Calculate the range and the standard deviation for the set of numbers. \( 6,3,2,6,7,6,1,5,2,7 \), The range is \( \square \). (Simplify your answer.)

It seems there was an issue with the calculation. Let's calculate the standard deviation manually. To calculate the standard deviation, we need to find the squared differences between each number and the mean, sum them up, and then divide by the total count of numbers. Finally, we take the square root of the result. Let's calculate the standard deviation manually. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{\left(6-4.5\right)^{2}+\left(3-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(6-4.5\right)^{2}+\left(7-4.5\right)^{2}+\left(6-4.5\right)^{2}+\left(1-4.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step1: Write as a multiplication: \(\sqrt{\frac{\left(6-4.5\right)^{2}\times 3+\left(3-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}+\left(1-4.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step2: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(3-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}+\left(1-4.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step3: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}+\left(1-4.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step4: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+\left(7-4.5\right)^{2}+\left(1-4.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step5: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(1-4.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step6: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+\left(5-4.5\right)^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step7: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(2-4.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step8: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+\left(7-4.5\right)^{2}}{10}}\) - step9: Subtract the numbers: \(\sqrt{\frac{1.5^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step10: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-1.5\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step11: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step12: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+2.5^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step13: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-3.5\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step14: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+0.5^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step15: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\left(-2.5\right)^{2}+2.5^{2}}{10}}\) - step16: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+2.5^{2}}{10}}\) - step17: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{2}\right)^{2}\times 3+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}}{10}}\) - step18: Multiply the terms: \(\sqrt{\frac{\frac{27}{4}+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}}{10}}\) - step19: Rewrite the expression: \(\sqrt{\frac{\frac{27}{4}+\left(-\frac{3}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(-\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}}{10}}\) - step20: Add the numbers: \(\sqrt{\frac{\frac{27}{4}+\left(-\frac{3}{2}\right)^{2}+\frac{25}{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}}{10}}\) - step21: Rewrite the expression: \(\sqrt{\frac{\frac{27}{4}+\left(-\frac{3}{2}\right)^{2}+\frac{25}{2}+\left(\frac{5}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}{10}}\) - step22: Add the numbers: \(\sqrt{\frac{\frac{27}{4}+\left(-\frac{3}{2}\right)^{2}+\frac{25}{2}+\frac{25}{2}+\left(-\frac{7}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}{10}}\) - step23: Add the numbers: \(\sqrt{\frac{\frac{93}{2}}{10}}\) - step24: Divide the terms: \(\sqrt{\frac{93}{20}}\) - step25: Use the properties of radicals: \(\frac{\sqrt{93}}{\sqrt{20}}\) - step26: Simplify the expression: \(\frac{\sqrt{93}}{2\sqrt{5}}\) - step27: Multiply by the Conjugate: \(\frac{\sqrt{93}\times \sqrt{5}}{2\sqrt{5}\times \sqrt{5}}\) - step28: Multiply the numbers: \(\frac{\sqrt{465}}{2\sqrt{5}\times \sqrt{5}}\) - step29: Multiply the numbers: \(\frac{\sqrt{465}}{10}\) The range of the given set of numbers is 6, and the standard deviation is approximately 2.156.