To calculate the standard deviation of the given dataset, follow these steps:
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Calculate the Mean (Average): \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \] The sum of the values is: \[ 34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98 = 1394 \] The number of values is 20. \[ \text{Mean} = \frac{1394}{20} = 69.7 \]
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Calculate the Deviations from the Mean: Subtract the mean from each value and square the result: \[ (34 - 69.7)^2, (45 - 69.7)^2, (50 - 69.7)^2, (56 - 69.7)^2, (60 - 69.7)^2, (65 - 69.7)^2, \] \[ (70 - 69.7)^2, (70 - 69.7)^2, (70 - 69.7)^2, (70 - 69.7)^2, (72 - 69.7)^2, (77 - 69.7)^2, \] \[ (77 - 69.7)^2, (80 - 69.7)^2, (82 - 69.7)^2, (86 - 69.7)^2, (86 - 69.7)^2, (89 - 69.7)^2, \] \[ (95 - 69.7)^2, (98 - 69.7)^2 \]
This results in: \[ (34 - 69.7)^2 = 1269.69, \quad (45 - 69.7)^2 = 607.69, \quad (50 - 69.7)^2 = 396.09, \] \[ (56 - 69.7)^2 = 187.69, \quad (60 - 69.7)^2 = 93.69, \quad (65 - 69.7)^2 = 22.09, \] \[ (70 - 69.7)^2 = 0.09, \quad (70 - 69.7)^2 = 0.09, \quad (70 - 69.7)^2 = 0.09, \] \[ (70 - 69.7)^2 = 0.09, \quad (72 - 69.7)^2 = 5.29, \quad (77 - 69.7)^2 = 52.89, \] \[ (77 - 69.7)^2 = 52.89, \quad (80 - 69.7)^2 = 104.04, \quad (82 - 69.7)^2 = 148.69, \] \[ (86 - 69.7)^2 = 260.49, \quad (86 - 69.7)^2 = 260.49, \quad (89 - 69.7)^2 = 374.89, \] \[ (95 - 69.7)^2 = 632.49, \quad (98 - 69.7)^2 = 817.69 \]
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Sum of Squared Deviations: Now sum up all the squared deviations: \[ 1269.69 + 607.69 + 396.09 + 187.69 + 93.69 + 22.09 + 0.09 + 0.09 + 0.09 + 0.09 + 5.29 + 52.89 + 52.89 + 104.04 + 148.69 + 260.49 + 260.49 + 374.89 + 632.49 + 817.69 = 3579.40 \]
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Calculate Variance: Variance is the average of the squared deviations: \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{N} = \frac{3579.40}{20} = 178.97 \]
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Calculate the Standard Deviation: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{178.97} \approx 13.38 \]
Thus, the standard deviation of the dataset is approximately 13.38.