What is a Zscore?
A Zscore (or standard score) represents how many standard deviations a given measurement deviates from the mean. In other words it merely rescales, or standardizes, your data. A Zscore serves to specify the precise location of each observation within a distribution. The sign of the Zscore (+ or − ) indicates whether the score is above (+) or below ( − ) the mean.
A Zscore is calculated by subtracting the mean value from the value of the observation, and dividing by the standard deviation. Commonly a known reference population mean and standard deviation are used.
Algebraically speaking 
Z = 
Y_{i} − μ 

σ 
Where
 Z is the number of standard deviations a given measurement deviates from the mean,
 μ is the true population mean,
 σ is the true population standard deviation.

Alternatively, Zscores can be calculated using the sample mean and the sample standard deviation:
Algebraically speaking 
Z = 
Y_{i} − 

s 
Where
 Z is the number of standard deviations a given measurement deviates from the mean
 is the mean of the sample,
 s is the standard deviation of the observations in the sample

The latter formulation is most commonly used when all the measurements in a sample are transformed to Zscores to give a Zscore distribution.
Characteristics of a Zscore distribution
If your Zscore distribution is based on the sample mean and sample standard deviation, then the mean and standard deviation of the Zscore distribution will equal zero and one respectively. If your Zscore distribution is based on the population mean and population standard deviation, then the mean and the standard deviation of the Zscore distribution will only approximate to zero and one if the sample is random.
The shape of a Zscore distribution will be identical to the original distribution of the raw measurements. If the original distribution is normal, then the Zscore distribution will be normal, and you will be dealing with a standard normal distribution. You can then make assumptions about the proportion of observations below or above specific Zvalues. If however, the original distribution is skewed, then the Zscore distribution will also be skewed. In other words converting data to Zscores does not normalize the distribution of that data!
In some applications (such as weightforage in nutritional studies), the Zscores are not based upon the known population mean and standard deviation, but on an external reference population. In this situation the Zscores are used to identify those individuals in the sample falling below a specified Zscore. Sometimes the distribution of the whole sample is examined, in which case the Zscores will not have a mean of zero and a standard deviation of one  what is of interest is the extent to which their distribution differs from the reference population.
Sometimes Zscores are themselves transformed to avoid negatives values. For example, the Tscore has a mean of 50 and a standard deviation of 10. You can transform Zscores to Tscores by multiplying each Zscore by 10 and adding 50.
Algebraically speaking 
Transformed standard score 
= 
μ_{new} + Z σ_{new} 
Where
 Z is the Zscore of an observation,
 μ_{new} is the new mean of the population,
 σ_{new} is the new standard deviation of the observations in that population.

Another commonly used transformed score is the (so called) intelligence quotient (IQ) score. This has a mean of 100 and (usually) a standard deviation of 15.
Uses of Zscores
To identify the position of observation(s) in a population distribution
This is the commonest use of Zscores. Converting a measurement to a Zscore indicates how far from the mean the observation lies in units of standard deviations. If the population distribution approximates to a normal distribution, then we can also estimate the proportion of the population falling above or below a particular value.
This application has been most developed in studies of nutritional state of children. The basic data are age, sex, weight and height. The three preferred indices are:
 Weightforheight
This is an indicator of wasting and is associated with failure to gain weight or a loss of weight. In other words acute malnutrition.
 Heightforage
Low height for age is an indicator of stunting, which is usually associated with poor economic conditions or repeated exposure to food shortage caused by drought or war. In other words chronic malnutrition.
 Weightforage
This reflects the effects of either wasting or stunting, or both.
Other measures used to assess condition of (especially) children severely affected with malnutrition are head circumference, midupper arm circumference, triceps skinfold and subscapular skinfold.
Another measure which may be calculated is body mass index (BMI) which is weight (in kg) divided by height squared (in metres). This is widely used to identify adults with a weight problem, either underweight (BMI less than 18.5), overweight (BMI in range 2530) or obese (BMI above 30). The measurement of waist circumference provides information on the distribution of body fat and may be a more accurate predictor of health risk when used with BMI.
Zscores may be computed for any or all of these measures. The Zscore of a child's 'weightforheight' (for example) is computed using the child's weight together with the population mean weight and standard deviation for that height derived from a set of reference values. Until recently this was the 1977 National Centre for Health Statistics/World Health Organization (NCHS/WHO) reference which was itself derived from two data sets: one from the NCHS and the other from the Fels Research Institute. These values were intended to reflect the growth rates achieved by children when not constrained by lack of food or disease. They were used for a number of years but were recognised as not being ideal, since they took no account of ethnic diversity in growth rates  if such exist. In 2006 and 2007 the World Health Organization released two sets of child growth standards to replace the NCHS standard.
We can only infer proportions for Zscores if they approximate a normal distribution. Height for age in the reference population is approximately normal, but weight for age and weight for height are skewed to the right, because of the presence of obesity. Hence the reference population is divided into two halves at the median, and the standard deviations are calculated for each half. Although this is a bit of a fudge, it does normalize any distribution of Zvalues calculated from these values  provided it is similarly skewed.
Reporting of nutrition Zscores should include the following, broken down by sex and age and giving sample sizes: (i) mean values and standard deviations of nutritional indicators. (ii) percentage of children with indicators below different cutoff levels. The recommended Zscore cutoff point to classify malnutrition for each of the three indices is 2, which is the lowest 2.3% of the reference population. Any child falling below 3 is suffering from severe malnutrition.
The WHO has strongly advocated the use of Zscores in nutrition studies, although other methods are still used in some countries.
To standardize data for subsequent display and/or analysis.
If you wish to compare the distributions of two variables that are measured in quite different units, then converting measurements to Zscores enables them to be displayed on the same axes. There are also a number of analyses you will meet (for example major axis regression, described in Unit 12 and cluster analysis) where you convert measurements to Zscores prior to analysis.
To eliminate a factor in the analysis by expressing data relative to the mean
In field studies, it is quite common to find that the response variable you are studying is affected by the one (or more) explanatory variables in addition to the variable you are most interested in. One way to remove the effect of an explanatory variable is to standardize the data to the mean value of each level of that variable.
For example, we may be interested in the growth rate of a number of fish in relation to their position in the feeding hierarchy. Under laboratory conditions the amount of food provided can be kept constant  but under natural conditions it may vary from month to month and greatly affect the rate of fish growth. If the growth rate of each fish is standardized to the mean growth rate each month, the effect of varying conditions is eliminated.
