Statistical Analysis for Education and Psychology Researchers

Table 1, Appendix A4. It has a mean μ of 0 and a σ standard deviation and variance of 1.

In notation this is N (0, 1) where N refers to Normal and 0=μ and 1=σ. This may also be

written as N(μ, σ^2 ).

Example 4.7: Use of the Standard Normal Distribution to Describe

Probabilities of Events

The following is the distribution of pulse rate for males in the author’s statistics classes
see Figure 4.3. The average pulse rate is 72 and the standard deviation is 12. For this
example, consider these values to represent the population of all male students at the
University of Manchester. We can use the sample mean and standard deviation to
estimate the corresponding population parameters.

We want to know the probability of observing a male with a pulse rate of

96 or more, and the probability of observing a male student’s pulse rate of

105 or more?

The standard normal distribution can be used to answer questions like those above. We
would proceed by first transforming the distribution of pulse rate scores to a standard
normal distribution, so we can use the normal deviate or Z tables. We want to be able to
say that a score xi from N(72, 144) is comparable to a score Zi from N(0, 1).

Figure 4.3: Distribution of male pulse rates

We can use a linear transformation, that is subtract a constant from each score and
divide by a constant. Look at the second row of scores (xi−μ) in Figure 4.3. This
distribution has a mean of zero. This distribution is transformed into a Z distribution by
dividing (xi−μ) by the σ of the original distribution.

Equation

for Z score

or

Probability and inference 105