A sample of size $$ n=10 $$ is drawn from a population. The data is shown below. $$ \begin{array}{ll}115 & 98.8 \quad 115 \quad 105.3 \ 115 \quad 87.9 & 92.2 \ ext { What is the range of this data set? } \ ext { Wh.9 } \ ext { When }\end{array} $$ range $$ = $$ What is the standard deviation of this data set? (Round to 2 decimal places) stdev $$ =\square $$

Question

A sample of size $$ n=10 $$ is drawn from a population. The data is shown below. $$ \begin{array}{ll}115 & 98.8 \quad 115 \quad 105.3 \ 115 \quad 87.9 & 92.2 \ 	ext { What is the range of this data set? } \ 	ext { Wh.9 } \ 	ext { When }\end{array} $$ range $$ = $$ What is the standard deviation of this data set? (Round to 2 decimal places) stdev $$ =\square $$

UpStudy AI · Accepted Answer

To find the range of the data set, we need to determine the difference between the largest and smallest values in the data set.

Given data set:
$$ \begin{array}{ll}115 & 98.8 \quad 115 \quad 105.3 \ 115 \quad 87.9 & 92.2 \ 	ext { Wh.9 } \ 	ext { When }\end{array} $$

The largest value in the data set is 115 and the smallest value is 87.9. Therefore, the range of the data set is:
Range = Largest value - Smallest value
Range = 115 - 87.9
Range = 27.1

To find the standard deviation of the data set, we can use the formula:
Standard Deviation (stdev) = $$ \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}} $$

Where:
- $$ x_i $$ is each individual data point
- $$ \bar{x} $$ is the mean of the data set
- $$ n $$ is the number of data points

First, we need to calculate the mean of the data set:
Mean = $$ \frac{\sum_{i=1}^{n}x_i}{n} $$

Let's calculate the mean and then the standard deviation.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(115+98.8+115+105.3+115+87.9+92.2+87.9ight)}{10}$$
- step1: Remove the parentheses:
  $$\frac{115+98.8+115+105.3+115+87.9+92.2+87.9}{10}$$
- step2: Write as a multiplication:
  $$\frac{115	imes 3+98.8+105.3+87.9+92.2+87.9}{10}$$
- step3: Multiply the numbers:
  $$\frac{345+98.8+105.3+87.9+92.2+87.9}{10}$$
- step4: Add the numbers:
  $$\frac{817.1}{10}$$
- step5: Convert the expressions:
  $$\frac{\frac{8171}{10}}{10}$$
- step6: Multiply by the reciprocal:
  $$\frac{8171}{10}	imes \frac{1}{10}$$
- step7: Multiply the fractions:
  $$\frac{8171}{10	imes 10}$$
- step8: Multiply:
  $$\frac{8171}{100}$$
The mean of the data set is 81.71.

