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The “maximum” and “minimum” can be used either to calculate the range of a data set in descriptive statistics, or to calculate the extreme values of a function in differential calculus. Here we talk about both uses.
The maximum and minimum in statistics
In statistics, the sample maximum and minimum, also called the largest and smallest observations, are the values of the largest and smallest elements in a data set (ie, the sample).
If there are outliers in the sample, they necessarily include the sample maximum or minimum, or both, depending on whether they are extremely high or low. However, if they are not abnormally far from the other observations, the sample maximum and minimum are not necessarily outliers.
Thus, the minimums and maximums are also useful for understanding a given set of data. Let’s take this example of the weight of 12 children.
38 50 13 110 26 42 81 22 36 49 77 98
Using the above data set of children’s weights we can find the minimum and maximum. The minimum is simply the lowest observation, while the maximum is the highest observation. The easiest way to know what is the minimum and maximum of a data set is to organize them from smallest to largest:
13 22 26 36 38 42 49 50 77 81 98 110
So, for our data, the minimum is 13 and the maximum is 110.
The maximum and minimum in calculation
In calculus, the terms maximum and minimum refer to the extreme values of a function, that is, the largest and smallest values that the function reaches.
Maximum means the upper limit or the largest amount possible. The absolute maximum of a function is the largest number contained in the interval of the function. In other words, if f(a) is greater than or equal to f(x) , for all x in the domain of the function, then f(a) is the absolute maximum.
For example, the function f(x) = -16×2 + 32x + 6 has a maximum value of 22 for x = 1 . Each value of x produces a value of the function that is less than or equal to 22, so 22 is an absolute maximum. In graphical terms, the absolute maximum of a function is the value of the function that corresponds to the highest point on the graph.
On the contrary, the minimum means the lower limit or the smallest amount possible. The absolute minimum of a function is the smallest number in its range and corresponds to the value of the function at the lowest point on its graph.
The theory to find the maximum and minimum values of a function is based on the fact that the derivative of a function is equal to the slope of the tangent. When the values of a function increase as the value of the independent variable increases, the tangent lines to the graph of the function have a positive slope, and the function is said to be increasing.
Conversely, when the values of the function decrease as the value of the independent variable increases, the tangent lines have a negative slope and the function is said to be decreasing. At the exact point where the function goes from increasing to decreasing or from decreasing to increasing, the tangent line is horizontal (slope 0) and the derivative is zero.
Sources
- Becerril, E. (sf). Increasing and decreasing functions .
- Franco, A. (2016). Statistics: maximum and minimum values.
- Requena, B. (2014). Maxima and minima of a function .
- Santiago , R., Gomez, J. & Parra, B. (2003). Theory of maximums and minimums .
