For the data set 3, 10, 10, 4, 7, 10 and 5, what is the mean deviation about the mean?

Introduction / Context:
This question is about a measure of dispersion called mean deviation about the mean. It tells you on average how far the observations lie from the arithmetic mean of the data. Understanding this helps in describing how spread out a data set is.

Given Data / Assumptions:
- The data points are 3, 10, 10, 4, 7, 10 and 5.
- We need to calculate mean deviation about the mean, not about the median.
- There are 7 observations in total.

Concept / Approach:
Mean deviation about the mean is defined as the average of the absolute deviations from the arithmetic mean. The steps are: find the mean, compute each observation minus the mean in absolute value, sum these absolute deviations, and finally divide by the number of observations.

Step-by-Step Solution:
Step 1: Compute the sum of the data: 3 + 10 + 10 + 4 + 7 + 10 + 5. Step 2: The sum is 49, so the mean is 49 / 7 = 7. Step 3: Find absolute deviations from the mean 7: |3 - 7| = 4, |10 - 7| = 3, |10 - 7| = 3, |4 - 7| = 3, |7 - 7| = 0, |10 - 7| = 3, |5 - 7| = 2. Step 4: Sum of absolute deviations = 4 + 3 + 3 + 3 + 0 + 3 + 2 = 18. Step 5: Mean deviation about the mean = 18 / 7.

Verification / Alternative check:
Check that the mean 7 lies between the smallest and largest values and that deviations are correctly computed. Also confirm that the total of 49 is correct. Once those are confirmed, 18 / 7 is the only possible correct value.

Why Other Options Are Wrong:
- Option 49/7: This equals 7, which would mean each value differs from the mean on average by 7 units, far too large for this data set.

- Option 19/7: This is approximately 2.714, which does not match the exact sum of absolute deviations, which is 18.

- Option 50/7: This is based on a mistaken sum or deviation calculation and does not correspond to the true total absolute deviation.

Common Pitfalls:
Learners often forget to take absolute values and instead use signed deviations that add up to zero. Another common error is to divide by n - 1 instead of n, which is used in variance estimation, not in mean deviation. Careful arithmetic with each deviation is essential.

Final Answer:
The mean deviation about the mean for this data set is 18/7.