The minimum size for the required dataset appears to be 6, made up of data which, when ordered would be represented by the letters a to f. The sum of the data = 6*83=498, so that the mean will be 83. Because there is an even number of data in the set, the median has to be the average of c and d, so (c+d)/2=81; also we need a mode, which requires a minimum of two data values of 80, so b and c can be 80 and d needs to be 82 so (80+82)/2=81, the required median. f=a+26 so that the range is 26. Now we can add all the elements together:

a+80+80+82+e+(a+26)=498. 2a+e=498-(80+80+82+26)=498-268=230. e>82 and needs to be an even number, so the smallest is 84, leaving 2a=146, so a=73. The ordered dataset is { 73 80 80 82 84 99 }. The data doesn't need to be ordered, so we could have { 99 80 84 73 80 82 }, for example.