Final answer:
The standard deviation of a data set remains unchanged if a constant value is added to each observation. In this case, the standard deviation stays at 6.5, not any of the provided options.
Step-by-step explanation:
Standard Deviation after Increasing Every Observation
If the standard deviation of a set of 50 observations is 6.5, and each observation is increased by 5, the new standard deviation remains unchanged.
This is because standard deviation is a measure of spread or variability around the mean of a data set, and adding or subtracting a constant to each observation does not change the spread of the data. Therefore, the new standard deviation is also 6.5.When we increase each value in a set of observations by a constant amount, the position of the data points shifts by that amount, but the spread or dispersion of the data does not change.
Standard deviation is a measure of this spread — it tells us how much the values in a set of data differ from the mean of the set. Since we're adding the same constant to each observation, each value is shifted equally, so the distance between any two points remains the same.
This implies that the variability or dispersion in the data does not change when you add (or subtract) a constant from every observation.
Therefore, the standard deviation remains unchanged. In the specific case where the original standard deviation is 6.5 and we increase each observation by 5, the new standard deviation is also 6.5. None of the options A (2.5), B (1.5), C (3.5), or D (4.5) reflect the correct value. The correct answer is that the standard deviation remains 6.5 after each observation is increased by 5.
The possible answers provided suggest a misconception, as the standard deviation does not decrease when a constant is added to every observation. The change affects the mean, not the variability. Considering this, the correct answer, which is not listed amongst the provided options, should be the original standard deviation of 6.5.