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