In a class of 25 students, 24 of them took an exam in class and 1 student took a make-up exam the following day. The professor graded the first batch of 24 exams and found an average score of 76 points with a standard deviation of 8.5 points. The student who took the make-up the following day scored 66 points on the exam.

a) Does the new student's score increase or decrease the average?
b) The new average is:
(round to two decimal places)
c) Does the new student's score increase or decrease the standard deviation of the scores?

At first we have an average of 76, if we add another score two things will happen: if the new score is greater than the average, the average will increase but, if the score is lower than the average, the average will decrease. In this case, as the new score 66<76, the average will decrease.

b)

1) At first we have:

Average = (score 1 + score 2 +....+ score 24)/ 24 (N)

⇒ 76 = (score 1 + score 2 +....+ score 24)/24

⇒ 76 x 24 = (score 1 + score 2 +....+ score 24)

⇒ 1824 = (score 1 + score 2 +....+ score 24)

2) When the student´s score (66) is added we have tha N=25 and

(score 1 + score 2 +....+ score 25) = 1824 + 66

⇒ (score 1 + score 2 +....+ score 25) = 1890.

Now we can calculate the new average

Average = (score 1 + score 2 +....+ score n)/25

⇒ Average = 1890/25 ⇒ Average = 75.6.

The new average is 75.6

c) The concept of standar deviation give us an idea of the range of the scores and is represented as (mean ± standard deviation) = [mean - standard deviation ; mean + standard deviation]. In this case we have (76 ± 8.5) = [ 67.5 ; 84.5]. This implies that all scores are included in this range. So if we add a score that it is not included in the range the standard deviation will increase.