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# Maximum

On this page: Example, with R,  Definition and Use,  Simple formula,  Tips and Notes,  Test yourself,  References  Download R  R is Free, very powerful, and does the boring calculations & graphs for scientists.

### Example, with R

Sometimes a maximum - the highest ranking number - is very obvious:

0   0   0   0   999999   0   0

Or you could find the maximum with

### Definition and Use

• The maximum is the highest number in a list of numbers - or, if none of the numbers are positive, it is the number closest to zero.
• The maximum is the highest ranking item in a list - assuming items of equal value are ranked equally.

### Simple formula

Assuming there are n different items in y, and r is their rank:

the maximum of y is y[r = n]

### Tips and Notes

• If the items' values are nominal, their rank is arbitrary and the maximum is equally arbitrary, misleading, or meaningless.
• If one or more items cannot be ranked, a maximum cannot be found unless those items are excluded - or their rank is provided.
• More than one item may share the maximum value. But if every item is identical the maximum is rather meaningless: their maximum, average and minimum are identical.
• If you are using the maximum in a sample to estimate the maximum in a population, you will usually underestimate the population maximum unless you are dealing with a large sample of a nominal variable that can only adopt a few values.
• Advertisers are much beloved of the maximum. Advertisers may claim that 'the efficacy of the product is UP TO 90%'. Put simply this means that the MAXIMUM efficacy is 90% - the MEAN efficacy may be 60% or 30% or ....Unfortunately the public (and it seems journalists) do not seem to understand the concept of a maximum, and persist in interpreting this as an 'average'. In UK there have been calls to ban "up to" speed claims by broadband providers, but it is most unlikely that it will ever happen!

### Test yourself

1. You can find the maximum by examining each item in turn,
2. or sort them into ascending order and select the last item,
3. or select the highest ranking item(s)...

To obtain the maximum of a non-numerical list, you must be able to rank its contents:

### Useful references

Barry Collins (2011) "Up to" broadband speeds "no longer credible". Full text
A consumer body has urged the UK Advertising Standards Authority to ban one type of "up to" claims, but the deception will continue!

Wikipedia: Sample maximum and minimum. Full text
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