Posted by ash on Sunday, November 21, 2010 at 12:04am.
A population is a collection of data, and a sample is a subset (i.e. a smaller number of the data, obtained randomly) of the population.
The population mean, μ, is the arithmetic average of ALL the data present in the population. The sample mean is the arithmetic mean of the particular sample. It can vary from sample to sample, but is generally close to the population mean.
For example:
Consider a class of 10 children with ages 8,8,9,9,9,10,10,8,9,10.
The population mean is
μ=(8+8+9+9+9+10+10+8+9+10)/10
=90/10
=9.0
If we take a random sample, say, every second child starting from the first, we get
sample mean
=(8+9+9+10+9)
=45/5
=9.0
Which happens to be identical to the population mean. This is why often we take the sample mean to represent the population mean.
The standard deviation requires a little more calculation, it is defined, for the population, as:
σ
=√((Σ(x-μ)²)/n)
In the above case, we have
σ
=√(((8-9)²+(8-9)²+(9-9)²+(9-9)²+(9-9)²++(10-9)²+(10-9)²+(8-9)²+(9-9)²+(10-9)²)/10)
=√((1²+1²+0²+0²+0²+1²+1²+1²+0²+1²)/10)
=√(6/10)
=0.7746...
The sample standard deviation, sn is calculated similarly, with the exception that the divisor n is usually replaced by (n-1) to give a better estimate of the population standard deviation.