For the data set 3, 10, 10, 4, 7, 10, 5, the mean deviation about the mean (i.e., the average of the absolute deviations of the observations from their mean) is equal to which of the following?

Introduction / Context:
This is a statistics question about mean deviation, also known as average absolute deviation. We are given a small data set and asked to compute the mean deviation about the mean, which involves first finding the arithmetic mean and then averaging the absolute deviations of each observation from that mean.

Given Data / Assumptions:

• Data set: 3, 10, 10, 4, 7, 10, 5.
• Total number of observations n = 7.
• Mean deviation about the mean is defined as: (1 / n) * Σ|xᵢ − mean|.
• We must compute this value exactly as a simplified fraction.

Concept / Approach:
There are two main steps:

• Compute the arithmetic mean of the data set.
• Compute the absolute deviation of each data point from that mean, sum these deviations, and divide by the number of observations.

Careful arithmetic and correct handling of absolute values are essential to avoid sign errors.

Step-by-Step Solution:
Step 1: Compute the mean of the data. Sum = 3 + 10 + 10 + 4 + 7 + 10 + 5. 3 + 10 = 13, +10 = 23, +4 = 27, +7 = 34, +10 = 44, +5 = 49. Mean = 49 / 7 = 7. Step 2: Compute 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 3: Sum all absolute deviations. Total deviation = 4 + 3 + 3 + 3 + 0 + 3 + 2. 4 + 3 = 7, +3 = 10, +3 = 13, +0 = 13, +3 = 16, +2 = 18. Step 4: Compute mean deviation about the mean. Mean deviation = total deviation / number of observations = 18 / 7.
Verification / Alternative check:
Recheck the mean: 49 / 7 = 7 clearly, and each deviation was computed carefully with absolute values. The sum 18 is consistent with multiple recounts, so 18/7 is reliable.
Why Other Options Are Wrong:
49/7 = 7 and 50/7 ≈ 7.14 are far too large to be average absolute deviations in this data set. 19/7 ≈ 2.71 and 17/7 ≈ 2.43 are close to 18/7 but do not match the actual total deviation. Only 18/7 exactly equals (4 + 3 + 3 + 3 + 0 + 3 + 2) / 7.
Common Pitfalls:
Forgeting to divide by the total number of observations after summing the absolute deviations can lead to answering 18 instead of 18/7. Another common mistake is to omit absolute values and simply sum signed deviations, which would always give zero when taken about the mean, completely missing the idea of average distance.
Final Answer:
The mean deviation of the given data about its mean is 18/7.