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. The following is an example data set:. 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. 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.
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 is added to the data set that we examined above. This would affect the maximum, and it would change from 41 to 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. Actively scan device characteristics for identification. Use precise geolocation data.
Select personalised content. Similarly, if this point right over here is d, f of d looks like a relative minimum point or a relative minimum value. Once again, over the whole interval, there's definitely points that are lower. And we hit an absolute minimum for the interval at x is equal to b. But this is a relative minimum or a local minimum because it's lower than the-- if we look at the x values around d, the function at those values is higher than when we get to d.
So let's think about, it's fine for me to say, well, you're at a relative maximum if you hit a larger value of your function than any of the surrounding values. And you're at a minimum if you're at a smaller value than any of the surrounding areas. But how could we write that mathematically? So here I'll just give you the definition that really is just a more formal way of saying what we just said.
So we say that f of c is a relative max, relative maximum value, if f of c is greater than or equal to f of x for all x that-- we could say in a casual way, for all x near c. So we could write it like that. But that's not too rigorous because what does it mean to be near c? And so a more rigorous way of saying it, for all x that's within an open interval of c minus h to c plus h, where h is some value greater than 0. So does that make sense?
Well, let's look at it. So let's construct an open interval. So it looks like for all of the x values in-- and you just have to find one open interval. There might be many open intervals where this is true. But if we construct an open interval that looks something like that, so this value right over here is c plus h. That value right over here c minus h. And you see that over that interval, the function at c, f of c is definitely greater than or equal to the value of the function over any other part of that open interval.
And so you could imagine-- I encourage you to pause the video, and you could write out what the more formal definition of a relative minimum point would be. Well, we would just write-- let's take d as our relative minimum. Similarly, the maximum and minimum of a function are the largest and smallest value that the function takes at a given point.
Together, they are known as the extrema the plural of extremum. Minimum means the least you can do of something. For example, if the minimum amount of dollars you must pay for something is seven, then you cannot pay six dollars or less you must pay at least seven. You can do more than the minimum, but no less.
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