Find the range, mean, and standard deviation for the data set. \( 21,19,19,22,22,19,22,19,22,18 \) The range is 4 . (Type an integer or decimal. Do not round.) The mean is 20.3 . (Type an integer or decimal. Do not round.) The standard deviation is (Type an integer or decimal

To find the range, mean, and standard deviation for the given data set, we can follow these steps: 1. Find the range: The range is the difference between the highest and lowest values in the data set. In this case, the highest value is 22 and the lowest value is 18. Therefore, the range is \(22 - 18 = 4\). 2. Find the mean: The mean is the average of all the values in the data set. To find the mean, we add up all the values and divide by the total number of values. In this case, the sum of the values is \(21 + 19 + 19 + 22 + 22 + 19 + 22 + 19 + 22 + 18 = 204\). There are 10 values in the data set. Therefore, the mean is \(204 / 10 = 20.4\). 3. Find the standard deviation: The standard deviation is a measure of the spread or dispersion of the data set. It is calculated using the formula: \[ \text{Standard Deviation} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \text{Mean})^2}{n}} \] where \(x_i\) represents each value in the data set, \(\text{Mean}\) is the mean of the data set, and \(n\) is the total number of values in the data set. Let's calculate the standard deviation using the given data set: \[ \text{Standard Deviation} = \sqrt{\frac{(21-20.4)^2 + (19-20.4)^2 + (19-20.4)^2 + (22-20.4)^2 + (22-20.4)^2 + (19-20.4)^2 + (22-20.4)^2 + (19-20.4)^2 + (22-20.4)^2 + (18-20.4)^2}{10}} \] Now, let's calculate the standard deviation. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{\left(21-20.4\right)^{2}+\left(19-20.4\right)^{2}+\left(19-20.4\right)^{2}+\left(22-20.4\right)^{2}+\left(22-20.4\right)^{2}+\left(19-20.4\right)^{2}+\left(22-20.4\right)^{2}+\left(19-20.4\right)^{2}+\left(22-20.4\right)^{2}+\left(18-20.4\right)^{2}}{10}}\) - step1: Write as a multiplication: \(\sqrt{\frac{\left(21-20.4\right)^{2}+\left(19-20.4\right)^{2}\times 4+\left(22-20.4\right)^{2}\times 4+\left(18-20.4\right)^{2}}{10}}\) - step2: Subtract the numbers: \(\sqrt{\frac{\left(21-20.4\right)^{2}+\left(-1.4\right)^{2}\times 4+\left(22-20.4\right)^{2}\times 4+\left(18-20.4\right)^{2}}{10}}\) - step3: Subtract the numbers: \(\sqrt{\frac{\left(21-20.4\right)^{2}+\left(-1.4\right)^{2}\times 4+1.6^{2}\times 4+\left(18-20.4\right)^{2}}{10}}\) - step4: Subtract the numbers: \(\sqrt{\frac{0.6^{2}+\left(-1.4\right)^{2}\times 4+1.6^{2}\times 4+\left(18-20.4\right)^{2}}{10}}\) - step5: Subtract the numbers: \(\sqrt{\frac{0.6^{2}+\left(-1.4\right)^{2}\times 4+1.6^{2}\times 4+\left(-2.4\right)^{2}}{10}}\) - step6: Convert the expressions: \(\sqrt{\frac{0.6^{2}+\left(-\frac{7}{5}\right)^{2}\times 4+1.6^{2}\times 4+\left(-2.4\right)^{2}}{10}}\) - step7: Convert the expressions: \(\sqrt{\frac{0.6^{2}+\left(-\frac{7}{5}\right)^{2}\times 4+\left(\frac{8}{5}\right)^{2}\times 4+\left(-2.4\right)^{2}}{10}}\) - step8: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{5}\right)^{2}+\left(-\frac{7}{5}\right)^{2}\times 4+\left(\frac{8}{5}\right)^{2}\times 4+\left(-2.4\right)^{2}}{10}}\) - step9: Convert the expressions: \(\sqrt{\frac{\left(\frac{3}{5}\right)^{2}+\left(-\frac{7}{5}\right)^{2}\times 4+\left(\frac{8}{5}\right)^{2}\times 4+\left(-\frac{12}{5}\right)^{2}}{10}}\) - step10: Multiply the terms: \(\sqrt{\frac{\left(\frac{3}{5}\right)^{2}+\frac{196}{25}+\left(\frac{8}{5}\right)^{2}\times 4+\left(-\frac{12}{5}\right)^{2}}{10}}\) - step11: Multiply the terms: \(\sqrt{\frac{\left(\frac{3}{5}\right)^{2}+\frac{196}{25}+\frac{256}{25}+\left(-\frac{12}{5}\right)^{2}}{10}}\) - step12: Add the numbers: \(\sqrt{\frac{\frac{121}{5}}{10}}\) - step13: Divide the terms: \(\sqrt{\frac{121}{50}}\) - step14: Use the properties of radicals: \(\frac{\sqrt{121}}{\sqrt{50}}\) - step15: Simplify the expression: \(\frac{11}{5\sqrt{2}}\) - step16: Multiply by the Conjugate: \(\frac{11\sqrt{2}}{5\sqrt{2}\times \sqrt{2}}\) - step17: Multiply the numbers: \(\frac{11\sqrt{2}}{10}\) The range of the data set is 4, the mean is 20.3, and the standard deviation is approximately 1.555635.