Statistical Analysis for Education and Psychology Researchers

There is often confusion about whether a standard deviation or a standard error should
be reported. This confusion generally stems from misunderstanding what a standard error
is. The sample standard deviation is a measure of spread of raw scores and should
therefore be reported, with the mean, when the purpose is to describe a data distribution.
The standard error is an index of the precision of an estimated parameter and should be
reported when the aim is to compare parameter estimates e.g., when comparing means for
different treatment effects. You should put standard error bars and not standard deviation
bars on a graph which compares treatment means.

Confidence Intervals

Estimation is when a sample statistic is used to estimate a population parameter; this is
sometimes called point estimation. A confidence interval (CI) defines a range of values
within which the parameter is likely to be found, this may be referred to as interval
estimation. It is important to realize that it is the parameter which is fixed and the
confidence interval which might vary from sample to sample. The idea of confidence is
the proposition that the stated interval actually includes the population parameter of
interest. A common confidence interval is the 95 per cent interval. We would expect the
95 per cent confidence interval, written as CI0.95, to encompass the estimated parameter
95 times out of 100. We can also evaluate confidence intervals for the difference between
two parameters, for example, the difference between two means.
In general a 95 per cent CI is defined as:

The formula therefore for a CI0.95 of a proportion is:

CI0.95

for a

Proport

ion—

4.5

Example 4.6: CI0.95 for a Proportion

The CI0.95 for the proportion of age 11 pupils who cannot correctly interpret graphs with
complex scales, see Example 4.3, is;

This gives a 95 per cent confidence interval for the population proportion π as 0.18 to
0.42, that is from 18 to 42 per cent. You may think this is rather a wide range for the
possible values of π. To narrow the range a larger sample size would be required.

In the author’s paper standard errors were not reported When you evaluate a confidence

Probability and inference 101