The arithmetic mean is the average of a set of data.
To find the mean: add up all the numbers and then divide by the number of items in the set. For example, the average of 1, 2, 6, 8, 10 is:
1 + 2 + 6 + 8 + 10 / 5 = 5.4.
Mean vs Median
Both are measures of where the center of a data set lies, but they are usually different numbers. For example, take this list of numbers: 10,10,20,40,70.
- The mean (average) is found by adding all of the numbers together and dividing by the number of items in the set: 10 + 10 + 20 + 40 + 70 / 5 = 30.
- The median is found by ordering the set from lowest to highest and finding the exact middle. The median is just the middle number: 20.
Sometimes the two will be the same number. For example, the data set 1,2,2,6,7 has an average of 1 + 2 + 4 + 6 + 7 / 5 = 4 and a median (a middle) of 4.
Mean vs Average: What’s the Difference?
When you first started out in mathematics, you were probably taught that an average was a “middling” amount for a set of numbers. You added up the numbers, divided by the number of items you can and voila! you get the average. For example, the average of 10, 5 and 20 is:
10 + 6 + 20 = 36 / 3 = 12.
The you started studying statistics and all of a sudden the “average” is now called the mean. What happened? The answer is that they are exactly the same word (they are synonyms).
That said, technically, the word mean is short for the arithmetic mean.
Terms that are sometimes used in statistics that have very narrow meanings:
- Mean of the sampling distribution: used with probability distributions, especially with the Central Limit Theorem. It’s an average of a set of distributions.
- Expected value: (used sometimes to refer to an average of a probability distribution)
- Sample mean: the average value in a sample.
- Population mean: the average value in a population.
Advantages and Disadvantages
The median can be heavily influenced by outliers — numbers that are very small or very large. Take the above example of 10,10,20,40,70. If we add 1500 to the set, the mean becomes 10 + 10 + 20 + 40 + 70 + 150 / 6 = 175, which is a poor reflection of the center of the set. The median on the other hand is less affected by outliers; the middle of this set is 30 (in between the 20 and 40). In general, if you have outliers, use the median as a measure of central tendency. If you have a large data set with no outliers, use the mean.
There are other uses of the word in mathematics, depending on what branch of math and what kind of data you’re working with. There are dozens of different types. Most have very narrow applications to fields like finance or physics. These are some of the most common you’ll come across.
These are fairly common in statistics, especially when studying populations. Instead of each data point contributing equally to the final average, some data points contribute more than others. If all the weights are equal, then this will equal the arithmetic mean. There are certain circumstances when this can give incorrect information, as shown by Simpson’s Paradox.
To find it:
- Add the reciprocals of the numbers in the set. To find a reciprocal, flip the fraction so that the numerator becomes the denominator and the denominator becomes the numerator. For example, the reciprocal of 6/1 is 1/6.
- Divide the answer by the number of items in the set.
- Take the reciprocal of the result.
This is used quite a lot in physics. In some cases involving rates and ratios it gives a better average than the arithmetic mean. You’ll also find uses in geometry, finance and computer science.
This type has has very narrow and specific uses in finance, social sciences and technology. For example, let’s say you own stocks that earn 5% the first year, 20% the second year, and 10% the third year. If you want to know the average rate of return, you can’t use the arithmetic average. Why? Because when you are finding rates of return you are multiplying, not adding. For example, the first year you are multiplying by 1.05.
This is used mostly in calculus and in machine computation (i.e. as the basic for many computer calculations). It’s related to the perimeter of an ellipse. When it was first developed by Gauss, it was used to calculate planetary orbits. The arithmetic-geometric is (not surprisingly!) a blend of the arithmetic and geometric averages. The math is quite complicated but you can find a relatively simple explanation of the math here.
Used in geometry to find the volume of a pyramidal frustrum. A pyramidal frustrum is basically a pyramid with the tip sliced off.
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