Chapter 10.3, Problem 4GP

To determine

To compare mean and standard deviation

Answer to Problem 4GP

Standard deviation of line up B is more than line up A

Explanation of Solution

Given information:

Lineup A
4.25	4.31	4.19	4.40	4.23
4.18	4.71	4.56	4.32	4.39

Lineup B
4.47	4.68	4.25	4.41	4.49
4.18	4.27	4.69	4.32	4.44

Formula used:Mean is, Mean= $\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}$

Where, n=total number of terms,

Standard deviation,

  ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$

Where $x_i^{}$ = each value from data

  $\mu$ =mean of data

  $N$ = size of data

Calculation:

For Lineup A

For mean,

Substituting the values,

  $mean$ = $\frac{4.25+4.31+4.19+4.40+4.23+4.18+4.71+4.56+4.32+4.39}{10}$

So,

  $mean=\frac{43.5}{10}$

On solving,

  $mean=4.35$

Hence, mean is 4.35

For Standard deviation,

Values	Difference= $x_{}^{}-\mu$	${(x_{}^{}-\mu )}^2$
4.25	-0.104	0.010816
4.31	-0.044	0.001936
4.19	-0.164	0.026896
4.40	0.046	0.002116
4.23	-0.124	0.015376
4.18	-0.174	0.030276
4.71	0.356	0.126736
4.56	0.206	0.042436
4.32	-0.034	0.001156
4.39	0.036	0.001296
Total		0.25904

Substituting the values,

  $=\sqrt{\frac{0.25904}{10}}$

On solving,

  $=0.025904$

Hence, standard deviation is 0.026

For Line up B

For mean,

Substituting values

  $mean$ = $\frac{4.47+4.68+4.25+4.41+4.49+4.18+4.27+4.69+4.32+4.44}{10}$

So,

  $mean=\frac{44.2}{10}$

On solving,

  $mean=4.42$

Hence, mean is 4.42

For standard deviation,

Values	Difference= $x_{}^{}-\mu$	${(x_{}^{}-\mu )}^2$
4.47	0.05	0.0025
4.68	0.26	0.0676
4.25	-0.17	0.0289
4.41	-0.01	0.0001
4.49	0.07	0.0049
4.18	-0.24	0.0576
4.27	-0.15	0.0225
4.69	0.27	0.0729
4.32	-0.1	0.01
4.44	0.02	0.0004
Total		0.2674

Substituting the values,

  $\sigma =\sqrt{\frac{0.2674}{10}}$

On solving,

  $=0.1635$

Hence, standard deviation is 0.1635

Comparing the mean of A and B, the mean of A is slightly greater than B. Comparing the standard deviation, line up B has a greater standard deviation than line up A, hence, the observations of B are spread over a larger range of numbers.