## Calculation of z-scores

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The second way to express the distance between an individual child's weight and the average weight of comparable children in the reference population is by z-score. This is a bit more complex but has certain advantages over percent of median and is widely used to present survey results. Some background is necessary before discussing anthropometric z-scores.

A z-score, or standard deviation, is a measure of the dispersion of data. Some data are quite dispersed, as shown on the left graph, where the values for individuals may be quite different from each other. Other data are not very dispersed, as shown on the right graph, meaning that the values for individuals are quite similar to each other. If, for example, the measurement plotted on the graphs below is weight, the population in the left graph has a wide range of weights. There are some very light people and some very heavy people. The people in the population on the right are much more similar to each other; their weights do not show such a wide range.

The peak of the curve is the average of all the measurements. The standard deviation is calculated from the actual collection of measurements. For example, you have probably seen measurements given as something like16.3 (SD = 2.3). The number after the "SD" is the standard deviation. If we add one standard deviation to the average and subtract one standard deviation from the average, 67% of all the measurements lie between these two numbers, as shown on the graph below. The area between the mean plus two standard deviations and the mean minus two standard deviations represents 95% of the measurements.

For example, is the average weight for a group of children is 10.0 kg and the standard deviation is 1.0 kg, then 67% of children in this group have a weight between 9.0 kg and 11.0 kg (the average plus and minus the standard deviation). 95% of the children in the group have a weight between 8.0 kg and 12.0 kg (the average plus and minus two standard deviations).

The graph below shows the distribution of weights for all 80 cm girls in the reference population, as shown before. It also shows where on this graph our individual 80 cm girl is located. At the bottom of the graph is shown the average minus two and the average minus three standard deviations, calculated from the average weight and standard deviation of all girls who are 80 cm tall, as given in the anthropometry tables. We can see that our girl is slightly above the average minus 3 standard deviations and below the average minus 2 standard deviations. Standard deviations can also be called z-score, hence the terminology of -2.0 z-scores (two standard deviations below the average) and -3.0 z-scores (three standard deviations below the average).

We can calculate the girl's exact z-score by using the formula:

We get the average and standard deviation of the reference population from the anthropometry tables. This table also shows the value for the average minus two and minus three standard deviations. This allows us to quickly tell where our individual child is relative to these two points without having to calculate an exact z-score. The table below is more like a real anthropometry table because it has the standard deviation, the average minus two standard deviations, and the average minus three standard deviations for this group of children in the WHO standard population.

Absence of acute protein-energy malnutrition, or normal nutritional status, is defined as having a weight-for-height z-score of -2.0 or greater. Moderate acute protein-energy malnutrition is defined as having a weight-for-height z-score of -3.0 to less than -2.0. Severe acute protein-energy malnutrition is defined as having a weight-for-height z-score less than -3.0.

Our girl therefore has moderate protein-energy malnutrition, as defined by weight-for-height z-score.

But we have to remember oedema. As with percent of median, if the child has pitting oedema, the weight-for-height z-score is not useful in determining nutritional status and such children are automatically defined as having severe acute protein-energy malnutrition regardless of his or her z-score.

But in fact, you probably will rarely have to calculate z-scores by hand. Several computer programs will do this for you if you include in the dataset each child's height, weight, sex, and the presence or absence of oedema. To calculate height-for-age and weight-for-age, you will also have to have either age or date of birth and date of interview, from which age can be calculated.

However, you should be careful using such computer programs. They often have options and settings which you need to be familiar with. For example, most programs will exclude from analysis those z-scores which fall beyond a certain range in order to not include z-scores calculated from obviously wrong anthropometric measurements.

Below is a list of such programs. If you are connected to the internet, click the name of the program to go to the website from which you can download the program without cost.

EpiInfo (cannot calculate indices using the WHO standard population)

NutriSurvey (cannot calculate indices using the WHO standard population)

SPSS, SAS, S-Plus, and Stata(The link does not provide the programs themselves. They are all commercial programs which are expensive. The website provides code to use in these programs to calculate z-scores using either NCHS:CDC:WHO reference and WHO standard populations.)