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'''Average''' is the sum of a group of numbers divided by the number of values in the group. For example, the average of 3, 5, and 7 is <math>15 \div 3=5</math>.
The '''average''' is the sum of a group of numbers divided by the number of values in the group. For example, the average of 3, 5, and 7 is <math>15 \div 3=5</math>.
Another term for average is the [[arithmetic mean]].
Another term for average is the [[arithmetic mean]].
The other two common types of average are the median and the mode.
==Other measures of central tendency==
An average takes a set of numbers and replaces it with a single number. The average has these properties:
*If all the numbers in the set are equal, then the average equal to every number in the group. The average of 15, 15, 15, 15, and 15 is 15.
*If the numbers in the set are not equal, the average always falls somewhere within the set. That is, it is higher than the lowest number and lower than the highest number.
Because of these characteristics, an average takes a group of numbers and replaces it with a single number that can be thought of as the center of the group, or as a representative value that can stand for the whole group.
The average is ''not'' the only way to do this. It is one of a number of ''measures of central tendency.'' All of them replace a group of numbers with a single number that falls somewhere within the group.
Because the average is familiar and easily calculated, it is often chosen as the single number that summarizes a group. Depending on how the number is to be used, sometimes the average is a good choice, sometimes it is not, so it is important to understand the other choices.
*The median is the value which splits the group of numbers in the middle: half are higher, half are lower.
*The median is the value which splits the group of numbers in the middle: half are higher, half are lower.
*The mode is the most common number.
*The mode is the exact value which occurs the most often in the group; in a continuous distribution, it is the highest peak.
==An "analog computer" for the average==
*The average is also known as the ''arithmetic mean.'' There are other ''means.''
**The [[geometric mean]] is used for "averaging" compound interest rates and in other percentage-growth situations.
**The [[harmonic mean]] is used when averaging speeds that are all measured along the same distance (rather than the same time).
**The [[root mean square]] is used in engineering power calculations, and heavily used in statistics in measurements of [[variance]].
==The average as a "balance point"==
[[Image:Ruler.jpg|right]]
[[Image:Ruler.jpg|right]]
If equal weights are hung from a ruler, and the weight of the ruler itself is small enough to be neglected, the distance marking at which the ruler will balance can be shown to be the average of the distance markings at which the weights are hung.
If equal weights are hung from a ruler, and the weight of the ruler itself is small enough to be neglected, the distance marking at which the ruler will balance can be shown to be the average of the distance markings at which the weights are hung.
In this diagram, weights are hung at the 1, 9, and 11 inch marks. The average of 1, 9, and 11 is 7. The ruler will balance if it is hung at the 7 inch mark. This can be considered as an example of an "analog computer" for the average.
In this diagram, weights are hung at the 1, 9, and 11 inch marks. The average of 1, 9, and 11 is 7. The ruler will balance if it is hung at the 7 inch mark. This can be considered as an example of an "analog computer" for the average.
When there are two numbers in a set, the average always "splits the difference" between them. For example, the average of 10 and 20 is 15. The two numbers, 10 and 20, are separated by ten units. The average, 15, is five units away from 10 and five units away from 20.
In the weight-and-ruler example, if we look at the distances of the weights to the left and right of the average, we see that
:One weight is 6 inches to the left of the average
:The other weights are 2 and 4 units to the right of the average
The distance of the weight on the left&dmash;6 inches;equals the total of the distances of the weights on the right.
It can be shown that this is always true. The ''sum of the distances''<ref>i.e. the absolute magnitude of the difference</ref> between the average and each of the numbers above it is always equal to the sum of the distances between the average and each of the numbers below it.
Compare this to the median. In the case of the median, the ''count'' of the numbers above the median equals the count of the numbers below the median.
==Weighted averages==
==Weighted averages==