Primary Mathematics/Average, median, and mode

Average, median, and mode

There are three primary measures of central tendency, and a couple less often used measures, which each, in their own way, tell us what a typical value is for a set of data. Generally, when finding the measures of central tendency, one would order the values of the data set from least to greatest.

Mode

The mode is simply the number which occurs most often in a set of numbers. For example, if there are seven 12-year olds in a class, ten 13-year olds, and four 14-year olds, the mode is 13, since there are more 13 year olds than any other age. In elections, the mode is often called the plurality, and the candidate who gets the most votes wins, even if they don't get the majority (over half) of the votes.

Median

The median is the middle value of a set of values. For example, if students scored 81, 84, and 93 on a test; we select the middle value of 84 as the median.

If you have an even number of values, the average of the two middle values is used as the median. For example, the median of 81, 84, 86, and 93 is 85, since that's midway between 84 and 86, the two middle values.

Average

The straight average, or arithmetic mean,(sometimes referred to simply as "average" or "mean"), is the sum of all values divided by the number of values. For example, if students scored 81, 84, and 93 on a test, the average is 86.

Mean

In mathematics, mean has several different definitions depending on the context.

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution.[1] In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving {\displaystyle \mu =\sum xP(x)} \mu =\sum xP(x).[2] An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value {\displaystyle 2^{n}} 2^{n} is {\displaystyle {\tfrac {1}{2^{n}}}} {\tfrac {1}{2^{n}}} for n = 1, 2, 3, ....

For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted by {\displaystyle {\bar {x}}} {\bar {x}}, pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted {\displaystyle {\bar {x}}} {\bar {x}}) to distinguish it from the population mean (denoted {\displaystyle \mu } \mu or {\displaystyle \mu _{x}} \mu _{x}).[3]

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.[4]

Outside of probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.

Weighted average

The weighted average or weighted mean, is similar to the straight average, with one exception. When totaling the individual values, each is multiplied by a weighting factor, and the total is then divided by the sum of all the weighting factors. These weighting factors allow us to count some values as "more important" in finding the final value than others.

Example

Let's say we had two school classes, one with 20 students, and one with 30 students. The grades in each class on a particular test were:

Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98

Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99

The straight average for the morning class is 80% and the straight average of the afternoon class is 90%. If we were to find a straight average of 80% and 90%, we would get 85% for the mean of the two class averages. However, this is not the average of all the students' grades. To find that, you would need to total all the grades and divide by the total number of students:

${\frac {4300\%}{50}}=86\%$

Or, you could find the weighted average of the two class means already calculated, using the number of students in each class as the weighting factor:

${\frac {(20)80\%+(30)90\%}{20+30}}=86\%$

Note that if we no longer had the individual students' grades, but only had the class averages and the number of students in each class, we could still find the mean of all the students grades, in this way, by finding the weighted mean of the two class averages.

Geometric mean

The geometric mean is a number midway between two values by multiplication, rather than by addition or subtraction. For example, the geometric mean of 3 and 12 is 6, because you multiply 3 by the same value (2, in this case) to get 6 as you must multiply by 6 to get 12. The mathematical formula for finding the geometric mean of two values is:

${\sqrt {AB}}$

Where:

A = one value
B = the other value

So, in our case:

${\sqrt {3(12)}}={\sqrt {36}}=6$

Note the new notation used to show multiplication. We now can omit the multiplication sign and show simply AB to mean A×B. However, when using numbers, 312 would be confusing, so we put parenthesis around at least one of the numbers to make it clear.

The geometric mean can be extended to additional values:

${\sqrt[{3}]{(2)(9)(12)}}={\sqrt[{3}]{216}}=6$

For three values, a cube root is performed instead of a square root. On four values, the fourth root is performed.