The arithmetic mean of a set of numbers is equal to the sum of that set of numbers divided by the number of elements in the set. As an example, for the set S = {1, 3, 5, 7, 9}, the arithmetic mean of S is equal to (1+3+5+7+9)/5 = 5.
The arithmetic mean of a set of numbers is sometimes referred to as the average of that set of numbers. In general, the formula for the arithmetic mean of a set of numbers S is:
A = ΣS/n
where ∑S is the sum of the numbers in that set and n is the number of elements in that set.
As with many elementary mathematical concepts, it is hard to say exactly when humans first started using the concept of the arithmetic mean, though it likely was first developed by ancient astronomers as a means of reducing observational errors in astronomical data. The first formal representation of the concept of arithmetic mean was provided in the 16th century, but historical texts dating back to at least the 7th century B.C. reference the technique of deriving the average of a set of numbers.
Mean, Median, and Mode
The concept of arithmetic mean is closely related to the concepts of the median and mode of a set of numbers. Mean, median, and mode are all considered measures of central tendency. In a way, one can view the mean, median, and mode as all different ways of “summarizing” the information in a dataset and giving a value that is “typical” of that dataset.
Median
The median of a set of numbers can be thought of the element that falls in the middle of that set. The median of a set is defined as the element that has an equal amount of elements larger than and smaller than that element. For instance, in the set S = {2, 4, 6, 8, 10, 12, 14} the median value is 8 as there are 3 values less than 8 (2, 4, and 6) while there are 3 elements greater than 8 (10, 12, and 14). Consequently, 8 is the 4th larger and 4th smallest element of the set.
If there are an odd amount of numbers in the set, then the median is just whatever number falls directly in the middle when arranging the numbers from least to greatest. In a set with an even amount of members, there will be no single central value. In these cases, the median value is normally calculated as the mean of the two most middle values. so in the set S = {1, 3, 5, 7, 9, 11} the median value is the mean of the two central values (5+7)/2 = 6.
Mode
The mode of a set of numbers corresponds to the most frequently recurring element in that set. For the set S = {1, 2, 2, 2, 3, 4, 7, 7} the mode is 2 as 2 is the most recurring element of the set (3 times). In other words, the mode of a set is the value one is most likely to get if they choose at random from the set. When graphing the probability distribution for the value of the elements in a set, the mode will correspond to any “peaks” in that distribution.
It is entirely possible that a set of numbers has more than one mode. in the set S = {2, 2, 2, 4, 5, 6, 6, 6, 7, 7, 8} the two modes are 2 and 6 as those two elements both recur the most. Sets that have 2 distinct modes are called bimodal. Sets that have more than 2 distinct modes are called multimodal. the set S = {12, 12, 15, 23, 23, 26, 27, 28, 28} has 3 modes, 12, 23, and 28, and so it multimodal.
As a consequence of this definition of mode, it is impossible to have a non-empty finite set with no mode. In a set where every element is different, it may seem at first glance that there is no mode. However, a set where every element is distinct is actually multimodal. This is because in a set where each element is different, every element is a mode because each element recurs with the same frequency (namely, once). IN the set S = {1, 2, 3, 4, 5, 6}, every element is a mode because each element recurs with the same frequency. However, some infinite sets, like the set of real numbers, do not have a well-defined mode.
Additionally, the concept of mode is one of the few measures of central tendency that makes sense in non-numerical contexts. For example, in taking a set of U.S. last names, one might find that “Smith” is the mode of the dataset. That is, among the set of U.S. last names, Smith is the most recurring name.
Mean, Median, And Mode Together
Combining all of these concepts together, we can individually give the mean, median, and mode of some set of numbers S. Lets say that S = {3, 7, 14, 14, 14, 14, 22, 22 36, 38, 56, 56, 63}.
- The mean of the set is (3+7+14+14+14+14+22+22+36+38+56+56+63)/13 = 27.62
- The median of the set is equal to the central value. In this case, the central value is 22 as 22 is the 7th largest and 7th smallest element.
- The mode of the set is the most recurring element, so in this case, it is 14.
Thus, for the set S = {3, 7, 14, 14, 14, 14, 22, 22 36, 38, 56, 56, 63}, the mean, median, and mode are 27.62, 22, and 14, repsectively.
For a number of datasets, it is possible that multiple of these measures will coincide. In the set S = {3, 3, 3, 3, 3, 3, 3} the mean, median, and mode are all the same value, 3.
For example, the above graph represents a variety of Gaussian probability distributions, also called bell curves. Normal distributions represent the expected distribution of the random variable in class. In a normal distribution, the mean, median, and mode of the dataset are all the same; the value that coincides with the peak of the probability distribution. Many datasets, such as the distribution of IQ scores in the population, height, blood pressure, and standardized test scores, take on the form of a normal distribution, making it a useful tool for statistical analysis.
Uses Of Mean, Median, And Mode
Mean, median, and mode are such fundamental mathematical concepts it is hard to find a situation where they would not apply in some way.
Uses Of Mean
One of the most common uses of the arithmetic mean is in figuring out the average velocity of an object over some period of time. Let’s say the velocities of an object over time is the set {3, 7, 9, 15, 18, 20}. One can figure out the average velocity of an object over that time by summing the individual velocities and dividing by the number of elements. So in this case, the average velocity is (3+7+9+15+18+20)/6 = 12 m/s. A similar process can be used to determine that average acceleration of a body over time. By summing the individual acceleration values over a period of time, one can determine the average acceleration of an object over that time.
Another context in which the arithmetic mean is used is in economics for calculating the gross domestic product per capita of a nation. The GDP per capita is a measure of the average economic output per individual of a population. The GDP per capita is calculated by summing the gross economic output of each individual and dividing by the total number of individuals. Say a hypothetical country has 5 individuals and their economic output of each individual is given by the set {$24,000, $36,000, $36,000, $49,000, $63,000}. The GDP per capita of that population would be (24,000+36,000+36,000+49,000+63,000)/5 = $41,600. Generally, the calculated value for GDP per capita is then corrected to account for inflation or monetary value over a period of time.
Uses Of Median
The most common use of the median is to determine the characteristics of a “typical” member of a set in cases where the arithmetic mean may be skewed by abnormally high or low values. Lets say we have a set S of people with ages S = {11, 13, 17, 22, 25, 92}. The mean of these ages is (11+13+17+22+25+92)/6 = 30. In this case, the mean of 30 is skewed by the presence of the outlier value of 92, which is much higher than the other values. In fact, only one member of the set has an age greater than 30, so an average age of 30 should not really be considered representative of the set. In this case, a value that is more representative of a typical member of that set is the median value, which is (17+22)/2 = 19.5.
One place medians are used is in calculating average incomes of populations. In countries with high amounts of economic inequality, the presence of individuals with much higher incomes than others can skew the mean calculation. In these sorts of cases, a median income measure might be a better way to determine the income of the average person, rather than the mean which can be distorted by very high or low outlier values.
Uses Of Mode
The most obvious use of the mode is to determine which member of a set a randomly selected element is likely to be. Since the mode is the most frequently occurring element in the set, any random member of the set is most likely to be a mode (or modes).
Like the median, the mode is a useful metric to represent a “typical” member of a dataset when the mean is skewed by abnormally high or low values. In the set S = {2, 2, 2, 2, 2, 2, 7, 18, 29, 54} a mean calculation would be skewed by the higher values at the end of the set. So instead, we can take the mode, which is 2, as an indicator of a typical member of the set. Any random member of a set has the highest percentage chance of being the mode of that set.