## Numerical Examples

for finding Averages

Example #1

Suppose the following grades were made by twenty-four students on a recent hour exam:

25, 35, 45, 45, 58, 62, 62, 62, 75, 78, 78, 83, 84, 88, 88, 92, 92, 92, 95, 95, 95, 97, 100, 100

The average score is the sum of these scores divided by the number of students:

25 + 35 + 45 + 45 + 58 + 62 + 62 + 62 + 75 + 78 + 78 + 83 + 84 + 88 + 88 + 92 + 92 + 92 + 95 + 95 + 95 + 97 + 100 + 100 = 1826average = 1826 / 24

average = 76.08

Another -- and fully equivalent method -- would be to multiply each score by the number of student who got that score and add up those numbers. For the scores we have above that means100 x 2 = 200 97 x 1 = 194 95 x 3 = 285 92 x 3 = 276 88 x 2 = 176 84 x 1 = 84 83 x 1 = 83 78 x 2 = 156 75 x 1 = 75 62 x 3 = 186 58 x 1 = 58 45 x 2 = 90 35 x 1 = 35 25 x1=2524 1826

From this list we can again find the average score,average score = 1826 / 24

average score = 76.08

We could write this asS=_{avg}N_{i}S_{i}/N_{i}

whereS_{avg}= average score

S_{i}= a particular score

N_{i}= number of students

receiving that score

Example #2

Suppose the students in a class have the following heights, measured in centimeters:

182 cm, 180, 180, 175, 174, 174, 174, 168, 168, 168, 166, 166, 162, 162, 162, 159, 140

What is the average height of students in this class?

We can find the average my adding these heights,

182 + 180 + 180 + 175 + 174 + 174 + 174 + 168 + 168 + 168 + 166 + 166 + 162 + 162 + 162 + 159 + 140 = 2860

and then dividing this by seventeen, the number of students,average height = 2860 / 17

average height = 168.2 cm

Or, we could take each particular height measurement and multiply it by the number of students with that particular height. Then we could make a table like this:182 x 1 = 182 180 x 2 = 360 175 x 1 = 175 174 x 3 = 522 168 x 3 = 504 166 x 2 = 332 162 x 3 = 486 159 x 1 = 159 140 x1=14017 2860

Now we can find the average height by dividing 2860 by 17;average height = 2860 / 17

average height = 168.2

h_{avg}= 168.2 cm

We could write this ash=_{avg}N_{i}h_{i}/N_{i}

whereh_{avg}= average height

h_{i}= a particular height

N_{i}= number of studentshaving that height

Center of MassRocket PropulsionReturn to ToC, Ch9, Momentum(c) Doug Davis, 2001; all rights reserved