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The table below reports birth weight and whether the mother smoked for 189 pregnancies. If Smoking and Birthweight were independent, then we would expect the number of low birth weight babies born to smoking mothers to be:
(???/???) * ??? = 23.1
The residual for the corresponding cell of the table is thus:
??? - 23.1 = 6.9.
Birth Weight
Smokes High Med Low Total
No Smoke 42 44 29 115
Smoke 13 31 30 74
Total 55 75 59 189

E = (row total)(column total)/n
...where E is the expected cell count.

E = (74)(59)/189 = 23.1

Residual = 30 - 23.1 = 6.9

I hope this helps and is what you were asking.

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