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When the inner shells on an electron are full, the outer electrons are shielded from the positive charge of the nucleus.
The larger an atom is in the same group, the further the outer electrons are from the nucleus. Due to the greater screening effect, it is easier to remove them. Note that the noble gases already have full valence shells, so they are (generally) not reactive. If you ignore the noble gases, electronegativity increase from left to right in the same group because the atom only needs a few more electrons to have a complete valence shell. The small halogens (F, Cl) are the most electronegative, and the heavy alkali metals (Cs, Rb) are the least electronegative.
The first ionization energy is generally inversely proportional to electronegativity for the same reasons.
This is a page on periodic trends with the kind of nice 3D graphs found in (some?) textbooks.
If you take the increase of atomic number across a period, electro negativity and Ionisation energy INCREASES. This is because of the fact that across a period the nuclear charge increases while radius decreases Thus the effective pull of the nucleus on the electrons is more. Hence this INCREASES the Electronegativity and IE.(Flourine is the most electronegative.)
But now suppose you take the increase in atomic number down a group. The electronegativity and IE decrease. Even though there is lot of increase in the nuclear charge, every time you go down one element a new shell is added this increases the size. The size increase counteracts this increase in size and hence the pull on the electrons is less. So electronegativity and IE are consequently less.