Asked by Alto Saxophone
Find the mean absolute deviation of the following dataset. Express your answer as a decimal.
{21,25,25,26,26,27,28,30}
{21,25,25,26,26,27,28,30}
Answers
Answered by
GPT-4o mini
To find the mean absolute deviation (MAD) of the dataset \(\{21, 25, 25, 26, 26, 27, 28, 30\}\), we can follow these steps:
1. **Calculate the mean of the dataset:**
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]
Calculating the sum:
\[
21 + 25 + 25 + 26 + 26 + 27 + 28 + 30 = 208
\]
The number of data points is \(8\). Thus, the mean is:
\[
\text{Mean} = \frac{208}{8} = 26
\]
2. **Calculate the absolute deviations from the mean:**
For each data point, subtract the mean and take the absolute value:
\[
|21 - 26| = | -5 | = 5
\]
\[
|25 - 26| = | -1 | = 1
\]
\[
|25 - 26| = | -1 | = 1
\]
\[
|26 - 26| = | 0 | = 0
\]
\[
|26 - 26| = | 0 | = 0
\]
\[
|27 - 26| = | 1 | = 1
\]
\[
|28 - 26| = | 2 | = 2
\]
\[
|30 - 26| = | 4 | = 4
\]
3. **List the absolute deviations:**
The absolute deviations are:
\[
\{5, 1, 1, 0, 0, 1, 2, 4\}
\]
4. **Calculate the mean of the absolute deviations:**
Now, calculate the sum of the absolute deviations:
\[
5 + 1 + 1 + 0 + 0 + 1 + 2 + 4 = 14
\]
Now, divide this sum by the number of data points to find the mean absolute deviation:
\[
\text{MAD} = \frac{14}{8} = 1.75
\]
Thus, the mean absolute deviation of the dataset is:
\[
\boxed{1.75}
\]
1. **Calculate the mean of the dataset:**
\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]
Calculating the sum:
\[
21 + 25 + 25 + 26 + 26 + 27 + 28 + 30 = 208
\]
The number of data points is \(8\). Thus, the mean is:
\[
\text{Mean} = \frac{208}{8} = 26
\]
2. **Calculate the absolute deviations from the mean:**
For each data point, subtract the mean and take the absolute value:
\[
|21 - 26| = | -5 | = 5
\]
\[
|25 - 26| = | -1 | = 1
\]
\[
|25 - 26| = | -1 | = 1
\]
\[
|26 - 26| = | 0 | = 0
\]
\[
|26 - 26| = | 0 | = 0
\]
\[
|27 - 26| = | 1 | = 1
\]
\[
|28 - 26| = | 2 | = 2
\]
\[
|30 - 26| = | 4 | = 4
\]
3. **List the absolute deviations:**
The absolute deviations are:
\[
\{5, 1, 1, 0, 0, 1, 2, 4\}
\]
4. **Calculate the mean of the absolute deviations:**
Now, calculate the sum of the absolute deviations:
\[
5 + 1 + 1 + 0 + 0 + 1 + 2 + 4 = 14
\]
Now, divide this sum by the number of data points to find the mean absolute deviation:
\[
\text{MAD} = \frac{14}{8} = 1.75
\]
Thus, the mean absolute deviation of the dataset is:
\[
\boxed{1.75}
\]
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