Questions: For the following set of data, find the number of data within 1 population standard deviation of the mean. 82,59,69,65,75,72,71,74,74

Solution Steps

The mean \( \mu \) of the dataset is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{641}{9} \approx 71.22 \]

The variance \( \sigma^2 \) is calculated as follows: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} = 37.73 \] The population standard deviation \( \sigma \) is then: \[ \sigma = \sqrt{37.73} \approx 6.14 \]

The lower and upper bounds for one standard deviation from the mean are: \[ \text{Lower Bound} = \mu - \sigma \approx 71.22 - 6.14 = 65.08 \] \[ \text{Upper Bound} = \mu + \sigma \approx 71.22 + 6.14 = 77.36 \]

The number of data points within the range \( [65.08, 77.36] \) is 6.

Final Answer