Difference between revisions of "Average"
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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>(3+5+7) \div 3=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>(3+5+7) \div 3=15 \div 3=5</math>. | ||
| − | Another term for average is the | + | Another term for average is the ''arithmetic mean''. |
==Other "measures of central tendency"== | ==Other "measures of central tendency"== | ||
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*The [[median]] is the value which splits the group of numbers in the middle: half are higher, half are lower. The median is the most commonly used [[quantile]]. | *The [[median]] is the value which splits the group of numbers in the middle: half are higher, half are lower. The median is the most commonly used [[quantile]]. | ||
*The [[mode]] is the exact value (or values) which occurs the most often in the group; in a continuous distribution, it is the highest peak. | *The [[mode]] is the exact value (or values) which occurs the most often in the group; in a continuous distribution, it is the highest peak. | ||
| − | |||
*The average most commonly used is known as the ''arithmetic mean.'' There are other ''means.'' | *The average most commonly used is 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 [[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 [[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 [[root mean square]] is used in engineering power calculations, and heavily used in statistics in measurements of [[variance]]. | ||
| + | *The symbol for a parabolic mean is "<!-- (number) ||" | ||
==An example of average, median, and mode== | ==An example of average, median, and mode== | ||
| − | Consider the number of chapters in each of the thirty-nine books of the Old Testament.<ref>Chosen because it is an example of a set of numbers that is far from having a normal | + | Consider the number of chapters in each of the thirty-nine books of the Old Testament.<ref>Chosen because it is an example of a set of numbers that is far from having a normal redistribution.</ref>. Ranked in order by number of chapters, the list is: |
:1, 2, 3, 3, 3, 3, 4, 4, 4, 5, 7, 8, 9, 10, 10, 12, 12, 13, 14, 14, 21, 22, 24, 24, 25, 27, 29, 31, 31, 34, 36, 36, 40, 42, 48, 50, 52, 66, 150 | :1, 2, 3, 3, 3, 3, 4, 4, 4, 5, 7, 8, 9, 10, 10, 12, 12, 13, 14, 14, 21, 22, 24, 24, 25, 27, 29, 31, 31, 34, 36, 36, 40, 42, 48, 50, 52, 66, 150 | ||
*The thirty-nine books contain a total of 929 chapters, so the ''average'' number of chapters is 23.8. | *The thirty-nine books contain a total of 929 chapters, so the ''average'' number of chapters is 23.8. | ||
Revision as of 00:49, March 22, 2008
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 
.
Another term for average is the arithmetic mean.
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 measurement with these characteristics. 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 mid-range is the value which falls exactly half way between the two extreme values in the the group.
- The median is the value which splits the group of numbers in the middle: half are higher, half are lower. The median is the most commonly used quantile.
- The mode is the exact value (or values) which occurs the most often in the group; in a continuous distribution, it is the highest peak.
- The average most commonly used is 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 symbol for a parabolic mean is "
