The sum of all deviations from the mean in a dataset is always:a.Zerob.Positivec.Negatived.Undetermined

Step 1: Break Down the Problem

To understand this question, we need to consider the concept of mean (average) and deviations from it. Specifically, we want to analyze what happens when we sum all the deviations of the data points from the mean.

Step 2: Relevant Concepts

1. Mean (Average): The mean of a dataset is calculated by adding all the data points together and dividing by the number of data points.
2. Deviation: The deviation of each data point from the mean is calculated as:
   $\text{Deviation} = x_i - \mu$ where $x_i$ is a data point and $\mu$ is the mean of the dataset.

Step 3: Analysis and Detail

When we calculate the total sum of deviations for all data points, we have: $\text{Sum of Deviations} = \sum (x_i - \mu) = \sum x_i - n\mu$ Here $n$ is the number of data points. Since $\mu = \frac{\sum x_i}{n}$, we can substitute and find: $\text{Sum of Deviations} = \sum x_i - n\left(\frac{\sum x_i}{n}\right) = \sum x_i - \sum x_i = 0$

Step 4: Verify and Summarize

After performing the calculation, we see that the sum of all deviations from the mean is indeed zero. This is a fundamental property in statistics that holds for any dataset.

Final Answer

a. Zero