Statistical Analysis for Education and Psychology Researchers

To answer this question we turn to consideration of standard errors and confidence
intervals, these are discussed in the next section.

4.5 From Sample to Population

Standard Errors

The standard deviation of a sampling distribution of a statistic is the standard error of
that statistic. It follows that the standard deviation of the sampling distribution of a
proportion is called the standard error of a proportion, represented as σp. Just as the
sample proportion, P, is an unbiased estimator of the population proportion (π), the
standard deviation of a sampling distribution of a proportion, S(p), is used to estimate the
standard error of a population proportion, σp. The standard error of a statistic is an index
of the precision of a parameter estimate.
The standard error of a proportion is evaluated as:

Standard

Error of a

Proportion

—4.4

Where P is the proportion of interest and n is the sample size.

Example 4.5: Standard Error of a Proportion

The standard error of the observed sample proportion of age 11 pupils who cannot
correctly interpret graphs with complex scales (see Example 4.4) is;

The standard error is an index of precision, that is an indication of the amount of error
that results when a single sample statistic, here a proportion, is used to estimate the
corresponding population parameter. The larger the standard error the less precise is an
estimate. The standard error is related to both sample size and heterogeneity in the
population. Larger sample sizes reduce the standard error; we divide by a larger
denominator in equation 4.4. Greater heterogeneity in the population, that is a larger
variance, increases the standard error. In this example, as the proportion of successes
approaches 0.5, the numerator in equation 4.4 increases, thereby increasing the size of the
standard error. As the population becomes more homogeneous, i.e., P tends towards 0 or
1, then sampling variability will reduce. In planning a study consideration should be give
to both sample size and the variances of important variables.

Statistical analysis for education and psychology researchers 100