Question

mr. jansen gives a unit test to his class. he determines that the mean mark on the test is 62% and the standard deviation is 8 marks. when he hands the tests back a student realized that one question is wrong and is actually impossible to solve. so the teacher adds 5 percent to all the students marks, what is the mean and standard deviation of the new set of marks?

Answer 1

$$\mu={1\over n}\displaystyle\sum_{i=1}^nx_i$$ $$\sigma^2={1\over n}\displaystyle\sum_{i=1}^n(x_i-\mu)^2$$

$$\sigma=\sqrt{\sigma^2}$$

$$\mu_1={1\over n}\displaystyle\sum_{i=1}^nx_{i1}=62$$

$$\sigma_1^2={1\over n}\displaystyle\sum_{i=1}^n(x_{i1}-\mu_1)^2=8^2$$

Given that

$$x_{i2}=x_{i1}+5,\ i=1,2,...,n$$

Then

$$\mu_2={1\over n}\displaystyle\sum_{i=1}^nx_{i2}=$$ $$={1\over n}\displaystyle\sum_{i=1}^n(x_{i1}+5)=$$ $$={1\over n}\displaystyle\sum_{i=1}^nx_{i1}+5=\mu_1+5=62+5=67$$

$$\sigma_2^2={1\over n}\displaystyle\sum_{i=1}^n(x_{i2}-\mu_2)^2=$$ $$={1\over n}\displaystyle\sum_{i=1}^n((x_{i1}+5)-(\mu_1+5))^2=$$ $$={1\over n}\displaystyle\sum_{i=1}^n(x_{i1}-\mu_1)^2=\sigma_1^2=8^2$$

The mean will increase by 5 (it will be $67$).

The standard deviation will remain the same (it will be $8$).