9,13,43,55 what is the mean absolute deviate (MAD) of their ages

9,13,43,55 what is the mean absolute deviate (MAD) of their ages

The Correct Answer and Explanation is:

To find the Mean Absolute Deviation (MAD) of the ages given, we will follow these steps:

1. Calculate the Mean Age: First, we need to find the average (mean) of the ages.
2. Calculate the Absolute Deviations: Next, we find the absolute deviation of each age from the mean.
3. Calculate the Mean of the Absolute Deviations: Finally, we find the average of these absolute deviations.

Step 1: Calculate the Mean Age

The ages provided are (9), (13), (43), and (55). We first calculate the mean:

[
\text{Mean} = \frac{\text{Sum of ages}}{\text{Number of ages}} = \frac{9 + 13 + 43 + 55}{4}
]

Calculating the sum:

[
9 + 13 + 43 + 55 = 120
]

Now divide by the number of ages:

[
\text{Mean} = \frac{120}{4} = 30
]

Step 2: Calculate the Absolute Deviations

Now we find the absolute deviation of each age from the mean:

• For age (9):
  [
  |9 – 30| = 21
  ]
• For age (13):
  [
  |13 – 30| = 17
  ]
• For age (43):
  [
  |43 – 30| = 13
  ]
• For age (55):
  [
  |55 – 30| = 25
  ]

Step 3: Calculate the Mean of the Absolute Deviations

Now we find the mean of these absolute deviations:

[
\text{Sum of Absolute Deviations} = 21 + 17 + 13 + 25 = 76
]

Now divide by the number of ages:

[
\text{MAD} = \frac{76}{4} = 19
]

Conclusion

The Mean Absolute Deviation (MAD) of the ages (9), (13), (43), and (55) is 19.

Explanation of the MAD

The Mean Absolute Deviation provides a measure of variability in a dataset by quantifying how much individual data points differ from the mean. In this case, the MAD of 19 indicates that, on average, the ages deviate from the mean age of 30 by 19 years. This statistic is particularly useful as it accounts for the magnitude of deviations without considering their direction (positive or negative). In contrast to the standard deviation, which squares the deviations and can be influenced more heavily by outliers, the MAD offers a more robust perspective on variability, especially in skewed datasets. Therefore, a MAD of 19 suggests a moderate spread of ages around the mean, providing insights into the age distribution of this group.