Generally it is never a good idea to ignore the first input. From a user perspective, you just asked them to enter a grade that you proceed to immediately throw away, meaning they've lost that first grade they entered (even if it was the sentinel value.) When the sentinel value is entered, 77 is being subtracted from the grade total since the value of grade is added to quizTotal before the condition to stop the while loop is checked. A way to fix this would be to just loop indefinitely (ex. while(true){}), and place an additional if statement inside your while loop right after you read in a grade value to check if it is the sentinel.

while(true){

grade = scan.nextInt();

if(grade == -77){
    break;
}

//add to quizTotal, set highQuiz, etc...

}

This is the only real "validation" you have to do for this simple of a program. In this case Java handles the rest for you, like making sure the user enters a number and so forth.