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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

1 answer

To calculate the standard deviation, we first need to find the mean (average) of the data set provided. The data points from the table and their respective counts (frequency) are as follows:

  • -1 minute: 1 day
  • 0 minutes: 2 days
  • 1 minute: 4 days
  • 2 minutes: 2 days
  • 3 minutes: 1 day

Next, let's write down the values and their frequencies:

  • Value: -1, Frequency: 1
  • Value: 0, Frequency: 2
  • Value: 1, Frequency: 4
  • Value: 2, Frequency: 2
  • Value: 3, Frequency: 1

Step 1: Calculate the Mean

To find the mean, we use the formula:

\[ \text{Mean} = \frac{\sum (x \cdot f)}{N} \]

Where \(x\) is the value, \(f\) is the frequency, and \(N\) is the total number of days.

Calculating total values:

  • \((-1) \cdot 1 = -1\)
  • \(0 \cdot 2 = 0\)
  • \(1 \cdot 4 = 4\)
  • \(2 \cdot 2 = 4\)
  • \(3 \cdot 1 = 3\)

Now summing these:

\[ \sum (x \cdot f) = -1 + 0 + 4 + 4 + 3 = 10 \]

Total number of days:

\[ N = 1 + 2 + 4 + 2 + 1 = 10 \]

Now we calculate the mean:

\[ \text{Mean} = \frac{10}{10} = 1 \]

Step 2: Calculate the Variance

Next, we calculate the variance using the formula:

\[ \text{Variance} = \frac{\sum (f \cdot (x - \text{Mean})^2)}{N} \]

Calculating each \( (x - \text{Mean})^2 \):

  • For \(x = -1\): \[ (-1 - 1)^2 = (-2)^2 = 4 \] Contribution: \(1 \cdot 4 = 4\)

  • For \(x = 0\): \[ (0 - 1)^2 = (-1)^2 = 1 \] Contribution: \(2 \cdot 1 = 2\)

  • For \(x = 1\): \[ (1 - 1)^2 = 0^2 = 0 \] Contribution: \(4 \cdot 0 = 0\)

  • For \(x = 2\): \[ (2 - 1)^2 = 1^2 = 1 \] Contribution: \(2 \cdot 1 = 2\)

  • For \(x = 3\): \[ (3 - 1)^2 = 2^2 = 4 \] Contribution: \(1 \cdot 4 = 4\)

Now summing these contributions:

\[ \sum (f \cdot (x - \text{Mean})^2) = 4 + 2 + 0 + 2 + 4 = 12 \]

Now we can calculate the variance:

\[ \text{Variance} = \frac{12}{10} = 1.2 \]

Step 3: Calculate the Standard Deviation

Finally, the standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sqrt{1.2} \approx 1.095 \]

Rounding to the nearest tenth, the standard deviation is approximately:

\[ \text{Standard Deviation} \approx 1.1 \]

So the answer is:

The standard deviation is 1.1.