Now, we can calculate the standard deviation using the formula provided earlier.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\sqrt{\left(115-81.71ight)^{2}+\left(98.8-81.71ight)^{2}+\left(115-81.71ight)^{2}+\left(105.3-81.71ight)^{2}+\left(115-81.71ight)^{2}+\left(87.9-81.71ight)^{2}+\left(92.2-81.71ight)^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step1: Write as a multiplication:
  $$\frac{\sqrt{\left(115-81.71ight)^{2}	imes 3+\left(98.8-81.71ight)^{2}+\left(105.3-81.71ight)^{2}+\left(87.9-81.71ight)^{2}+\left(92.2-81.71ight)^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step2: Subtract the numbers:
  $$\frac{\sqrt{33.29^{2}	imes 3+\left(98.8-81.71ight)^{2}+\left(105.3-81.71ight)^{2}+\left(87.9-81.71ight)^{2}+\left(92.2-81.71ight)^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step3: Subtract the numbers:
  $$\frac{\sqrt{33.29^{2}	imes 3+17.09^{2}+\left(105.3-81.71ight)^{2}+\left(87.9-81.71ight)^{2}+\left(92.2-81.71ight)^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step4: Subtract the numbers:
  $$\frac{\sqrt{33.29^{2}	imes 3+17.09^{2}+23.59^{2}+\left(87.9-81.71ight)^{2}+\left(92.2-81.71ight)^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step5: Subtract the numbers:
  $$\frac{\sqrt{33.29^{2}	imes 3+17.09^{2}+23.59^{2}+6.19^{2}+\left(92.2-81.71ight)^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step6: Subtract the numbers:
  $$\frac{\sqrt{33.29^{2}	imes 3+17.09^{2}+23.59^{2}+6.19^{2}+10.49^{2}+\left(87.9-81.71ight)^{2}}}{9}$$
- step7: Subtract the numbers:
  $$\frac{\sqrt{33.29^{2}	imes 3+17.09^{2}+23.59^{2}+6.19^{2}+10.49^{2}+6.19^{2}}}{9}$$
- step8: Convert the expressions:
  $$\frac{\sqrt{\left(\frac{3329}{100}ight)^{2}	imes 3+17.09^{2}+23.59^{2}+6.19^{2}+10.49^{2}+6.19^{2}}}{9}$$
- step9: Convert the expressions:
  $$\frac{\sqrt{\left(\frac{3329}{100}ight)^{2}	imes 3+\left(\frac{1709}{100}ight)^{2}+23.59^{2}+6.19^{2}+10.49^{2}+6.19^{2}}}{9}$$
- step10: Convert the expressions:
  $$\frac{\sqrt{\left(\frac{3329}{100}ight)^{2}	imes 3+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+6.19^{2}+10.49^{2}+6.19^{2}}}{9}$$
- step11: Convert the expressions:
  $$\frac{\sqrt{\left(\frac{3329}{100}ight)^{2}	imes 3+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}+10.49^{2}+6.19^{2}}}{9}$$
- step12: Convert the expressions:
  $$\frac{\sqrt{\left(\frac{3329}{100}ight)^{2}	imes 3+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}+\left(\frac{1049}{100}ight)^{2}+6.19^{2}}}{9}$$
- step13: Convert the expressions:
  $$\frac{\sqrt{\left(\frac{3329}{100}ight)^{2}	imes 3+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}+\left(\frac{1049}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}}}{9}$$
- step14: Multiply the terms:
  $$\frac{\sqrt{3	imes \frac{3329^{2}}{100^{2}}+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}+\left(\frac{1049}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}}}{9}$$
- step15: Rewrite the expression:
  $$\frac{\sqrt{3	imes \frac{3329^{2}}{100^{2}}+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}+\left(\frac{619}{100}ight)^{2}+\left(\frac{1049}{100}ight)^{2}}}{9}$$
- step16: Add the numbers:
  $$\frac{\sqrt{3	imes \frac{3329^{2}}{100^{2}}+\left(\frac{1709}{100}ight)^{2}+\left(\frac{2359}{100}ight)^{2}+\frac{619^{2}}{5000}+\left(\frac{1049}{100}ight)^{2}}}{9}$$
- step17: Add the numbers:
  $$\frac{\sqrt{\frac{3	imes 3329^{2}+1709^{2}+2359^{2}+766322+1049^{2}}{10000}}}{9}$$
- step18: Simplify the root:
  $$\frac{\frac{\sqrt{3	imes 3329^{2}+1709^{2}+2359^{2}+766322+1049^{2}}}{100}}{9}$$
- step19: Multiply by the reciprocal:
  $$\frac{\sqrt{3	imes 3329^{2}+1709^{2}+2359^{2}+766322+1049^{2}}}{100}	imes \frac{1}{9}$$
- step20: Multiply the fractions:
  $$\frac{\sqrt{3	imes 3329^{2}+1709^{2}+2359^{2}+766322+1049^{2}}}{100	imes 9}$$
- step21: Multiply:
  $$\frac{\sqrt{3	imes 3329^{2}+1709^{2}+2359^{2}+766322+1049^{2}}}{900}$$
The standard deviation of the data set is approximately 7.34 when rounded to 2 decimal places.

Therefore, the range of the data set is 27.1 and the standard deviation is approximately 7.34.
The range of the data set is 27.1, and the standard deviation is approximately 7.34.

A sample of size is drawn from a population. The data is shown below.

range
What is the standard deviation of this data set? (Round to 2 decimal places)
stdev

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A sample of size is drawn from a population. The data is shown below. range What is the standard deviation of this data set? (Round to 2 decimal places) stdev