The following data represent the length of time (in minutes) between eruptions of Old Faithful in Yellowstone National Park.
b. Approximate the standard deviation length of time between eruptions.

Verified step by step guidance

1. Step 1: Identify the midpoints of each time interval. For each class interval, calculate the midpoint by averaging the lower and upper bounds. For example, for the interval 40-49, the midpoint is \(\frac{40 + 49}{2} = 44.5\). Repeat this for all intervals.
2. Step 2: Multiply each midpoint by its corresponding frequency to find the weighted values. This will help in calculating the mean. For each class, compute \(\text{midpoint} \times \text{frequency}\).
3. Step 3: Calculate the mean length of time between eruptions using the formula for grouped data: \[ \bar{x} = \frac{\sum (f_i \times x_i)}{\sum f_i} \] where \(f_i\) is the frequency and \(x_i\) is the midpoint of each class.
4. Step 4: Calculate the squared deviations from the mean for each midpoint, multiply each by the corresponding frequency, and sum these values. This is done using: \[ \sum f_i (x_i - \bar{x})^2 \] This step measures the total squared distance of data points from the mean.
5. Step 5: Finally, approximate the standard deviation by taking the square root of the variance. The variance for grouped data is given by: \[ \sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}} \] This will give you the standard deviation of the length of time between eruptions.

Key Concepts

Grouped Data and Frequency Distribution

Grouped data organizes raw data into intervals or classes with corresponding frequencies, summarizing large datasets efficiently. Understanding frequency distributions helps in estimating measures like mean and standard deviation when individual data points are unavailable.

Approximate Mean for Grouped Data

The approximate mean of grouped data is calculated by using the midpoints of each class interval multiplied by their frequencies, then dividing the sum by the total frequency. This estimate is essential for further calculations like variance and standard deviation.

Standard Deviation for Grouped Data

Standard deviation measures the spread of data around the mean. For grouped data, it is approximated by calculating the squared deviations of class midpoints from the mean, weighted by frequencies, then taking the square root of the average squared deviation.