Question
Rin incorrectly calculated the mean absolute deviation for the data set 25,25,31,36,38.Her work is shown here.
1. Find the mean.
Mean =
25 + 25 + 31 + 36 + 38
5
=
155
5
= 31
2. Find each absolute deviation.
Absolute deviations:6,6,0,5,7
3. Find the mean absolute deviation.
MAD =
6 + 0 + 5 + 7
4
=
18
4
= 4.5
In which step did Rin make an error?
1. Find the mean.
Mean =
25 + 25 + 31 + 36 + 38
5
=
155
5
= 31
2. Find each absolute deviation.
Absolute deviations:6,6,0,5,7
3. Find the mean absolute deviation.
MAD =
6 + 0 + 5 + 7
4
=
18
4
= 4.5
In which step did Rin make an error?
Answers
GPT-4o mini
Let's go through Rin's calculations step by step to identify where the error occurred.
1. **Calculate the Mean**:
- The data set is \(25, 25, 31, 36, 38\).
- The sum is calculated as \(25 + 25 + 31 + 36 + 38 = 155\).
- The number of data points is \(5\).
- The mean calculation is \( \frac{155}{5} = 31\).
- **This step is correct.**
2. **Calculate each Absolute Deviation**:
- The absolute deviations from the mean (31) for each data point are:
- For \(25\): \( |25 - 31| = 6 \)
- For another \(25\): \( |25 - 31| = 6 \)
- For \(31\): \( |31 - 31| = 0 \)
- For \(36\): \( |36 - 31| = 5 \)
- For \(38\): \( |38 - 31| = 7 \)
- Thus, the absolute deviations are correctly noted as \(6, 6, 0, 5, 7\).
- **This step is also correct.**
3. **Calculate the Mean Absolute Deviation (MAD)**:
- Rin wrote \(MAD = \frac{6 + 0 + 5 + 7}{4}\).
- However, the correct numerator is \(6 + 6 + 0 + 5 + 7 = 24\), not \(6 + 0 + 5 + 7\).
- The number of deviations should also be \(5\) because there are \(5\) data points.
- So, MAD should be calculated as:
- \( MAD = \frac{24}{5} = 4.8 \).
**Conclusion**: Rin made an error in calculating the sum of the absolute deviations and mistakenly using the number \(4\) instead of \(5\) as the denominator. Hence, the error occurred in the calculation of the mean absolute deviation (MAD).
1. **Calculate the Mean**:
- The data set is \(25, 25, 31, 36, 38\).
- The sum is calculated as \(25 + 25 + 31 + 36 + 38 = 155\).
- The number of data points is \(5\).
- The mean calculation is \( \frac{155}{5} = 31\).
- **This step is correct.**
2. **Calculate each Absolute Deviation**:
- The absolute deviations from the mean (31) for each data point are:
- For \(25\): \( |25 - 31| = 6 \)
- For another \(25\): \( |25 - 31| = 6 \)
- For \(31\): \( |31 - 31| = 0 \)
- For \(36\): \( |36 - 31| = 5 \)
- For \(38\): \( |38 - 31| = 7 \)
- Thus, the absolute deviations are correctly noted as \(6, 6, 0, 5, 7\).
- **This step is also correct.**
3. **Calculate the Mean Absolute Deviation (MAD)**:
- Rin wrote \(MAD = \frac{6 + 0 + 5 + 7}{4}\).
- However, the correct numerator is \(6 + 6 + 0 + 5 + 7 = 24\), not \(6 + 0 + 5 + 7\).
- The number of deviations should also be \(5\) because there are \(5\) data points.
- So, MAD should be calculated as:
- \( MAD = \frac{24}{5} = 4.8 \).
**Conclusion**: Rin made an error in calculating the sum of the absolute deviations and mistakenly using the number \(4\) instead of \(5\) as the denominator. Hence, the error occurred in the calculation of the mean absolute deviation (MAD).