What is the mean absolute deviation of the following set of data 10 8 10 6 6 2 10 4
The Correct Answer and Explanation is :
To find the mean absolute deviation (MAD) of the data set (10, 8, 10, 6, 6, 2, 10, 4), we can follow these steps:
Step 1: Calculate the Mean
First, we need to find the mean of the data set.
[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
]
Calculating the sum:
[
10 + 8 + 10 + 6 + 6 + 2 + 10 + 4 = 56
]
The number of data points is 8. Therefore:
[
\text{Mean} = \frac{56}{8} = 7
]
Step 2: Calculate the Absolute Deviations
Next, we need to calculate the absolute deviation of each data point from the mean. The absolute deviation is found using the formula:
[
\text{Absolute Deviation} = |x – \text{Mean}|
]
Calculating the absolute deviations:
- For 10: (|10 – 7| = 3)
- For 8: (|8 – 7| = 1)
- For 10: (|10 – 7| = 3)
- For 6: (|6 – 7| = 1)
- For 6: (|6 – 7| = 1)
- For 2: (|2 – 7| = 5)
- For 10: (|10 – 7| = 3)
- For 4: (|4 – 7| = 3)
Step 3: Calculate the Mean Absolute Deviation
Now we calculate the mean of these absolute deviations:
[
\text{Sum of Absolute Deviations} = 3 + 1 + 3 + 1 + 1 + 5 + 3 + 3 = 18
]
The mean absolute deviation is then calculated as follows:
[
\text{MAD} = \frac{\text{Sum of Absolute Deviations}}{\text{Number of data points}} = \frac{18}{8} = 2.25
]
Conclusion
The mean absolute deviation of the data set (10, 8, 10, 6, 6, 2, 10, 4) is 2.25. The MAD provides a measure of the dispersion of the data points around the mean. A lower MAD indicates that the data points are closer to the mean, while a higher MAD signifies a greater spread of values. This measure is particularly useful in fields such as statistics and data analysis for assessing variability without being overly affected by extreme values.