Standard scores and normal distributions

Introduction

In this chapter we will discuss standard distributions. Standard scores represent the position of a score or measurement in relation to an overall set of scores. Standard distributions are also useful for comparing scores from different sets of measurements. Standard scores are used in both clinical practice and research in the health sciences. In clinical practice the score of a patient is often compared with a known distribution to interpret the score. Measurements such as blood pressure or cholesterol levels are compared with a distribution to interpret the patient’s result.

The aims of this chapter are to:

1. Define ‘standard’ scores.

2. Describe the characteristics of normal and standard normal curves.

3. Show how standard normal curves can be used for calculating percentile ranks.

4. Show how standard normal curves can be used to compare scores from different distributions.

Standard scores (z scores)

Consider this example: infant A walked unaided at the age of 40 weeks, while infant B is 65 weeks old but still cannot walk. What sense can we make of these measurements? Could infant B need further clinical investigation in case he has some neurological abnormality? The fact that infant B is unable to walk at the age of 65 weeks is not very informative in the absence of additional information about how this compares with norms for other children. However, say that it is known that the distribution of walking ages is such that µ = 50 weeks and σ = 5. Assuming that the frequency distribution is normal, the frequency polygon representing the population would look something like that shown in Figure 17.1.

Figure 17.1 Age at which children walk unaided.

In this instance, infant B’s score is clearly above the mean. In fact, by inspection, we can see the infant’s score at this point of time was three standard deviations above (+3) the mean (65 = 50 + (3 × 5)). In contrast, infant A began walking earlier than the mean, his score of 40 being two standard deviations below (−2) the mean. In general, any ‘raw’ score in a frequency distribution can be described in terms of its distance from the mean. The process of transforming a score into a measurement based on its distance from the mean in standard deviations is called standardizing the score. Such ‘transformed’ scores are called z scores or standard scores.

A z score represents how many standard deviations a given raw score is above or below the mean. The equation for transforming specific raw scores into z scores is given as:

For the above equation, x is the raw score, or µ is the mean of the distribution from which the score was drawn and s or σ is the standard deviation of the distribution. That is, when we know the mean and standard deviation of a distribution, we can transform any raw score into a z score. Conversely, when the z score is known, we can use the above equations to calculate the corresponding raw scores.

In the above example, the z scores corresponding to the infants’ raw scores are:

These calculations support our previous observations that A’s score was two standard deviations below the mean and B’s score was three standard deviations above the mean. In other words, A walked very early and B was a very late starter. The particular value of standardizing scores for understanding clinical or research evidence will be discussed in the context of the concepts of normal and standard normal distributions.