How much information, in bits, is conveyed by a 3-bit output representing how many 1s in a 7-bit input?
Assuming the 7-bit input is chosen randomly, the entropy of the number of 1s is
$$
 \sum_{i=0}^7 \frac{\binom{7}{i}}{2^7} \log \frac{2^7}{\binom{7}{i}} \approx 2.45 \text{ bits}.
$$
This is because the probability that the number of 1s is $i$ equals $\binom{7}{i}/2^7$.

You are asking how much information is conveyed by particular outputs, and this can be interpreted as the loss in entropy: if $X$ is the random 7-bit input and $Y$ is its sum, you could define the information conveyed by $Y=i$ as $H(X)-H(X|Y=i)$. Since $X$ has uniform distribution,
$$
 H(X) - H(X|Y=i) = \log 2^7 - \log \binom{7}{i} = \log \frac{2^7}{\binom{7}{i}}.
$$
The information conveyed by different outputs, under this definition, works out to be roughly (in bits):
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline
i &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; 6 &amp; 7 \\\hline
H(X) - H(X|Y=i) &amp; 7 &amp; 4.19 &amp; 2.61 &amp; 1.87 &amp; 1.87 &amp; 2.61 &amp; 4.19 &amp; 7
\\\hline
\end{array}
$$
The weighted average of these values equals the entropy calculated above.