Average
| sum of the numbers | |
| average = | number of numbers |
Example:
What is the average of 16 , 19 , and 25?
Solution:
average = (16 + 19 + 25) / 3 = 20
Check this out:
16 , 19 , 25
-4 -1 +5 difference from the average of 20
Notice that the differences add up to zero.
This is always the case. We can use this to our advantage.
Example:
| # of lawn mowers | 752 | 747 | 755 | 754 |
| day | Monday | Tuesday | Wednesday | Thursday |
The table above shows the number of lawmowers produced on every weekday except Friday. If 750 mowers were produced on average on the five days including Friday how many lawnmowers were produced on Friday?
a)736
b)739
c)742
d)750
e)758
Solution:
We could use the average formula but this could be tedious. We know the average is 750.
| 2 | -3 | 5 | 4 | ||
| # of lawn mowers | 752 | 747 | 755 | 754 | ? |
| day | Monday | Tuesday | Wednesday | Thursday | Friday |
We know that the differences including Friday must add up to zero.
Let x = Fridays’s difference
2 - 3 + 5 + 4 + x = 0
x = -8
Friday # of lawmowers has to be a difference of 8 from the average.
750 - 8 = 742
The correct answer is C.
Example:
If the average of 12, 23, and 25 is 20, what is the average of 19, 30, and 32?
Solution:
We could just use the average formula but there is a quicker way.
Notice:
12 23 25
+7 +7 +7
19 30 32
If our sum was originally S and we add 7(3) to S our new average is:
| S + 21 | S | 21 | S | 7 | ||||
| 3 | = | 3 | + | 3 | = | 3 | + |
S/3 is our old average or just 20.
The new average is 20 + 7
Concept: When you add exactly the same number to each of the numbers the average goes up by the same amount. Thus, the average is 20 + 7 = 27
Example:
If the average of 12, 23, and 25 is 20, what is the average of 19, 30, and 67?
Solution:
We are adding a difference of 67 - 25 = 42 to the sum.
| S + 42 | S | 42 | S | 14 | ||||
| 3 | = | 3 | + | 3 | = | 3 | + |
S/3 is the original average, so the new average is just 20 + 14 or 34
In general,
| change in average | change in the sum | |
| = | number of numbers |
Weighted Averages
Example:
In a room there are 30 cats with an average weight of 12lbs and 40 dogs with an average weight of 33lbs. What is the average weight of the animals in the room?
a)12
b)22.5
c)24
d)35
e)37
Solution:
First of we cannot just average the two averages. This would be incorrect as there are different amount of dogs and cats.
We can use the average formula:
| 30(12) + 40(33) |
| 70 |
1680/70 = 24
We can also solve the problem with weighted averages as follows. We write the average of the cats on one side and the average of dogs on the other side. If there were exactly the same amount of cats and dogs the average would be in the middle.
However, since we have more dogs than cats the average is pulled toward the dog’s average. Let’s work are way backwards since we know the average is 24.
The difference between the cat’s average weight and the average is 12. The difference between the dog’s average weight and the average is 9. 12:9 ican be reduced to 4:3.The ratio between the distances in 4:3 in favor of the cats, which is exactly the reverse ratio of the amounts. There are 30 cats and 40 dogs.
The ratio of cats to dogs is 3:4
The ratio of the distances is 4:3
Example:
A club sold an average of 92 raffle tickets per member. Among the females the average was 84, and among the males the average was 96. What was the ratio of male to female members in the club?
a)1:1
b)1:2
c)1:3
d)2:1
e)3:1
Solution:
The ratio of the distances is 2:1
The ratio of females to males is the inverse of the distances or 1:2
The ratio of the males to is 2:1.
The correct answer is D.
Median, Mode, and Range
The “median” is the “middle” value in the list of numbers. To find the median, your numbers have to be listed in numerical order. The “mode” is the value that occurs most often.
The “range” is just the difference between the largest and smallest values.
Example:
Find median, mode, and range for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
The median is the middle value.
13, 13, 13, 13, 14, 14, 16, 18, 21
There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:
13, 13, 13, 13, 14, 14, 16, 18, 21
The median is 14. The mode is the most frequent number in the set, so 13 is the mode.
The largest value in the list is 21 and the smallest is 13. The range is 21 -13 = 8
Standard Deviation
The standard deviation is a number that expresses the degree to which the number vary from the mean, either above or below it. The grater the standard deviation the greater the degree in variation.
Example:
If a certain sample of data has a mean of 20 and SD of 3.Which of the following is more than 2.5 SDs from the mean?
a)12
b)13.5
c)17
d)23.5
e)26.5
Solution:
2.5 standard deviations is 2.5 x 3 = 7.5 in each direction
12.5 ← 20 → 27.5
Only A)12 is greater than this.
The correct answer is A.
(46)
(28)
(2)