Hammett $\sigma_m$ and $\sigma_p$ constants are based upon the acid dissociation of benzoic acid and m- and p-substituted benzoic acids $(\ce{X-C6H4-CO2H})$ in water at $\pu{25 ^\circ C}$. They are correlated with $K_\mathrm{a}$ values of these benzoic acids as shown in few examples in the following table:
$$
\begin{array}{c|cccc}
\hline
\text{Substituent }(\ce{X}) & \sigma_p & K_\mathrm{a} & \mathrm{p}K_\mathrm{a} & \text{property} \\
\hline
\ce{p-OCH3} & -0.27 & 3.5 \times 10^{-05} & 4.46 & \text{EDG} \\
\ce{p-CH3} & -0.17 & 4.3 \times 10^{-05} & 4.34 & \text{EDG} \\
\ce{H} & 0.00 & 6.46 \times 10^{-05} & 4.19 & \text{Standard} \\
\ce{p-Cl} & +0.23 & 1.10 \times 10^{-04} & 3.96 & \text{EWG} \\
\ce{p-NO2} & +0.78 & 3.90 \times 10^{-04} & 3.41 & \text{EWG} \\
\hline
\end{array}\\
\text{Source: http://research.cm.utexas.edu/nbauld/unit4.htm}
$$
It is safe to say that $\sigma_p$ constants are more reflected on mesomeric effect while $\sigma_m$ constants are more reflected on inductive effect (Ref.1).
Thus, let's look at some of $\sigma_p$ and $\sigma_m$ values:
$$
\begin{array}{c|cccc}
\hline
\text{Substituent }(\ce{X}) & \sigma_p & \sigma_m \\
\hline
\ce{N(CH3)2} & -0.83 & -0.15 \\
\ce{NH2} & -0.66 & -0.16 \\
\ce{OH} & -0.37 & +0.12 \\
\ce{OCH3} & -0.27 & +0.12 \\
\ce{CH3} & -0.17 & -0.07 \\
\ce{NHC(=O)CH3} & -0.15 & +0.07 \\
\ce{C6H5}* & -0.01 & +0.06 \\
\ce{H} & 0.00 & 0.00 \\
\ce{OC(=O)Ph} & +0.13 & +0.21 \\
\ce{Cl} & +0.23 & +0.37 \\
\ce{C(=O)CH3} & +0.50 & +0.38 \\
\ce{NO2} & +0.78 & +0.71 \\
\hline
\end{array}\\
\text{*: Indicates that the group is in the most sterically hindered conformation.}\\
\text{Source: Journal of Medicinal Chemistry Reference and others; see Ref.2 & 3 below}
$$
According to $\sigma_p$ values, $+M$ strength of groups are:
$$\ce{-NH2 \gt -OH \gt -OR \gt -CH3 \gt -NH(C=O)R \gt -Ph \gt -OC(=O)R}$$
References:
- Corwin Hansch, Albert Leo, Robert W. Taft, "A survey of Hammett substituent constants and resonance and field parameters," Chem. Rev. 1991, 91(2), 165–195 (DOI: https://doi.org/10.1021/cr00002a004).
- Corwin Hansch, Sharon D. Rockwell, Priscilla Y. C. Jow, Albert Leo, and Edward E. Steller, "Substituent Constants for Correlation Analysis," J. Med. Chem. 1977, 20(2), 304-306 (DOI: https://doi.org/10.1021/jm00212a024).
- Corwin Hansch and Albert Leo, In Substituent Constants for Correlation Analysis in Chemistry and Biology; Wiley-Interscience: New York, NY, 1979 (ISBN-13: 978-0-471-05062-9).