Q 3.61. - Elementary Statistics

Question

Consider the data set 1,2,3,4,5,6,7,8,9.

Part (a)  Use the defining formula to obtain the sample standard deviation.

Part (b) Replace the 9 in the data set by 99, and again use the defining formula to compute the sample standard deviation.

Part (c) Compare your answer in parts (a) and (b). The lack of what property of the standard deviation accounts for its extreme sensitivity to the change of 9 to 99? 

Step-by-Step Solution

Answer

Part (a) $\sigma \approx 2.7389$

Part (b) $\sigma \approx 31.5832$

Part (c) The lack of opposition.

Part (a) Step 1. Given information.

The data points are as follows:

1,2,3,4,5,6,7,8,9. 

Part (a) Step 2. The sample standard deviation

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}}$

Determine the mean of the given data set.

$$\begin{gathered} \overline{x}=\frac{1+2+3+4+5+6+7+8+9}{9} \\ \overline{x}=\frac{45}{9} \\ \overline{x}=5 \end{gathered}$$

Data	Data Mean	${(Data Mean)}^2$
1	-4	16
2	-3	9
3	-2	4
4	-1	1
5	0	0
6	1	1
7	2	4
8	3	9
9	4	16

The variance

$$\begin{gathered} \sigma =\sqrt{\frac{\sum _{}{\left(x_i-\overline{\overline{x}}\right)}^2}{n-1}} \\ \sigma =\sqrt{\frac{60}{9-1}} \\ \sigma \approx 2.7389 \end{gathered}$$

As a result, the required value is 2.7289

Part (b) Step 1. Given information.

The data points are as follows:

1,2,3,4,5,6,7,8,99. 

Part (b) Step 2. The given data's sample standard deviation.

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}}$

Determine the mean of the given data set.

$$\begin{gathered} \overline{x}=\frac{1+2+3+4+5+6+7+8+99}{9} \\ \overline{x}=\frac{135}{9} \\ \overline{x}=15 \end{gathered}$$

Data	Data Mean	${(Data Mean)}^2$
1	-14	196
2	-13	169
3	-12	144
4	-11	121
5	-10	100
6	-9	81
7	-8	64
8	-7	49
9	-6	36

The variance

$$\begin{gathered} \sigma =\sqrt{\frac{\sum _{}{\left(x_i-\overline{\overline{x}}\right)}^2}{n-1}} \\ \sigma =\sqrt{\frac{7980}{9-1}} \\ \sigma \approx 31.5832 \end{gathered}$$

As a result, the required value is 31.5832

Part (c) Step 1. Given information.

The data points are as follows:

1,2,3,4,5,6,7,8,99.4. 

Part (c) Step 2. The cause of the extreme deviation in both data sets.

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}}$

Sensitivity to outlier 99 is caused by a lack of resistance.