One class of 75 students has an average midterm score of 67 and an sd of 12. another class of 325 students has an average of 72 and an sd of 15. find the average and the sd of the combined list of 400 students. hint: this uses the sd of a long list formula. one class of 75 students has an average midterm score of 67 and an sd of 12. another class of 325 students has an average of 72 and an sd of 15. find the average and the sd of the combined list of 400 students. hint: this uses the sd of a long list formula.

- average: 71.0625
- standard deviation: 

Explanation:

Let 

a = average score of 325 students = 72
b = average score of 75 students = 67
S = total score of 325 students
T = total score of 75 students
u = standard deviation of the score of 325 students = 15
v = standard deviation of the score of 75 students = 12
C = average of the squares of the scores of 325 students 
D = average of the squares of the scores of 75 students 

The total score of 325 students is calculated as:

S = 325a
   = 325(72) 
S = 23,400

While the total score of 75 students is calculated in similar manner:

T = 75b
   = 75(67)
T = 5,025

To get the total score of 400 students, we just add the total score of 325 students and the total score of 75 students. So, the total score of 400 students is given by

S + T = 23,400 + 5,025
S + T = 28,425

To get the average score of  400 students, we just divide the total score of 400 students by the number of students, which is 400. In terms of equation,



To calculate the standard deviation of the scores of 400 students, we use the following formula:



where

= standard deviation of the scores of 400 students
= average of the squares of the scores of 400 students
= average of the scores of 400 students = 71.0625

Because

C = average of the squares of the scores of 325 students 
D = average of the squares of the scores of 75 students 
u = standard deviation of the score of 325 students = 15
v = standard deviation of the score of 75 students = 12
a = average score of 325 students = 72
b = average score of 75 students = 67

We have the following 

  (1)

   (2)

So, using equations (1) and (2),



Hence, the standard deviation of the scores of 400 students is calculated as 

Is it multiple choice

x^2 + 81

we can factor this using difference of square binomials.

rewrite the equation:

x^2 - (-81)

factor:

(a^2 - b^2) = (√a^2 + √b^2)(√a^2 - √b^2)

(x^2 - (-81) = (x + √-81)(x - √-81)

we can remove the - from under the radical by multiplying it by i.

(x^2 - (-81) = (x + i√81)(x - i√81)

(x^2 - (-81) = (x + 9i)(x - 9i)

(x + 9i)(x - 9i)

we can check our answer by using foil:

x^2 - 9ix + 9ix - 81i^2

x^2 - 81^2

i^2 = -1

x^2 + 81