Answer to Question #219421 in Statistics and Probability for fanscil

Question

11 8 15 12 11 12 11 14 15 5 6 11

16 14 13 14 13 9 11 19 12 10 13 5

13 7 3 14 13 16 20 15 5 15 15 5

15 7 9 11 17 18 10 14 9 7 4 15

13 12 16 8 12 12

(a)  Find:

i.              Mean

ii.             Median

iii.            Mode

iv.            Interquartile range

v.             Range

vi.            Variance

vii. Standard deviation

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} X & f \\ \hline 3 & 1 \\ \hdashline 4 & 1\\ \hdashline 5 & 4\\ \hdashline 6 & 1\\ \hdashline 7 & 3\\ \hdashline 8 & 2\\ \hdashline 9 & 3\\ \hdashline 10 & 2\\ \hdashline 11 & 6 \\ \hdashline 12 & 6\\ \hdashline 13 & 6\\ \hdashline 14 & 5\\ \hdashline 15 & 7\\ \hdashline 16 & 3\\ \hdashline 17 & 1\\ \hdashline 18 & 1\\ \hdashline 19 & 1\\ \hdashline 20 & 1\\ \hdashline & \sum f=54\\ \hdashline \end{array}$$

i.

$$mean=\bar{X}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nX_i=\dfrac{630}{54}=\dfrac{35}{3}\approx11.666667$$

ii.

$$X_{27}=12=X_{28}$$

$$median=12$$

iii.

$$mode=15,$$

appeared 7 times.

iv.

Lower quartile $(X_L):9$

Upper quartile $(X_U):15$

Interquartile range

$$IQR=X_U-X_L=15-9=6$$

v.

$$Range=20-3=17$$

vi.

$$Var(X)=\sigma^2=\dfrac{1}{n}\displaystyle\sum_{i=1}^n(X_i-\bar{X})^2=\dfrac{830}{54}=\dfrac{415}{27}$$

$$\approx15.370370$$

vii.

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{415}{27}}\approx3.920506$$