Interpreting blood test results

When a doctor suspects that a patient has diabetes, they recommend having the patient's blood glucose level measured—that is, the concentration of sugar in the blood. Statistical studies show that a healthy person's normal blood glucose level lies between 0.7 and 1.1 grams per liter (g/L), defining what is known as the corresponding reference interval, here [0.7, 1.1]. If the result falls outside this interval, it may be considered "pathological," depending on the patient's profile. To construct this interval for a biological parameter X, such as blood glucose level, the parameter is measured in a sample of subjects presumed to be healthy, under the assumption that the variable follows a normal distribution with mean μ and variance σ² (see box).

We then check whether the observed value X = x₀ lies in the reference interval, which contains 95% of the values in the population in question. For a normal distribution, this interval is known to be bounded by μ – 1.96σ and μ + 1.96σ. In practice, the population parameters μ and σ are generally unknown. They are therefore estimated using the observed mean m and observed standard deviation s of a sample of people presumed to be healthy (with a sample size considered "sufficiently large"). Thus, the reference interval actually used in practice is [m – 1.96s, m + 1.96s].

The Z-score ----------

Mean, standard deviation and the normal distribution ----------------------------------

Consider a numerical variable measured in a sample of n individuals drawn from a population assumed to be homogeneous (see the article "Statistics and probability revisited"). All the measurements can be viewed as a point X in an n-dimensional space, with coordinates X = (x₁, x₂ … *x_n). By definition, the observed mean m of the sample is given by m = (x*₁ + x₂ + … + *x_n) / n*.

The mean of the squared deviations from this mean gives the observed variance, a standard measure of dispersion. Thus, s² = [(x₁ – m)² + (x₂ – m)² +… + (*x_n – m*)²] / n.

The observed standard deviation is defined as the square root of the observed variance. All these concepts are formalized in probability theory.

In probability theory, a real-valued random variable X is said to follow a normal distribution with expected value μ and standard deviation σ > 0 if X has the probability density function (probability per unit of measurement) p, defined for every real number x by:

p(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp ( -\frac{1}{2}(\frac{x-\mu}{\sigma})^2).

In this case, the probability that X lies in an interval [a, b] is given by the definite integral of the function p from a to b. Such a random variable is also called a Gaussian random variable.

Another way to measure how far an observation lies from a statistic computed from the data, or from a theoretical model (a probability distribution), is to use the observation's Z-score. Suppose we have a dataset with observed mean m and observed standard deviation s, or, in the theoretical case, a probability distribution with mean m and standard deviation s. Let x₀ be a particular observation. We can standardize this observation by expressing it as a "deviation from the mean in standard-deviation units": this measures the signed distance d(X) between this observation and the others in the empirical case, or between this observation and a model in the theoretical case. This distance is the quantity z₀ = (x₀ – m) / s when comparing the observation with a statistic, or z₀ = (x₀ – m) / s when comparing it with a theoretical probability distribution (the notation z is conventionally associated with standardized variables). This quantity is called the Z-score.

Interpreting blood test results

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