When computing the mean, the smallest value
a. can never be negative
b. can never be zero
c. can never be less than the mean
d. can be any value
The correct answer and explanation is :
The correct answer is d. can be any value.
Explanation:
The mean, also called the average, is computed by adding up all the values in a dataset and then dividing the sum by the number of values. The formula for the mean ($\bar{x}$) of a dataset of $n$ values $x_1, x_2, \dots, x_n$ is:
$$
\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}
$$
The smallest value in the dataset, or the “minimum,” can take on any value, including negative numbers, zero, or even values that are smaller than the mean.
Here’s why option d is correct:
- The smallest value can be any value because the minimum depends entirely on the data provided. It can be larger, smaller, or equal to the mean. If the dataset includes negative numbers, the smallest value could be negative. For instance, in a dataset like ${-5, 1, 3}$, the mean is $\frac{-5 + 1 + 3}{3} = -\frac{1}{3}$, and the smallest value is $-5$, which is smaller than the mean.
Why the other options are incorrect:
- a. can never be negative: This is false because the smallest value can be negative. For example, in the dataset ${-5, 2, 4}$, the smallest value is $-5$, which is negative.
- b. can never be zero: This is false because the smallest value can be zero. For example, in the dataset ${0, 3, 5}$, the smallest value is $0$.
- c. can never be less than the mean: This is also false. The smallest value can definitely be less than the mean. For example, in a dataset like ${1, 2, 3, 10}$, the mean is $\frac{1+2+3+10}{4} = 4$, but the smallest value is 1, which is less than the mean.
Therefore, the smallest value in a dataset is not constrained by the mean, and can be any value, including negative, zero, or smaller than the mean itself.