To calculate the standard deviation of the bus arrival times, we'll follow these steps:
- List the data: Extract the bus arrival times (in minutes before or after 8:00) from the table.
- Calculate the mean (average) of the data.
- Find the squared deviations from the mean.
- Calculate the variance (average of the squared deviations).
- Find the standard deviation (the square root of the variance).
Assuming the bus arrival times from the table (which you didn't provide) are:
- -5, -3, -1, 0, 2, 4, 6, 3, -2, -4 (example data; replace with actual data from your table for accuracy)
Now, let’s go through the calculations step by step:
Step 1: Find the Mean
\[ \text{Mean} = \frac{\sum \text{data}}{n} \]
If we use the example data: \[ \text{Mean} = \frac{(-5) + (-3) + (-1) + 0 + 2 + 4 + 6 + 3 + (-2) + (-4)}{10} \] \[ = \frac{0}{10} = 0 \]
Step 2: Find the Squared Deviations from the Mean
Now, calculate the squared deviations from the mean for each data point: \[ (-5 - 0)^2 = 25 \] \[ (-3 - 0)^2 = 9 \] \[ (-1 - 0)^2 = 1 \] \[ (0 - 0)^2 = 0 \] \[ (2 - 0)^2 = 4 \] \[ (4 - 0)^2 = 16 \] \[ (6 - 0)^2 = 36 \] \[ (3 - 0)^2 = 9 \] \[ (-2 - 0)^2 = 4 \] \[ (-4 - 0)^2 = 16 \]
Step 3: Calculate the Variance
Next, sum the squared deviations and divide by \(n\) (the number of data points): \[ \text{Variance} = \frac{(25 + 9 + 1 + 0 + 4 + 16 + 36 + 9 + 4 + 16)}{10} = \frac{120}{10} = 12 \]
Step 4: Calculate the Standard Deviation
Lastly, take the square root of the variance: \[ \text{Standard Deviation} = \sqrt{12} \approx 3.464 \]
Rounding
Rounding to the nearest tenth: \[ \text{Standard Deviation} \approx 3.5 \]
So, if the provided data matches what I used for calculation, the standard deviation is 3.5. Please substitute the actual arrival time values from the table into this process to obtain your accurate standard deviation.