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Arithmetic mean

Definition. The arithmetic mean of several numbers is the ratio of the sum of all this numbers to their total number.

Arithmetic meansum of numbers
total number
Arithmetic mean formula. If we have a data set containing the values a1, a2, ... ,an , then the arithmetic mean A is defined by the formula

Aa1 + a2 + ... + an
n
For example:
for two numbers
a
and
b
arithmetic mean is

a
+
b
2

for three numbers
a
,
b
and
c
arithmetic mean is

a
+
b
+
c
3

and so on.

Examples of tasks with arithmetic mean

Example 1. Bob plucked from the tree 4 pears, Kate - 2 pears, and John - 6. Children fruits piled together and divided equally. How many pears got each of them?
Solution. Find the arithmetic mean:
4 + 2 + 6 = 12 = 4
33
Answer: each got 4 pears.


Example 2. On programming courses 15 people came on Monday, 10 on Tuesday,12 on Wednesday, 11 on Thursday, 7 on Friday, 14 on Saturday, 8 on Sunday. Find the mean attendance of courses for the week.
Solution. Find the arithmetic mean:
15 + 10 + 12 + 11 + 7 + 14 + 8 = 77 = 11
77
Answer: an average programming courses visit 11 people a day.


Example 3. The driver drove two hours at a speed of 120 km/h and hour at a speed of 90 km/h. Find the average speed of the car.
Solution. Let us find arithmetic mean of car speeds for each hour of the way:
120 + 120 + 90 = 330 = 110
33
Answer: the average speed of the car wos 110 km/h.


Example 4. Arithmetic mean of 3 numbers is equal to 6, and arithmetic mean of other 7 numbers is equal to 3. Find the arithmetic mean of all 10 numbers.
Solution. Since arithmetic mean of 3 numbers is equal to 6 then their sum is equal to 6 · 3 = 18, similar the sum of remaining 7 numbers is equal to 7 · 3 = 21.
So the sum of all 10 numbers is equal to 18 + 21 = 39, and arithmetic mean is equal to
39 = 3.9
10
Answer: arithmetic mean of 10 numbers is equal to 3.9.

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