all students in a pe class completed a basketball free throw shooting event and the highest number of shots made was 32. the next day a student who had transferred into the school completed the event, making 35 shots. indicate whether adding the new student's score to the rest of the data made each of these summary statistics increase, decrease, or stay the same?
Minimum stays the same.
Q1 might increase or stay the same.
Median might increase or stay the same.
Q3 might increase or stay the same.
Maximum will increase.
Mean will increase.
Standard deviation will increase.
Minimum and maximum should be obvious. Q1, median and Q3 are based on position in the distribution and will go up if the next data point (in ascending value order) is different AND the addition of a new maximum changes the position of the quartile boundaries in the distribution.
Here is an interesting bonus problem to get more understanding of this: Suppose you have a distribution with n values and no two of them are equal, by adding 1 data point that is higher than the maximum Q1, the median, and Q3 change. What is the lowest possible value for n?
The Mean must go up any time you add a value greater that the mean the mean increases and a number greater than the maximum is certainly greater than the mean.
The standard deviation will increase because a number greater than the maximum will deviate from the mean more than the maximum and increase the overall spread.
To determine whether adding the new student's score affected the summary statistics, we need to consider which summary statistics are affected by the new score. In this case, the relevant summary statistics are the highest number of shots made and the mean/average number of shots made.
1. Highest number of shots made:
The highest number of shots made was initially 32. By adding the new student's score of 35, this would increase the highest number of shots made to 35, as the new score is higher than the previous highest score.
2. Mean/Average number of shots made:
To determine if the mean/average number of shots made increased, decreased, or stayed the same, we need to calculate it both before and after adding the new student's score.
Before adding the new score:
Let's say there were 'n' students in the PE class, and their total number of shots made is 'T' (excluding the new student's score). Therefore, the mean/average number of shots made is T/n.
After adding the new score:
Now, the number of students in the PE class is 'n+1', and the total number of shots made is 'T + 35' (including the new student's score). Therefore, the mean/average number of shots made would be (T + 35)/(n+1).
If the new mean is greater than the previous mean, then adding the new student's score increased the mean. If the new mean is lower, then it decreased the mean. If both means are equal, then it stayed the same.
Summary:
Adding the new student's score increased the highest number of shots made but whether it increased or decreased the mean/average number of shots made depends on the specific values of T, n, and the new student's score.