# Standard Distribution and Standard Error

A value (called a parameter) in a population of individuals may vary from one to the other. This variability may or may not follow a normal distribution, a mathematical term that describes a range of values where the middle value (median), the average value (mean) and the most commonly occurring value (mode) are the same, and values greater or less than this central value occur in the population with symmetrically diminishing frequency the further they are away from the central value. The standard deviations from the mean divide the normal distribution into probability segments. So all values above the mean have a combined probability of 0.5 (i.e. 50% of the distribution of values), values above 1 SD have p=0.16 (0.14+0.02) and values above 2 SD have p=0.02 (2% of values). The same applies to segments below the mean. Values between 1 SD above and 1 SD below the mean have a combined probability of 0.68, while those within 2 SD of the mean have a combined probabillity of 0.96.

When we use the term population, we do not necessarily mean all the people in the whole world, but merely the entirety of the collection of subjects in which we are interested. It could be all people with a certain medical condition, or all inanimate objects of a certain type. This term population is distinct from a sample, which is a subgroup of the population. A sample is considered representative of a population, to save tracking down and measuring every single subject in the given population.

Height is an example of a parameter that in many populations, (e.g. all adults, all members of a certain profession) has an approximately normal distribution, with a “popularity” range of heights being distributed as in the diagram; most people’s height is near the average, and extremes in height are rare. Statistically, popularity is called probability, i.e. it is more probable that a randomly selected person will be measured to have an intermediate height than a low or high extreme. The Standard Distribution (SD) is a measure of how widely spread this distribution is about the mean value (see figure above). It is calculated such that approximately 68% (34% + 34%) of values will be within one SD of the mean; in other words somewhere between the mean minus the SD and the mean plus the SD.

The actual calculation involves defining a value known as the variance. This is the sum of the squares of the differences between each of the values of the parameter and the mean value, divided by the number of values. Squaring the differences is convenient because it makes all the negative differences positive, as we are interested in the overall magnitude of differences. The SD is the square root of the variance, so it is back in the measurement unit of the original data.

Still considering height, if we take a sample of people rather than measuring everyone in the population in question, we can appreciate that their mean height will be slightly different from the true population mean just by chance because of the people who happened to be selected. If we consider a number of different samples, more of them will have an observed mean near the true mean than far from it. In fact, the probability distribution of the samples’ mean heights will itself follow a normal distribution pattern about the true mean height. The spread of this distribution that encompasses 68% of samples is called the standard error of the mean (SE).

In other words, standard deviation is the spread of values of a parameter, while standard error is the spread of the mean values of a parameter determined by taking a number of different sample groups.