Question

Use the table to answer the question.

Minutes the Bus Came Late −1
−
1
0 1 2 3
Number of Days 1 2 4 2 1
Addie’s bus pickup time is 8:00. The table shows the minutes before and after 8:00 the bus arrived at the stop for the last 10 days. Negative values represent minutes before 8:00. Calculate the standard deviation for the number of minutes the bus arrived before or after 8:00. Round your answer to the nearest tenth.

(1 point)
The standard deviation is

Answer 1

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**.