Precision of apparatus

If precision if the closeness of agreement between independent values, then if you measure values to be 1.56 1.59 1.61

But using a higher precision apparatus you get 1.567 1.598 1.612
I know the uncertainty is reduced but surely the closeness decreases....
I know it sounds counter intuitive but could anyone clear this up for me please?A
And also, does precision determine the reliability of an experiment due to the decrease in random error?
And lastly do repeating measurements reduce uncertainty and why?

Thanks!

The range of values you have there from 1.56 to 1.61 is 0.05, gives you an idea of the (random) uncertainty/error in the readings.
Using an instrument with a higher resolution (precision) is not going to reduce this random error, because the precision of +- 0.001 is a smaller value than the actual uncertainty. The readings with the 2nd instrument still have the same range of values due to the same random error being present.

The precision of an instrument is only one factor in the uncertainty in a measurement. There is also the actual act of taking the measurement. You have already asked about parallax etc.

Taking repeated measurements reduces the random error in readings. The more measurements you take, the closer the mean value should be to the true value. If you only take a single measurement, all you know is the precision of the instrument. In the case above, if you had taken a single measurement with the 2nd instrument and got, say, 1.612, you could easily be fooled into thinking you have measured that value to an accuracy of +- 0.001
When you take repeated measurements and get a range of values of 0.05, you can see immediately that the uncertainty is much greater than 0.001

Oh, ok thanks.
But random error decreases precision?
And also since precision is the agreement of the closeness of individual points, is this a measure of the reliability?

Thanks!

Yes but you need to be careful. Are you talking about the precision (resolution) of an instrument or the precision of a set of readings?
In the example above you are talking about the precision (reliability) a set of data points. In your original post you were talking about the precision (resolution) of an instrument.