Bias is any factor that tends to sway the availability of data, its interpretation, or the data itself in a particular direction, potentially leading us to draw incorrect conclusions. For example, studies with positive findings are more likely to be published, and so there is a publication bias towards studies with positive results. This may lead us to conclude, for example, that an intervention has health benefits, when in reality there is an equal number of studies that show no health benefits.
Noise is the variation within the data that has no meaning or value for the interpretation of said data. For example, suppose you measure your weight once a week. Even if you don’t lose or gain any bodyweight, the result of your weighing will be slightly different each week because you might be more or less hydrated, fed, or wearing heavier/lighter clothes, for example. The direction of this effect is random, and so could either obscure or exaggerate any real change in your weight from week to week – this is noise. If the scales instead gave a reading that always underestimated your weight, they would be biased.
Bias can be mitigated by recognising where it has the potential to occur and either taking precautions to prevent it or accounting for it after the fact. For example, you could use your scales to weigh some objects of known weight. You could then determine by how much the scales underestimated your weight and adjust accordingly. Noise can be mitigated by techniques such as averaging. For example, rather than weighing yourself once, you could weigh yourself multiple times throughout the day and calculate the average. Some statistical methods may also be able to reduce noise from a set of data.