The mean of six numbers is 23. If one of the numbers is excluded, the mean of the remaining numbers is 20.

Let the numbers be x1, x2,..., x6.
Mean = 23
We know:

Mean $=\frac{\text { Sum of observa tions }}{\text { Number of observations }}$

Thus, we have:

$23=\frac{x_{1}+x_{2}+\ldots+x_{6}}{6}$

$x_{1}+x_{2} \ldots \ldots \ldots x_{6}=138$

If one number, say, x6, is excluded, then we have:

$20=\frac{x_{1}+x_{2}+\ldots+x_{5}}{5}$

$x_{1}+x_{2} \ldots \ldots+x_{5}=100 \ldots \ldots \ldots \ldots \ldots \ldots$ (ii)

Using (i) and (ii), we get:

$138=x_{1}+x_{2}+\ldots+x_{5}+x_{6}$

$\Rightarrow 138=100+x_{6} \quad \ldots$ (i)

$\Rightarrow x_{6}=38$

Thus, the excluded number is 38