Answer:
The dot displays data representing a data set whose mode is 0.5, mean 2.5 and the variability in the distribution is 0.53.
Mean Absolute Deviation (MAD) is the average absolute distance between each data and the mean of the data set. This value represents the variability in a data set.
MAD = ( (|x₁ - mean| x n₁) + (|x₂ - mean| x n₂) + ... + (| xₙ - mean| x nₙ) ) / number of data
Now we take a look at the problem.
First, we put the dot displays data into a table:
Number of sports played Number of samples
1 1
1.5 1
2 3
2.5 4
3 3
3.5 2
The mode of the data set is 2.5 because it has the most samples.
The mean of the data set can be calculated as follows:
Mean = sum of all data / number of data
= ( (1 x 1) + (1.5 x 1) + (2 x 3) + (2.5 x 4) + (3 x 3) + (3.5 x 2) / 14 )
= ( 2 + 1.5 + 6 + 10 + 9 + 7) / 14
= 35.5 / 14
= 2.54 ≈ 2.5
The variability in the distribution can be calculated from the MAD:
MAD = ( (|x₁ - mean| x n₁) + (|x₂ - mean| x n₂) + ... + (|xₙ - mean| x nₙ) ) / number of data
= ( (|1 - 2.5| x 1) + (|1.5 - 2.5| x 1) + (|2 - 2.5| x 3) + (|2.5 - 2.5| x 4) + (|3 - 2.5| x 3) + (|3.5 - 2.5| x 2) ) / 14
= (1.5 + 1 + 1.5 + 0 + 1.5 + 2) / 14
= 7.5 / 14
= 0.535 ≈ 0.53
Thus:
The mode of the data set is 0.5;
The mean number of sports is 2.5; and
The variability in the distribution is 0.53
Explanation: