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The chord-axial force coefficient is the dimensionless chord-axial aerodynamic force:Here $S$ is the stated aerodynamic reference area, $\rho$ the reference mass density and $U$ the reference speed. The chord axis and its positive direction accompany the normalization. A signed coefficient can be negative even when the flow-relative drag coefficient is positive.

An original example uses $\rho =1.2$ kg/m${}^3$, $U=25$ m/s, $S=0.40$ m${}^2$ and aft-directed load $A=9$ N. Then $q=375$ Pa and $C_A=9/(375\cdot 0.40)=0.060$. Reporting that load with area 0.80 m${}^2$ instead gives 0.030. The reference-area scaling of aerodynamic coefficients changes the reported coefficient without changing the balance reading.

For a section, axial load per unit span $A^{\prime }$ is instead normalized as $c_a=A^{\prime }/(qc)$ using airfoil chord $c$. The denominator $qc$ has units N/m, whereas $qS$ has units N. A per-span load divided by a finite area does not produce the same dimensionless quantity.

With common references, the inverse airfoil force-axis transformation isFor stipulated $C_D=0.040$, $C_L=0.60$ and $\alpha =5{}^{\circ }$, this gives $C_A\simeq -0.01245$. The subtraction shows why labeling a body-axis coefficient as drag can change even the sign of the apparent result. NASA defines lift and drag relative to the flow, which is the physical basis for keeping these axes distinct.^[ref]

At fixed area, density and measured force, a small fractional speed error contributes approximately $-2\delta U/U$ to the coefficient's fractional error. Normalization consequently does not replace measurement calibration or an uncertainty assessment of the reference flow.

The chord-normal force coefficient normalizes the chord-normal aerodynamic force:The reference dynamic pressure, wing area, body axis and sign convention accompany the coefficient. For sectional force per unit span, use $c_n=N^{\prime }/(qc)$ with airfoil chord $c$. Mixing total force and sectional normalization gives different units.

With stipulated $q=500\mathrm{Pa}$, $S=0.40{\mathrm{m}}^2$ and $N=100\mathrm{N}$, $C_N=0.50$. Reporting the same force with reference area $0.20{\mathrm{m}}^2$ gives $C_N=1.0$. This reference-area scaling of aerodynamic coefficients changes a reported number without changing the measured load.

When chord-axial force coefficient $C_A=A/(qS)$ uses the same references, the airfoil force-axis transformation gives
For $C_A=0.10$, $C_N=0.50$ and $\alpha =30{}^{\circ }$, the drag coefficient is approximately $0.3366$ and the lift coefficient is $0.3830$. Neither equals the normal coefficient.

Normalization does not remove measurement error. At fixed mass density and reference area, a small fractional speed error contributes approximately twice that fractional error to $q$, and hence the opposite fractional error to $C_N$ if $N$ is otherwise fixed. The usefulness of a dimensionless coefficient depends on retaining the measured references and their measurement uncertainty.