You are not alone, I am lost somewhere between Wolfram Language, math and physics 97% of the time. I used ConstantArray as a easy way to generate repeated frequency values the length of the mass span. ConstantArray[x,4] -> {x,x,x,x} so I just repeated the frequencies for the length of the mass span. That way, all data is weighted properly. I would not recommend trying this with large data sets but for lists of under several hundred elements it is a straightforward method.

Note that I erred in the data table so the proper result is 39. Technically, you can not get more significant digits than the least of your data, so you can only resolve to 39. Here is a corrected code block...

Block[{x = {{{20, 24}, 20}, {{25, 29}, 24}, {{30, 39}, 45}, {{40, 54},
      30}, {{55, 59}, 5}, {{60, 69}, 4}}, m, f, a},
 m = Table[Range[x[[i, 1, 1]], x[[i, 1, 2]]], {i, 1, Length@x}];
 f = Table[ConstantArray[x[[i, 2]], Length[m[[i]]]], {i, 1, Length@x}];
 a = Flatten[m].Flatten[f]/Total[Flatten[f]];
 Column[{
   StringForm["Given a mass-frequency table: ``", 
    MatrixForm[x\[Transpose]]],
   StringForm["The table range of masses m \[Rule] ``", 
    m // MatrixForm],
   StringForm["Corresponding frequencies f \[Rule] ``", 
    f // MatrixForm],
   StringForm[
    "Thus m*f/(Total@f) evaluates average a \[TildeTilde] ``", 
    Round[N@a, 1]]
   }]
 ]

In an effort to contribute something (useful, I hope) to the community after all these years, I offer this schoolboy approach with yet another result for the weighted average... 34.

But, I have no idea how to add the code from my cloud directory, so I attach the nb file in hopes someone can see it.

Attachments:

aveMaria.nb

"My doubt is whether my representation of the uneven frequency distribution using Piecewise and ProbabilityDistribution appropriate to estimate the mean and median?" Yes, but other representations are probably better. It's an issue of Density Estimation.

Your dist estimates the density by a piecewise constant function, while the solution proposed by D.L. is the midpoint density estimation. More sophisticated methods consist in fitting by some F the associated distribution function in a convenient basis (polynomials (basic, trigonometrical, Berstein,..), splines, wavelets). Then f:=F' is some density estimation, Integrate[x f[x]] is the estimation of the mean, x0 such that F[x0]==0.5 is an estimation of the median, etc... There is a large literature about this topic.

Hello, your mean is correct, since

Integrate[ x PDF[dist, x], {x, 19.5, 69.5}]=38.9198

The above mean computation is not properly weighting for interval lengths. So it would actually be:

vals = {20, 24, 45, 45, 30, 30, 30, 5, 4, 4};
midpts = {22, 27, 32, 37, 42, 47, 52, 57, 62, 67};
midpts.vals/Total[vals] // N

(* Out[169]= 38.9198 *)