Question

Calculate the average deviation from the mean for the following dataset:

{23, 24, 26, 27, 28, 30, 31}

Round your answer to the nearest hundredth.

(1 point)
The average deviation from the mean is
.

Answer 1

To calculate the average deviation from the mean for the dataset {23, 24, 26, 27, 28, 30, 31}, we'll follow these steps:

1. **Calculate the mean** of the dataset.
2. **Calculate the absolute deviations** from the mean.
3. **Calculate the average of the absolute deviations**.

### Step 1: Calculate the Mean
The mean (average) is calculated by taking the sum of all the values and dividing by the number of values.

\[
\text{Mean} = \frac{23 + 24 + 26 + 27 + 28 + 30 + 31}{7}
\]
\[
\text{Mean} = \frac{189}{7} = 27
\]

### Step 2: Calculate the Absolute Deviations from the Mean
Now, we find the absolute deviation for each data point from the mean (27):

- For 23: \( |23 - 27| = 4 \)
- For 24: \( |24 - 27| = 3 \)
- For 26: \( |26 - 27| = 1 \)
- For 27: \( |27 - 27| = 0 \)
- For 28: \( |28 - 27| = 1 \)
- For 30: \( |30 - 27| = 3 \)
- For 31: \( |31 - 27| = 4 \)

Now we list the absolute deviations:
- 4, 3, 1, 0, 1, 3, 4

### Step 3: Calculate the Average of the Absolute Deviations
Next, we compute the average of these absolute deviations:

\[
\text{Sum of absolute deviations} = 4 + 3 + 1 + 0 + 1 + 3 + 4 = 16
\]

Now, divide by the number of data points (which is 7):

\[
\text{Average deviation} = \frac{16}{7} \approx 2.2857
\]

### Final Step: Round the Answer
Rounding to the nearest hundredth:

\[
\text{Average deviation from the mean} \approx 2.29
\]

Thus, the average deviation from the mean is **2.29**.