Question

which data set is the most clustered around its mean?
4, 10, 8, 6
11, 3, 10, 4
2, 9,13, 4
11, 2, 9, 6

Explanation:

To determine which data set is most clustered around its mean, we calculate the mean and then the mean absolute deviation (MAD) for each set. A smaller MAD indicates more clustering around the mean.

Step 1: Calculate the mean for each data set

• Set 1: \(4, 10, 8, 6\)

Mean \( \bar{x}_1 = \frac{4 + 10 + 8 + 6}{4} = \frac{28}{4} = 7 \)

• Set 2: \(11, 3, 10, 4\)

Mean \( \bar{x}_2 = \frac{11 + 3 + 10 + 4}{4} = \frac{28}{4} = 7 \)

• Set 3: \(2, 9, 13, 4\)

Mean \( \bar{x}_3 = \frac{2 + 9 + 13 + 4}{4} = \frac{28}{4} = 7 \)

• Set 4: \(11, 2, 9, 6\)

Mean \( \bar{x}_4 = \frac{11 + 2 + 9 + 6}{4} = \frac{28}{4} = 7 \)

Step 2: Calculate MAD for each set (absolute deviation from mean)

• Set 1: \(|4 - 7|, |10 - 7|, |8 - 7|, |6 - 7|\)

Deviations: \(3, 3, 1, 1\)
MAD \( = \frac{3 + 3 + 1 + 1}{4} = \frac{8}{4} = 2 \)

• Set 2: \(|11 - 7|, |3 - 7|, |10 - 7|, |4 - 7|\)

Deviations: \(4, 4, 3, 3\)
MAD \( = \frac{4 + 4 + 3 + 3}{4} = \frac{14}{4} = 3.5 \)

• Set 3: \(|2 - 7|, |9 - 7|, |13 - 7|, |4 - 7|\)

Deviations: \(5, 2, 6, 3\)
MAD \( = \frac{5 + 2 + 6 + 3}{4} = \frac{16}{4} = 4 \)

• Set 4: \(|11 - 7|, |2 - 7|, |9 - 7|, |6 - 7|\)

Deviations: \(4, 5, 2, 1\)
MAD \( = \frac{4 + 5 + 2 + 1}{4} = \frac{12}{4} = 3 \)

Answer:

The data set \(4, 10, 8, 6\) has the smallest MAD (2), so it is the most clustered around its mean.