RESEARCH METHOD COHEN ok

SAMPLING ERROR 107
Box 4.2
Distribution of sample means showing the spread
of a selection of sample means around the
population mean
Mpop Population mean
Ms Ms Ms Ms Mpop Ms Ms Ms Ms
Source: Cohen and Holliday 1979
Ms
Sample means
Chapter 10). Rose and Sullivan (1993: 144)
remind us that 95 per cent of all sample means
fall between plus or minus 1.96 standard errors
of the sample and population means, i.e. that we
have a 95 per cent chance of having a single
sample mean within these limits, that the sample
mean will fall within the limits of the population
mean.
By drawing a large number of samples of equal
size from a population, we create a sampling
distribution. We can calculate the error involved
in such sampling (see http://www.routledge.
com/textbooks/9780415368780 – Chapter 4, file
4.5.ppt). The standard deviation of the theoretical
distribution of sample means is a measure of
sampling error and is called the standard error
of the mean (SE M ). Thus,
SE = SD s
√
N
where SD S = the standard deviation of the sample
and N = the number in the sample.
Strictly speaking, the formula for the standard
error of the mean is:
SE = SD pop
√
N
where SD pop = the standard deviation of the
population.
However, as we are usually unable to ascertain the
SD of the total population, the standard deviation
of the sample is used instead. The standard error
of the mean provides the best estimate of the
sampling error. Clearly, the sampling error depends
on the variability (i.e. the heterogeneity) in the
population as measured by SD pop as well as the
sample size (N) (RoseandSullivan1993:143).
The smaller the SD pop the smaller the sampling
error; the larger the N, thesmallerthesampling
error. Where the SD pop is very large, then N
needs to be very large to counteract it. Where
SD pop is very small, then N, too,canbesmall
and still give a reasonably small sampling error.
As the sample size increases the sampling error
decreases. Hopkins et al. (1996:159)suggestthat,
unless there are some very unusual distributions,
samples of twenty-five or greater usually yield a
normal sampling distribution of the mean. For
further analysis of steps that can be taken to cope
with the estimation of sampling in surveys we refer
the reader to Ross and Wilson (1997).
The standard error of proportions
We said earlier that one answer to ‘How big a
sample must I obtain’ is ‘How accurate do I want
my results to be’ This is well illustrated in the
following example:
Aschoolprincipalfindsthatthe25studentsshetalks
to at random are reasonably in favour of a proposed
change in the lunch break hours, 66 per cent being in
favour and 34 per cent being against. How can she be
sure that these proportions are truly representative of
the whole school of 1,000 students
Asimplecalculationofthestandarderrorof
proportions provides the principal with her answer.
√
P × Q
SE =
N
where
P = the percentage in favour
Q = 100 per cent − P
N = the sample size
Chapter 4