Find the range, mean, and standard deviation for the data set. \( 20,19,19,21,22,19,21,23,23,21 \) The range is 4 . (Type an integer or decimal. Do not round.) The mean is 20.8. (Type an integer or decimal. Do not round.) The standard deviation is (Type an integer or decimal rounded to two decimal places as needed.)

To find the range, mean, and standard deviation for the given data set \( 20, 19, 19, 21, 22, 19, 21, 23, 23, 21 \), we will follow these steps: 1. **Range**: The range is the difference between the maximum and minimum values in the data set. 2. **Mean**: The mean is the average of all the values in the data set. 3. **Standard Deviation**: The standard deviation measures the amount of variation or dispersion of a set of values. Given: - Range = 4 - Mean = 20.8 Let's calculate the standard deviation step by step: 1. **Range**: - Range = Maximum value - Minimum value - Range = 23 - 19 = 4 2. **Mean**: - Mean = Sum of all values / Number of values - Mean = (20 + 19 + 19 + 21 + 22 + 19 + 21 + 23 + 23 + 21) / 10 = 208 / 10 = 20.8 3. **Standard Deviation**: - Standard Deviation = √[Σ(x - Mean)² / n] - Where x is each value in the data set, Mean is the mean of the data set, and n is the number of values in the data set. Let's calculate the standard deviation using the formula above. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{\left(\left(20-20.8\right)^{2}+\left(19-20.8\right)^{2}+\left(19-20.8\right)^{2}+\left(21-20.8\right)^{2}+\left(22-20.8\right)^{2}+\left(19-20.8\right)^{2}+\left(21-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(21-20.8\right)^{2}\right)}{10}}\) - step1: Remove the parentheses: \(\sqrt{\frac{\left(20-20.8\right)^{2}+\left(19-20.8\right)^{2}+\left(19-20.8\right)^{2}+\left(21-20.8\right)^{2}+\left(22-20.8\right)^{2}+\left(19-20.8\right)^{2}+\left(21-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(21-20.8\right)^{2}}{10}}\) - step2: Write as a multiplication: \(\sqrt{\frac{\left(20-20.8\right)^{2}+\left(19-20.8\right)^{2}\times 3+\left(21-20.8\right)^{2}\times 3+\left(22-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}}{10}}\) - step3: Subtract the numbers: \(\sqrt{\frac{\left(20-20.8\right)^{2}+\left(-1.8\right)^{2}\times 3+\left(21-20.8\right)^{2}\times 3+\left(22-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}}{10}}\) - step4: Subtract the numbers: \(\sqrt{\frac{\left(20-20.8\right)^{2}+\left(-1.8\right)^{2}\times 3+0.2^{2}\times 3+\left(22-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}}{10}}\) - step5: Subtract the numbers: \(\sqrt{\frac{\left(-0.8\right)^{2}+\left(-1.8\right)^{2}\times 3+0.2^{2}\times 3+\left(22-20.8\right)^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}}{10}}\) - step6: Subtract the numbers: \(\sqrt{\frac{\left(-0.8\right)^{2}+\left(-1.8\right)^{2}\times 3+0.2^{2}\times 3+1.2^{2}+\left(23-20.8\right)^{2}+\left(23-20.8\right)^{2}}{10}}\) - step7: Subtract the numbers: \(\sqrt{\frac{\left(-0.8\right)^{2}+\left(-1.8\right)^{2}\times 3+0.2^{2}\times 3+1.2^{2}+2.2^{2}+\left(23-20.8\right)^{2}}{10}}\) - step8: Subtract the numbers: \(\sqrt{\frac{\left(-0.8\right)^{2}+\left(-1.8\right)^{2}\times 3+0.2^{2}\times 3+1.2^{2}+2.2^{2}+2.2^{2}}{10}}\) - step9: Convert the expressions: \(\sqrt{\frac{\left(-0.8\right)^{2}+\left(-\frac{9}{5}\right)^{2}\times 3+0.2^{2}\times 3+1.2^{2}+2.2^{2}+2.2^{2}}{10}}\) - step10: Convert the expressions: \(\sqrt{\frac{\left(-0.8\right)^{2}+\left(-\frac{9}{5}\right)^{2}\times 3+\left(\frac{1}{5}\right)^{2}\times 3+1.2^{2}+2.2^{2}+2.2^{2}}{10}}\) - step11: Convert the expressions: \(\sqrt{\frac{\left(-\frac{4}{5}\right)^{2}+\left(-\frac{9}{5}\right)^{2}\times 3+\left(\frac{1}{5}\right)^{2}\times 3+1.2^{2}+2.2^{2}+2.2^{2}}{10}}\) - step12: Convert the expressions: \(\sqrt{\frac{\left(-\frac{4}{5}\right)^{2}+\left(-\frac{9}{5}\right)^{2}\times 3+\left(\frac{1}{5}\right)^{2}\times 3+\left(\frac{6}{5}\right)^{2}+2.2^{2}+2.2^{2}}{10}}\) - step13: Convert the expressions: \(\sqrt{\frac{\left(-\frac{4}{5}\right)^{2}+\left(-\frac{9}{5}\right)^{2}\times 3+\left(\frac{1}{5}\right)^{2}\times 3+\left(\frac{6}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}+2.2^{2}}{10}}\) - step14: Convert the expressions: \(\sqrt{\frac{\left(-\frac{4}{5}\right)^{2}+\left(-\frac{9}{5}\right)^{2}\times 3+\left(\frac{1}{5}\right)^{2}\times 3+\left(\frac{6}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}}{10}}\) - step15: Multiply the terms: \(\sqrt{\frac{\left(-\frac{4}{5}\right)^{2}+\frac{243}{25}+\left(\frac{1}{5}\right)^{2}\times 3+\left(\frac{6}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}}{10}}\) - step16: Multiply the terms: \(\sqrt{\frac{\left(-\frac{4}{5}\right)^{2}+\frac{243}{25}+\frac{3}{25}+\left(\frac{6}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}+\left(\frac{11}{5}\right)^{2}}{10}}\) - step17: Add the numbers: \(\sqrt{\frac{\frac{108}{5}}{10}}\) - step18: Divide the terms: \(\sqrt{\frac{54}{25}}\) - step19: Use the properties of radicals: \(\frac{\sqrt{54}}{\sqrt{25}}\) - step20: Simplify the expression: \(\frac{3\sqrt{6}}{5}\) The standard deviation for the given data set is approximately 1.47 when rounded to two decimal places.