To calculate the average deviation from the mean for the dataset {23, 24, 26, 27, 28, 30, 31}, we will follow these steps:
Step 1: Calculate the Mean
First, we sum all the numbers and divide by the total count of numbers.
\[ \text{Mean} = \frac{23 + 24 + 26 + 27 + 28 + 30 + 31}{7} \] \[ = \frac{ 23 + 24 + 26 + 27 + 28 + 30 + 31}{7} = \frac{ 189 }{ 7 } = 27 \]
Step 2: Calculate Deviations from the Mean
Next, we will calculate the absolute deviation for each data point from the mean (27).
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For 23: \(|23 - 27| = 4\)
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For 24: \(|24 - 27| = 3\)
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For 26: \(|26 - 27| = 1\)
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For 27: \(|27 - 27| = 0\)
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For 28: \(|28 - 27| = 1\)
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For 30: \(|30 - 27| = 3\)
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For 31: \(|31 - 27| = 4\)
Step 3: List the Deviations
The deviations are:
- 4, 3, 1, 0, 1, 3, 4
Step 4: Calculate the Average Deviation
Now, we calculate the average of these deviations.
\[ \text{Average Deviation} = \frac{4 + 3 + 1 + 0 + 1 + 3 + 4}{7} \] \[ = \frac{16}{7} \approx 2.2857 \]
Rounding to the nearest hundredth:
\[ \text{Average Deviation} \approx 2.29 \]
Final Answer
The average deviation from the mean for the dataset {23, 24, 26, 27, 28, 30, 31} is approximately 2.29.