The minimum is the smallest value in the data set. The maximum is the largest value in the data set. Learn more about how these statistics may not be so trivial.

### Background

A set of quantitative data has many features. One of the goals of statistics is to describe these features with meaningful values and to provide a summary of the data without listing every value of the data set. Some of these statistics are quite basic and almost seem trivial. The maximum and minimum provide good examples of the type of descriptive statistic that is easy to marginalize. Despite these two numbers being extremely easy to determine, they make appearances in the calculation of other descriptive statistics. As we have seen, the definitions of both of these statistics are very intuitive.

### The Minimum

We start by looking more closely at the statistics known as the minimum. This number is the data value that is less than or equal to all other values in our set of data. If we were to order all of our data in ascending order, then the minimum would be the first number in our list. Although the minimum value could be repeated in our data set, by definition this is a unique number. There cannot be two minima because one of these values must be less than the other.

### The Maximum

Now we turn to the maximum. This number is the data value that is greater than or equal to all other values in our set of data. If we were to order all of our data in ascending order, then the maximum would be the last number listed. The maximum is a unique number for a given set of data. This number can be repeated, but there is only one maximum for a data set. There cannot be two maxima because one of these values would be greater than the other.

Example

The following is an example data set:

23, 2, 4, 10, 19, 15, 21, 41, 3, 24, 1, 20, 19, 15, 22, 11, 4

We order the values in ascending order and see that 1 is the smallest of those in the list. This means that 1 is the minimum of the data set. We also see that 41 is greater than all of the other values in the list. This means that 41 is the maximum of the data set.

### Uses of the Maximum and Minimum

Beyond giving us some very basic information about a data set, the maximum and minimum show up in the calculations for other summary statistics.

Both of these two numbers are used to calculate the range, which is simply the difference of the maximum and minimum.

The maximum and minimum also make an appearance alongside the first, second, and third quartiles in the composition of values comprising the five number summary for a data set. The minimum is the first number listed as it is the lowest, and the maximum is the last number listed because it is the highest. Due to this connection with the five number summary, the maximum and minimum both appear on a box and whisker diagram.

### Limitations of the Maximum and Minimum

The maximum and minimum are very sensitive to outliers. This is for the simple reason that if any value is added to a data set that is less than the minimum, then the minimum changes and it is this new value. In a similar way, if any value that exceeds the maximum is included in a data set, then the maximum will change.

For example, suppose that the value of 100 is added to the data set that we examined above. This would affect the maximum, and it would change from 41 to 100.

Many times the maximum or minimum are outliers of our data set. To determine if they indeed are outliers, we can use the interquartile range rule.