What is the mean absolute deviation of 12 4 6 12 10 8 4 4
The Correct Answer and Explanation is :
To calculate the mean absolute deviation (MAD), we follow these steps:
1. Calculate the mean of the data set:
The data set is: 12, 4, 6, 12, 10, 8, 4, 4
[
\text{Mean} = \frac{12 + 4 + 6 + 12 + 10 + 8 + 4 + 4}{8} = \frac{60}{8} = 7.5
]
2. Calculate the absolute deviations from the mean:
Now, subtract the mean from each data point and take the absolute value (ignore the negative signs):
- |12 – 7.5| = 4.5
- |4 – 7.5| = 3.5
- |6 – 7.5| = 1.5
- |12 – 7.5| = 4.5
- |10 – 7.5| = 2.5
- |8 – 7.5| = 0.5
- |4 – 7.5| = 3.5
- |4 – 7.5| = 3.5
3. Find the average of these absolute deviations:
Now, sum all the absolute deviations and divide by the number of data points (8):
[
\text{MAD} = \frac{4.5 + 3.5 + 1.5 + 4.5 + 2.5 + 0.5 + 3.5 + 3.5}{8} = \frac{24}{8} = 3
]
Explanation:
The mean absolute deviation (MAD) is a measure of how spread out the numbers in a data set are. Unlike variance or standard deviation, which focus on squared deviations, MAD focuses on the average of absolute deviations from the mean, which makes it more intuitive and less sensitive to extreme outliers.
In this calculation, we first find the mean of the data points, which is the sum of the values divided by the number of data points. Then, we calculate how much each data point differs from the mean. Taking the absolute value ensures that all deviations are positive. Finally, we average these absolute deviations to get the MAD.
The result, 3, tells us that, on average, each data point in this data set deviates from the mean by 3 units. MAD gives a sense of how close the numbers in the data set are to the mean and is useful in understanding data variability in a simpler, more robust manner compared to other statistical measures like variance.