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That private is unnecessary, the default access specifier for a class is already private:

 class number_bag {
 /*private*/
     size_t        m_size;
     FloatingPoint m_sum;
     FloatingPoint m_square_sum;

Mark functions that do not throw noexcept (in your case, all of them). This helps the compiler optimize if you compile with exceptions.

Use std::move to avoid unnecessary copies:

 number_bag(size_t size, FloatingPoint sum, FloatingPoint square_sum) :
     m_size{size},
     m_sum{std::move(sum)},
     m_square_sum{std::move(square_sum)} {}

That private is unnecessary, the default access specifier for a class is already private:

 class number_bag {
 /*private*/
     size_t        m_size;
     FloatingPoint m_sum;
     FloatingPoint m_square_sum;

Mark functions that should not throw noexcept. This helps the compiler optimize if you compile with exceptions.

Use std::move to avoid unnecessary copies:

 number_bag(size_t size, FloatingPoint sum, FloatingPoint square_sum) :
     m_size{size},
     m_sum{std::move(sum)},
     m_square_sum{std::move(square_sum)} {}

Compile with -Wall (or every warning turned on) - As posted, your code generates 2 warnings:

 main.cpp: In function 'int main(int, const char**)':
 main.cpp:113:14: warning: unused parameter 'argc' [-Wunused-parameter]
  int main(int argc, const char * argv[]) {
       ^~~~
 main.cpp:113:38: warning: unused parameter 'argv' [-Wunused-parameter]
  int main(int argc, const char * argv[]) {
                               ^

If you compile with gcc, use -Wall -Wextra -pedantic (or every warning turned on, with extensions disabled) - As posted, your code generates 2 warnings:

 main.cpp: In function 'int main(int, const char**)':
 main.cpp:113:14: warning: unused parameter 'argc' [-Wunused-parameter]
  int main(int argc, const char * argv[]) {
       ^~~~
 main.cpp:113:38: warning: unused parameter 'argv' [-Wunused-parameter]
  int main(int argc, const char * argv[]) {
                               ^

Even if fp_type was necessary, please don't define it to just use it once, that's not really efficient in terms of code size.

Use auto to simplify operator+'s and operator-'s return type.

Don't use C-style casts, use C++ style casts instead. In your case, you can even use long double literals instead of an ugly cast:

auto bag3 = bag2 + 1.0l; // or 1.l

Always use pre-increment, unless you have a specific reason not to.

this-> is in every case you use it redundant. Remove it, it just clutters the view. :)

I don't see why you implemented your own copy constructor and copy assignment operator. You literally have no reason to, they do the same as the implicit defined ones.

Even if fp_type was necessary, please don't define it to just use it once, that's not really efficient in terms of code size.

Use auto to simplify operator+'s and operator-'s return type.

Don't use C-style casts, use C++ style casts instead. In your case, you can even use long double literals instead of an ugly cast:

auto bag3 = bag2 + 1.0l; // or 1.l

Always use the pre-increment operator (++i;) instead of the post-increment operator (i++;), unless you have a specific reason not to. Although a good compiler might generate in almost every case the same assembly code, there can exist a class that has side effects in its construction, and as such a compiler is not allowed to remove those.

this-> is in every case you use it redundant. Remove it, it just clutters the view. :)

I don't see why you implemented your own copy constructor and copy assignment operator. You literally have no reason to, they do the same as the implicit defined ones.