To calculate the standard deviation of the bus arrival times from the table, we first need to find the mean of the data, then find the squared differences from the mean, average those squared differences, and finally take the square root.
Step 1: Calculate the Mean
Using the given data:
- Minutes the bus came late: \[-1, 0, 1, 2, 3\]
- Number of days: \[1, 2, 4, 2, 1\]
We can create a frequency distribution:
- \(-1\) minute: 1 day
- \(0\) minutes: 2 days
- \(1\) minute: 4 days
- \(2\) minutes: 2 days
- \(3\) minutes: 1 day
Calculating the total number of days: \[ 1 + 2 + 4 + 2 + 1 = 10 \text{ days} \]
Next, we calculate the weighted sum of the minutes: \[ (-1 \times 1) + (0 \times 2) + (1 \times 4) + (2 \times 2) + (3 \times 1) = -1 + 0 + 4 + 4 + 3 = 10 \]
Now we calculate the mean: \[ \text{Mean} = \frac{\text{Total}}{\text{Number of Days}} = \frac{10}{10} = 1 \]
Step 2: Calculate the Squared Differences from the Mean
Now we find the squared differences from the mean for each value:
- For \(-1\): \(( -1 - 1 )^2 \times 1 = (-2)^2 \times 1 = 4 \times 1 = 4\)
- For \(0\): \((0 - 1)^2 \times 2 = (-1)^2 \times 2 = 1 \times 2 = 2\)
- For \(1\): \((1 - 1)^2 \times 4 = (0)^2 \times 4 = 0 \times 4 = 0\)
- For \(2\): \((2 - 1)^2 \times 2 = (1)^2 \times 2 = 1 \times 2 = 2\)
- For \(3\): \((3 - 1)^2 \times 1 = (2)^2 \times 1 = 4 \times 1 = 4\)
Now, we sum these values: \[ 4 + 2 + 0 + 2 + 4 = 12 \]
Step 3: Calculate the Variance
To find the variance, we divide the sum of squared differences by the number of days: \[ \text{Variance} = \frac{12}{10} = 1.2 \]
Step 4: Calculate the Standard Deviation
Finally, we take the square root of the variance to find the standard deviation: \[ \text{Standard Deviation} = \sqrt{1.2} \approx 1.095 \]
Rounding to the nearest tenth: \[ \text{Standard Deviation} \approx 1.1 \]
Final Answer
The standard deviation is 1.1.