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You could use the Bhattacharyya coefficient $w=e^{-d}$ $$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}, $$ as the measure $\in[0,1]$ for the closeness of two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$).

As the measure of the closeness of two distributions $p_A$ and $p_B$ You could use the Bhattacharyya coefficient $$w=\int \sqrt{p_A(x)p_B(x)}\,dx\in[0,1],$$ which for two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$) is given by $w=e^{-d}$ with $$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}. $$

You could use the Bhattacharyya coefficient $w=e^{-d}$ $$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}, $$ as the measure for the closeness of two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$).

You could use the Bhattacharyya coefficient $w=e^{-d}$ $$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}, $$ as the measure $\in[0,1]$ for the closeness of two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$).

You could use the Bhattacharyya distance $$w= \frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}, $$ as the measure $\in[0,1]$ for the closeness of two normal distributions (means $a,b$; variances $s^2$, $m^2$).

You could use the Bhattacharyya coefficient $w=e^{-d}$ $$d=\frac{1}{4} \ln \left [ \frac 1 4 \left( \frac{s^2}{m^2}+\frac{m^2}{s^2}+2\right ) \right ] +\frac{1}{4} \frac{(a-b)^{2}}{s^2+m^2}, $$ as the measure for the closeness of two Gaussian distributions (means $a,b$; variances $s^2$, $m^2$).