Final answer:
The correct answer is option a, zero. This is because for every value above the mean there is a corresponding value below it, canceling out when calculating the sum of deviations. The standard deviation being zero means all data points are identical.
Step-by-step explanation:
The correct answer is option a, zero. In statistics, when considering the deviations from the mean (the differences between each data point and the mean of the dataset), they will always sum to zero. This is because the mean is the central point of the data, and for every data point above the mean, there is typically a data point below the mean that balances it out. If you sum these positive and negative deviations, the result is zero, indicating that the mean is the 'balance point' of the data.
To clarify, if you are working with a data set where all values deviate from the mean, you calculate the deviation of each value by subtracting the mean from each value. Summing all these deviations will give you zero because the mean is the average value of the set. This is a fundamental property of the mean in statistics and it's unrelated to the standard deviation, which is a measure of the spread of data points around the mean. A standard deviation of zero signifies that all values in the data set are equal.