Needs["ErrorBarPlots`"]

(Subscript[En, α])={{22,Quantity[{4.6,.2},"Kiloelectronvolts"]},
{26,Quantity[{6.5,.2},"Kiloelectronvolts"]},
{28,Quantity[{7.6,.2},"Kiloelectronvolts"]},
{29,Quantity[{8.1,.2},"Kiloelectronvolts"]},
{30,Quantity[{8.7,.2},"Kiloelectronvolts"]},
{40,Quantity[{15.8,.2},"Kiloelectronvolts"]},
{42,Quantity[{17.4,.2},"Kiloelectronvolts"]},
{47,Quantity[{21.8,.2},"Kiloelectronvolts"]}}

(Subscript[En, β])={{26,Quantity[{7.1,.2},"Kiloelectronvolts"]},
{28,Quantity[{8.4,.2},"Kiloelectronvolts"]},
{29,Quantity[{9.0,.2},"Kiloelectronvolts"]},
{30,Quantity[{9.7,.2},"Kiloelectronvolts"]},
{40,Quantity[{17.6,.2},"Kiloelectronvolts"]},
{42,Quantity[{19.4,.2},"Kiloelectronvolts"]},
{47,Quantity[{24.5,.3},"Kiloelectronvolts"]}}

Ry=Quantity[13.6,"Electronvolts"]

a=With[{En=First[#],ΔEn=Last[#]},{Sqrt[En/Ry],((ΔEn)/(2Sqrt[En Ry]))}]&/@Last/@(Subscript[En, α])

b=With[{En=First[#],ΔEn=Last[#]},{Sqrt[En/Ry],((ΔEn)/(2Sqrt[En Ry]))}]&/@Last/@(Subscript[En, β])

sa=With[{n1=1,n2=2,En=First[Last[#]],ΔEn=Last[Last[#]],Z=First[#]},{Z-Sqrt[(En/(Ry ((1/(n1^2))-(1/(n2^2)))))],((ΔEn)/(2Sqrt[En Ry ((1/(n1^2))-(1/(n2^2)))]))}]&/@(Subscript[En, α])

sb=With[{n1=1,n2=3,En=First[Last[#]],ΔEn=Last[Last[#]],Z=First[#]},{Z-Sqrt[(En/(Ry ((1/(n1^2))-(1/(n2^2)))))],((ΔEn)/(2Sqrt[En Ry ((1/(n1^2))-(1/(n2^2)))]))}]&/@(Subscript[En, β])

(Subscript[σ, α])=Table[With[{Z=First[(Subscript[En, α])[[i]]],σ=First[sa[[i]]],Δσ=Last[sa[[i]]]},{{Z,σ},ErrorBar[Δσ]}],{i,Length[sa]}]

(Subscript[σ, β])=Table[With[{Z=First[(Subscript[En, β])[[i]]],σ=First[sb[[i]]],Δσ=Last[sb[[i]]]},{{Z,σ},ErrorBar[Δσ]}],{i,Length[sb]}]

ErrorListPlot[{(Subscript[σ, α]),(Subscript[σ, β])},Joined->True,PlotRange->{{20,50},{0,2.5}},AxesLabel->{"Z","σ"},PlotLegends->{"(Subscript[σ, α])","(Subscript[σ, β])"},ImageSize->Full,GridLines->Automatic,GridLinesStyle->Directive[Lighter[LightGray]]]

Style[Grid[Join[{{"Z","(Subscript[σ, α])","(Subscript[σ, β])"}},Table[With[{Z=First[(Subscript[En, α])[[i]]],σa=First[sa[[i]]],Δσa=Last[sa[[i]]],σb=If[i≥2,First[sb[[i-1]]]],Δσb=If[i≥2,Last[sb[[i-1]]]]},{Z,PlusMinus[NumberForm[σa,{2,2}],NumberForm[Δσa,{2,2}]],If[i≥2,PlusMinus[NumberForm[σb,{3,2}],NumberForm[Δσb,{2,2}]],""]}],{i,Length[sa]}]],Alignment->{Center,Automatic},Dividers->{None,{2->GrayLevel[0.7]}},Spacings->{{2->2,3->2},{2->0.75}}],"DialogStyle",FontSize->11]