There is one point in Bas's instructions I would like to clarify for beginners. While "Alternating Current" is technically (and 100%) correct, the reference to "current" could lead a beginner to assume they are measuring current, thereby setting their meters (photo aside) to Current (Amps) mode which, depending upon the particular meter can result in: 1) nothing much because current testing requires moving the positive test lead to another socket, 2) an anti-climatic finish that leaves the meter temporarily unusable (as in always reading 0.00 or over-range) due to a blown fuse, or 3) in [mostly] cheap meters it may result in a spectacular and catastrophic failure as the entire meter instantaneously goes up in smoke and flames. I feel it is safer to instruct beginners to set their meters to "AC Volts" for autoranging meters and a range of at least 200VAC for manual ranging meters (for 120V outlets, select a range of 400+VAC for 240V outlets).
Although some corrective action is almost certainly necessary for this issue, I feel the provided instructions are incomplete. While we are certainly interested in the initial unloaded voltage on this problem receptacle, the number that really matters is the voltage under maximum load -- in other words, the voltage drop. For a 120V receptacle we generally want to see between 110V and 125V on an unloaded receptacle. If you measure voltage signicantly more or less than this then it warrants further investigation elsewhere in your house to determine if the over- or under-voltage condition is endemic or specific to that receptacle. If the issue is endemic (as in all -- or atleast 1/2 -- your receptacles) then you need to contact your local power provider to resolve the issue, which will likely involve replacing your electrical drop's transformer.
If you have established that your home's voltage is within the expected range of 110-125V (for a 120V receptacle) then we can proceed with the knowledge that this issue is isolated to a single circuit (a circuit may consist of anywhere from a single outlet to a series of outlets and/or lights or other household accessories all of which reside on a single circuit breaker).
Once you have determined the unloaded voltage on the problem receptacle, go ahead and connect your compressor (use no extension cords) and power it on WHILE STILL MONITORING voltage. If you see a voltage drop of more than about 5-7% (5-8VAC) less than the unloaded voltage OR if either of these voltages appears to be intermittently dropping out (as in loose connection) then you have a definite problem with this circuit (either at this receptacle, the circuit breaker/fuse, or somewhere in between). You should register the maximum voltage drop as the compressor approaches peak [amp] load just before it's high-pressure cutoff. If you saw wild fluctuations in voltage then you likely have a loose connection at the receptacle, circuit breaker/fuse or anywhere in between. However, if you have excessive, but essentially steady, voltage drop then you have excessive resistance within the circuit which may be as simple as a loose connection or as complicated as an illegal tap (if your home has ever been remodelled -- psuedo-professionally or DIY), undersized wiring or even knob-and-tube (if an older home) wiring.
Go ahead and TURN OFF THE CIRCUIT BREAKER/FUSE for this receptacle, open the outlet box and try tightening down all the receptacle's screws and make certain that the all the wires make firm contact with their respective screws. If this outlet makes loose contact with the plug (i.e. you can easily pull a cord out of the receptacle with little effort) then the internal contacts may have lost their spring pressure in which case this is the perfect time to replace the receptacle since loose contact can cause the same symptoms and lead to an overheated outlet and plug. If you replace the receptacle, the white wire goes to the nickel screw (on same side as the wide receptacle prong) the black or red wire goes to the brass screw (short prong side) and the bare or green wire to the green ground screw. Afterward, go ahead and close up the receptacle outlet and flip the circuit breaker/fuse back on. Retest to see if this solved your problem.
If the above did not solve your problem then I really -- and strongly -- recommend you SECURE THE SERVICES OF A QUALIFIED ELECTRICIAN to troubleshoot and solve the problem for you. Loose and/or high-resistance connections are not something you want to ignore because they lead to overheated wiring and junctions when under load WHICH CAN RESULT IN FIRE! Residential wiring often serves multiple receptacles, lights, switches, smoke detectors and other household accessory loads which can make tracing the connection back to the circuit breaker/fuse panel challenging -- especially in homes with hidden junction boxes, illegal taps or knob-and-tube wiring -- and outside the realm of what the average homeowner is equipped to deal with.
