The signal-to-noise ratio (S/N) is a measure of the strength of the signal relative to the noise. In this case, the signal is the absorbance reading of 0.002 absorbance units, and the noise is the standard deviation of the noise, which is 0.0005 absorbance units.
The S/N for the 1 cm path length measurement is:
S/N = (signal) / (noise) = 0.002 absorbance units / 0.0005 absorbance units = 4
If the path length is increased to 5 cm and the number of scans is increased to 65, the signal will also increase by a factor of 5, but the noise will not change. Therefore, the signal will be 5 times larger and the noise will be the same, so the S/N will increase.
The signal for the new measurement is 5 * 0.002 = 0.01 absorbance units
The S/N for the new measurement is:
S/N = (signal) / (noise) = 0.01 absorbance units / 0.0005 absorbance units = 20
When you increase the path length by 5 times, the signal increases by 5 times, but the noise remains the same. Therefore, the signal-to-noise ratio will increase by a factor of 5. Similarly when you increase the number of scans, you are averaging the signal, which will reduce the noise, but the signal will remain the same. So the signal-to-noise ratio will increase.