Standard Deviation Zero Implication

The standard deviation of a set of numbers is a measure of how spread out the numbers are from their mean. A standard deviation of zero is a very specific case. It means there is absolutely no spread or variation among the numbers. In simpler terms, if the standard deviation of a set of numbers is zero, all the numbers in that set must be identical.

Let the five numbers be \(n_1, n_2, n_3, n_4, n_5\). If their standard deviation is zero, then it must be true that \(n_1 = n_2 = n_3 = n_4 = n_5\). Let's call this common value \(k\).

So, the set of five numbers is simply \(\{k, k, k, k, k\}\) for some value \(k\).

Analyzing the Options

Now let's examine each given option based on the fact that all five numbers are equal to \(k\):

1. All the numbers are zero

   This is not necessarily true. For example, if the numbers are \(\{5, 5, 5, 5, 5\}\), their standard deviation is zero, but the numbers are not zero. They are all equal to 5. So, this statement is false.

2. The median is zero

   The median is the middle value when the numbers are arranged in ascending order. For the set \(\{k, k, k, k, k\}\), the ordered set is also \(\{k, k, k, k, k\}\). The median is the third number, which is \(k\). The median is zero only if \(k=0\). This is not always true; the numbers could be \(\{10, 10, 10, 10, 10\}\), where the median is 10. So, this statement is false.

3. If at least one number is positive, then at least one number is negative

   If at least one number is positive, say \(n_1 > 0\), then since all numbers are equal (\(n_1 = n_2 = n_3 = n_4 = n_5 = k\)), it means \(k > 0\). If \(k > 0\), then all five numbers \(\{k, k, k, k, k\}\) are positive. There would be no negative numbers in the set. So, this statement is false.

4. The difference between the median and mode of these numbers is zero

   For the set \(\{k, k, k, k, k\}\):

   • The median is the middle value, which is \(k\).
   • The mode is the value that appears most frequently. Since all numbers are \(k\), the value \(k\) appears 5 times, which is the maximum possible frequency. So, the mode is \(k\).

   The difference between the median and the mode is Median − Mode \( = k - k = 0\). This is always true whenever all the numbers in the set are identical, which is exactly the condition for the standard deviation to be zero.

Conclusion

Based on the analysis, the only statement that must be true when the standard deviation of five numbers is zero is that the difference between their median and mode is zero. This is because zero standard deviation implies all the numbers are the same, and for a set of identical numbers, the median and the mode are equal to that common value, making their difference zero.