $$90\times 1=90$$

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.

   $$\frac{90\times 1}{1\times 5}$$

5. Calculate these two products separately.

   $$\frac{90}{5}$$

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.

    $$18\times 1=18$$

14. Show the two divisions explicitly.

    $$\frac{\frac{90}{5}}{\frac{5}{5}}$$

15. Evaluate the numerator division.

    $$\frac{90}{5}=18$$

16. Evaluate the denominator division.

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

17. Write the reduced fraction.

    $$18\times \frac{1}{1}$$

18. Calculate this operation before combining the remaining terms.

    $$90\times \frac{1}{5}=18$$

19. Variance=sum of squares/n−mean².

    $$\frac{1670}{5}-{18}^2$$

20. Calculate this operation before combining the remaining terms.

    $$1670\times 1=1670$$

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

    $$\frac{1670\times 1}{1\times 5}$$

22. Calculate these two products separately.

    $$\frac{1670}{5}$$

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

    $$\mathrm{GCD}=5$$

24. Calculate this operation before combining the remaining terms.

    $$334\times 1=334$$

25. Show the two divisions explicitly.

    $$\frac{\frac{1670}{5}}{\frac{5}{5}}$$

26. Evaluate the numerator division.

    $$\frac{1670}{5}=334$$

27. Evaluate the denominator division.

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

28. Write the reduced fraction.

    $$334\times \frac{1}{1}$$

29. Calculate this operation before combining the remaining terms.

    $$1670\times \frac{1}{5}=334$$

30. Calculate this operation before combining the remaining terms.

    $${18}^2=324$$

31. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 324=-324$$

32. Calculate this operation before combining the remaining terms.

    $$334-324=10$$

33. Standard deviation=positive square root of variance.

    $$\sqrt{10}=\sqrt{10}$$

Answer

mean=18; variance=10; standard deviation=√(10)≈3.16228