Statistical Analysis for Education and Psychology Researchers

a finite population, the sampling distribution is described by a theoretical probability
model.
3 The sampling distribution of a statistic is the link between probability and inference.
4 The sampling distribution of a statistic, just like a sample distribution of observations,
has a mean and variance. The standard deviation of the distribution of a statistic is
known as the standard error.
5 Sample statistics can be used to estimate corresponding population parameters. An
index of the precision of the estimation is given by the standard error of the statistic.
Standard errors are also used in calculating test statistics.
6 The 95 per cent confidence interval captures the true population parameter for 95
simple random samples out of a hundred (assuming the same n for each sample).
7 You cannot be certain of estimating a population parameter, the 95 per cent CI is not
100 per cent. There is a 5 per cent chance that the parameter of interest is not captured
by the CI0.95.

4.6 Continuous Random Variables

A continuous random variable denoted as X, can take any value in a range depending
upon the sensitivity of the measuring instrument. A continuous probability distribution
represents the distribution of probabilities assigned to an interval of a continuous random
variable. The idea of a discrete probability distribution where the distribution can be
represented by a bar chart, with each bar of unit width, and the height of bars representing
the probability of outcomes for the discrete random variable, can be extended to a
situation where the random variable is continuous. If you could imagine having a large
number of bars with the mid-point of the top of each bar joined, then as the number of
bars increases towards infinity, the line joining the tops of the bars would approximate
closer and closer to a smooth continuous curve. As we near infinity the curve becomes a
normal curve that is the normal curve becomes the limit of the binomial distribution.
The normal curve is a continuous probability distribution. Other examples of continuous
probability distributions include; F, t and χ^2 distributions.
With a continuous probability distribution probabilities are assigned not to discrete
outcomes but to an area under the density curve. This area equals the interval between
two values of the continuous random variable.

Normal Distributions

Many random continuous variables in the social sciences have an approximate normal
distribution in the population. For example, if distributions of measurements of a person’s
height, weight and reaction times are plotted, they would approximate to a normal curve.
Some psychological and achievement measures are specifically designed to have a
normal distribution, for example, IQ and standardized achievement tests.
We can think of a normal distribution as either a model describing a probability
distribution for a defined population or an empirically determined distribution. The
normal curve, more correctly termed a normal probability distribution, is important not
only because many variables of interest are assumed to be normally distributed in the

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