Math help! Algebra 1 and 2

Question

A teacher gives a test for which the mean of the scores is 68 and the standard deviation is 15. The teacher decides to scale the test scores by adding 10 points to each score. What are the mean and standard deviation of the scaled test scores?

Also please show work so i can understand

Thanks!

Answer 1

The mean merely shifts from 68 to 78. This is because 10 points are added to each score, so when there was the same number of scores above and below 68 [that is, the mean], there are now the same number of scores above and below 78 [red curve]

The standard deviation does not change. It is still 15 because each of the new scores is still the same distance from the new mean as the old scores were from the old mean.

Answer 2

mean = 68 + 10 = 78

s.d. = 15

How do I get this answer? It helps to think about the definition of mean and standard deviation. Let's suppose the class is very small (3 students) and students' scores are represented by the variables x, y and z. To calculate the mean we take the sum of the 3 scores and divide by the number of students:

mean = (x+y+z)/3

Now what happens if we add 10 to each score?

(x+10+y+10+z+10)/3 = (x+y+z+30)/3 = (x+y+z)/3 + 30/3 = (x+y+z)/3 + 10

Notice that adding 10 to each individual score has the same result as adding 10 to the mean! Now we can make a similar argument for the standard deviation, but I will spare the fine details in favor of a more conceptual answer. By definition, the standard deviation tells us how spread out the test scores are. But since we are only adding 10 to each score, this is going to shift each score by the same amount and therefore the standard deviation will not change.