Questions: Using your calculator, find the range and standard deviation, round to two decimals places: The table below gives the number of hours spent watching TV last week by a sample of 24 children. 21, 51, 50, 86, 46, 81, 69, 29, 42, 53, 20, 48, 82, 56, 46, 27, 69, 90, 60, 37, 59, 26, 94, 16 Range = Standard Deviation =

Solution Steps

To find the range and standard deviation of the given data set, follow these steps:

1. Range: The range is the difference between the maximum and minimum values in the data set.
2. Standard Deviation: The standard deviation measures the amount of variation or dispersion in the data set. Use the formula for standard deviation, which involves calculating the mean, the squared differences from the mean, and then taking the square root of the average of those squared differences.

The range of a data set is the difference between the maximum and minimum values. Given the data set: \[ \{21, 51, 50, 86, 46, 81, 69, 29, 42, 53, 20, 48, 82, 56, 46, 27, 69, 90, 60, 37, 59, 26, 94, 16\} \]

The maximum value is \(94\) and the minimum value is \(16\). Therefore, the range is: \[ \text{Range} = 94 - 16 = 78 \]

The standard deviation measures the amount of variation or dispersion in a data set. The formula for the population standard deviation is: \[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \] where \(N\) is the number of data points, \(x_i\) are the data points, and \(\mu\) is the mean of the data set.

First, calculate the mean (\(\mu\)): \[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i = \frac{1}{24} \sum_{i=1}^{24} x_i = \frac{1}{24} \times 1332 = 55.5 \]

Next, calculate the squared differences from the mean and their average: \[ \frac{1}{24} \sum_{i=1}^{24} (x_i - 55.5)^2 = 513.1 \]

Finally, take the square root of this average to find the standard deviation: \[ \sigma = \sqrt{513.1} \approx 22.67 \]

Final Answer