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466 Mastering Technical Mathematics
all 10 questions correctly; there are others who did not get a single answer right. To
determine the mean score for the whole class on this test—that is, the population mean,
called p
—you must add up the scores of each and every student and then divide by
100. First, sum the products of the numbers in the first and second columns. This will
give you 100 times the population mean:
(10 · 5) (9 · 6) (8 · 19) (7 · 17) (6 · 18) (5 · 11) (4 · 6)
(3 · 4) (2 · 4) (1 · 7) (0 · 3)
50 54 152 119 108 55 24 12 8 7 0
589
Dividing this by 100, the total number of test scores (one for each student who turns in
a paper), you obtain p
589/100 5.89.
The teacher in this class has assigned letter grades to each score. Students who scored
9 or 10 correct received grades of A; students who got scores of 7 or 8 received grades
of B; those who got scores of 5 or 6 got grades of C; those who got scores of 3 or 4
got grades of D; those who got less than 3 correct answers received grades of F. The
assignment of grades, informally known as the “curve,” is a matter of teacher temperament.
It would doubtless seem arbitrary to the students who took this test. (Some
people might consider the curve in this case to be too lenient, while a few might think
it is too severe.)
Now suppose you want to find the sample means for each grade in the test whose results
are tabulated in Table 29-7, rounding off the answers to two decimal places. Call the
sample means sa
for the grade of A, sb
for the grade of B, and so on down to sf
for
the grade of F. To calculate sa
, note that 5 students received scores of 10, while 6 students
got scores of 9, both scores good enough for an A. This is a total of 5 6, or 11,
students getting the grade of A. Calculating, you get
µ sa
[(5 · 10) (6 · 9)] / 11
(50 54) / 11
104/11
9.45
To find sb
, observe that 19 students scored 8, and 17 students scored 7. Therefore, 19
17, or 36, students received grades of B. Calculating, you get
sb
[(19 · 8) (17 · 7)] / 36
(152 119) / 36
271/36
7.53