Let's call the data list in your notebook weatherData.

Then you can use Cases (an alternative is Select, but Cases is more natural here) to pull out the data in your list that corresponds to the pattern with a June date (advice--time to start learning about patterns: go here and start with the tutorial links at the bottom; this is also in Mathematica itself in the Documentation Ceneter and it's better to learn it there as all the notebooks are Live-- http://reference.wolfram.com/language/ref/Pattern.html ).

juneWeatherData = Cases[weatherData, {{_, 6, _}, _}];

then you can use the earlier approach to pull out just the temperatures from this:

juneTemperatures=juneWeatherData[[All, 2]];

and then compute the mean using:

 Mean[juneTemperatures]

Which, for the data in your notebook gives 10.296

David, Notebook attached. Thanks!

Attachments:

David , works perfect! Thanks! The other problem is that since is a list with dates and real numbers (date in Date format) I not know how I can sort certain dates.For example within that list only looks for mean T in June. How I can Select dates with June within this large list and geenrate a new list only with June data over the timespan considered. I am just learning!!

If your data is in a parameter called weatherData then the largest temperature value will be

Max[weatherData[[All,2]]]

If you have Mathematica version 10.1 you can also use TakeLargestBy to get the largest value for temperature along with the associated date.

TakeLargestBy[weatherData,Last,1]

That 1 in the expression tells TakeLargestBy to only give you the largest value. If you want more of them then change it to another integer.

Of course if your list has more than one date with the same largest value then you will only get one of the values.