A sample has a mean of M = 25. If one person with a score of X = 25 is added to the sample, what effect will it have on the sample mean?
Why and how is there no effect on the sample mean? I am not really understanding this?
Study this explanation of mean. The answer should be clear to you then.
http://www.mathsisfun.com/mean.html
The mean functions as a fulcrum/balance point for the distribution. Since the score is added at the balance point, the mean does not change.
When a single person with a score of X = 25 is added to the sample, it will have no effect on the sample mean. This is because the mean represents the average value of the scores in the sample, and adding a score that is equal to the current mean will not change the overall average.
To understand why there is no effect on the sample mean, let's consider the formula to calculate the mean:
Mean (M) = Sum of all scores / Number of scores
When the person with a score of 25 is added to the sample, the sum of all scores will increase by 25, but at the same time, the number of scores will also increase by 1. Since the new score is equal to the current mean, the increase in the sum is balanced by the increase in the number of scores.
For example, let's say we have a sample of 4 scores: {20, 25, 30, 35}. The current mean is 27.5 ((20 + 25 + 30 + 35) / 4 = 110 / 4 = 27.5). Now, if we add a person with a score of 25 to the sample, the new sum of all scores will be 135 (110 + 25), and the new number of scores will be 5. The new mean will still be 27 ((135 / 5) = 27), which is exactly the same as the previous mean.
So, adding a score that is equal to the current mean has no effect on the sample mean because it does not change the overall average.