Misconception in difference between Accuracy and precision?

Question

I know that The more systematic errors > the less accurate The more random errors > the less precise

So lets say we have a stopwatch which has a zero error -1 (every reading less then 1 the actual value)

and an old man is supposed to measure 60s

His readings are 65 ,50,54,62 mean value = 65+50+54+62/4 = 57.75

lets say we remove the systematic error(by adding 1 to each) and the readings become 66 ,51 ,55,63 mean value = 58.75

Although Now mean is more closer to the true value but why is not mean = 60 what am i getting wrong?

Answer 1

What you're getting "wrong" is the fact that your final answer does not have an error associated with it, so you cannot say "my value of $58.75$ is wrong". How do you know? You need to estimate an error associated with it. In this case, the easiest way to assign an error is to find the standard deviation in your measurement values, and then divide by the square root of the number of values (this is a standard result in experimental physics) - i.e. the error in your mean value should be: $$\sigma_\text{mean}=\frac{\sigma}{\sqrt{N}}$$ For this experiment, after correcting for your systematic error (which in this case doesn't actually affect $\sigma$) you get $\sigma_\text{mean}\approx3.5$, and so your final answer based on this experiment is $$\bar{t}=58.8\pm3.5 \text{s}$$ Based on that alone, that is a perfectly reasonable result - the "actual answer" of $60$ s lies within that error bound (which we expect should happen ~60% of the time).

How to "improve" this value? Well if it truly is wrong only because of random errors, then the way to get a more accurate and precise result would be to take more readings (4 readings is not very many!) Unfortunately in practice, systematic errors (errors which are not random) are always present and are in general very difficult to deal with (e.g. the old man might have slow reflexes, which would affect the result etc).

Just to answer the question posed in the title and to summarise my answer, there are in general two sorts of errors:

• Random errors are random, and are calculated in the way given above. They can be reduced by making more measurements, and this will increase both the precision and (assuming no systematic errors) accuracy of your reading.
• Systematic errors are all other errors (non-random). They are in general difficult to deal with. You can take as many measurements as you like and get a precise result, but if all of your readings are systematically affected, they will in general give an inaccurate mean.