Questions: Given the following data, find the following: Round your answers to 2 decimal places as needed 6 9 18 23 25 34 38 50 56 64 76 Mean = Median = Range = Sample standard deviation =

Solution Steps

The mean (\(\mu\)) is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} \] Given the data: \([6, 9, 18, 23, 25, 34, 38, 50, 56, 64, 76]\), we have: \[ \mu = \frac{6 + 9 + 18 + 23 + 25 + 34 + 38 + 50 + 56 + 64 + 76}{11} = \frac{399}{11} = 36.27 \]

The median is the value that separates the higher half from the lower half of the data set. For a dataset of size \(N\), the median is at position: \[ \text{Rank} = Q \times (N + 1) \] For \(Q = 0.5\) (the median) and \(N = 11\): \[ \text{Rank} = 0.5 \times (11 + 1) = 6.0 \] The sorted data is: \([6, 9, 18, 23, 25, 34, 38, 50, 56, 64, 76]\). The value at position 6 is 34.

The range is the difference between the maximum and minimum values in the dataset: \[ \text{Range} = \max(x_i) - \min(x_i) = 76 - 6 = 70 \]

The sample standard deviation (\(s\)) is calculated using the formula: \[ s = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}} \] Given the mean \(\mu = 36.27\), we calculate the variance (\(\sigma^2\)): \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 523.02 \] Thus, the standard deviation is: \[ s = \sqrt{523.02} = 22.87 \]

Final Answer