Questions: Find the standard deviation of the following data: (report 3 decimal places) 8.7, 7.7, 6.1, 5.6, 9.3, 9.1, 7.7

Solution Steps

To find the standard deviation of a dataset, first calculate the mean of the data. Then, find the squared differences between each data point and the mean. Calculate the average of these squared differences, which gives the variance. Finally, take the square root of the variance to obtain the standard deviation.

To find the mean of the dataset \([8.7, 7.7, 6.1, 5.6, 9.3, 9.1, 7.7]\), sum all the data points and divide by the number of data points: \[ \text{mean} = \frac{8.7 + 7.7 + 6.1 + 5.6 + 9.3 + 9.1 + 7.7}{7} = 7.7429 \]

The variance is the average of the squared differences from the mean. First, calculate each squared difference: \[ (8.7 - 7.7429)^2, (7.7 - 7.7429)^2, (6.1 - 7.7429)^2, (5.6 - 7.7429)^2, (9.3 - 7.7429)^2, (9.1 - 7.7429)^2, (7.7 - 7.7429)^2 \] Then, find the average of these squared differences: \[ \text{variance} = \frac{(8.7 - 7.7429)^2 + (7.7 - 7.7429)^2 + (6.1 - 7.7429)^2 + (5.6 - 7.7429)^2 + (9.3 - 7.7429)^2 + (9.1 - 7.7429)^2 + (7.7 - 7.7429)^2}{7} = 1.782 \]

The standard deviation is the square root of the variance: \[ \text{standard deviation} = \sqrt{1.782} = 1.335 \]

Final Answer