# 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

Savg = Ni Si / Ni

where

Savg = average score

Si = a particular score

Ni = 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 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

havg = 168.2 cm

We could write this as

havg = Ni hi / Ni

where

havg = average height

hi = a particular height

Ni = number of students

having that height