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 = 1826

average = 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 means

100 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 x 1 =  25  
     24  1826

From this list we can again find the average score,

average score = 1826 / 24

average score = 76.08

We could write this as

S_avg = N_i S_i / N_i

where

S_avg = average score

S_i = a particular score

N_i = number of students

receiving that score

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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 x 1 = 140  
     17  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 as

h_avg = N_i h_i / N_i

where

h_avg = average height

h_i = a particular height

N_i = number of students

having that height

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(c) Doug Davis, 2001; all rights reserved