Questions: What is the variance and standard deviation of these groups of numbers? 24,67,17,20,31,44,12,23,16,37

Solution Steps

To find the mean \( \mu \) of the dataset, we use the formula: \[ \mu = \frac{\sum x_i}{n} \] where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points. For our dataset: \[ \mu = \frac{24 + 67 + 17 + 20 + 31 + 44 + 12 + 23 + 16 + 37}{10} = \frac{291}{10} = 29.1 \]

The variance \( \sigma^2 \) is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \] We first compute \( (x_i - \mu)^2 \) for each data point, sum these values, and then divide by \( n \): \[ \sigma^2 = \frac{(24 - 29.1)^2 + (67 - 29.1)^2 + (17 - 29.1)^2 + (20 - 29.1)^2 + (31 - 29.1)^2 + (44 - 29.1)^2 + (12 - 29.1)^2 + (23 - 29.1)^2 + (16 - 29.1)^2 + (37 - 29.1)^2}{10} = 248.09 \]

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{248.09} = 15.75 \]

Final Answer