Questions: Why is it important to consider all the measures of location in reporting statistics? Each of the measures has advantages and disadvantages in representing the data. The mean is the best representation for skewed data. For a symmetrical distribution they all have the same value.

Solution Steps

The mean \(\mu\) is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} \] Given the data: \([10, 20, 20, 30, 40, 50, 60, 70, 80, 90]\), we have: \[ \mu = \frac{470}{10} = 47.0 \]

The median is the 0.5 quantile. For a dataset of size \(N = 10\), the rank is calculated as: \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (10 + 1) = 5.5 \] The median is the average of the 5th and 6th values in the sorted data: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{40 + 50}{2} = 45.0 \]

The variance \(\sigma^2\) for a sample is calculated as: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \] Using the mean \(\mu = 47.0\), the variance is: \[ \sigma^2 = 756.67 \] The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{756.67} = 27.51 \]

Final Answer