Measures of dispersion in statistics

Dispersion measures are important because they tell us about the variability that we find in a certain sample or population. When we talk about a sample, this dispersion is important because it determines the error that we will have when making inferences for measures of central tendency, such as the mean.

In a data distribution, dispersion measures play a very important role; given that complement those of central position, characterizing the variability of the data.

So, Measures of central tendency indicate values around which data appear to cluster. They are recommended to infer the behavior of variables in populations and samples. Some examples of them are the arithmetic mean, the mode or the median (1).

Measures of dispersion complement these measures of central tendency. Besides, are essential in a data distribution. This is because they characterize the variability of the data. Its relevance in statistical training has been pointed out by Wild and Pfannkuch (1999).

In these measures, the perception of data variability is one of the basic components in statistical thinking. Well It gives us information about the dispersion of the data with respect to an average or mean.

The arithmetic mean is widely used in practice, but it can often be misinterpreted. This will happen when the values of the variable are very dispersed. On these occasions, it is when it is necessary to monitor the average of the dispersion measures (2).

What are dispersion measures?

In general terms, dispersion measures They are numerical values that indicate the level of variability of a variable. In other words, they are those values that reflect the degree of separation between the values of a statistical distribution, with respect to the measures of central tendency considered.

Likewise, dispersion measures are expressed in non-negative real numbers, and the value will be 0 when all the data in the distribution are equal. Therefore, the greater their dispersion, the greater the numerical value of the dispersion.

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What are dispersion measures used for?

In a statistical study, when generalizing the data from a sample, dispersion measures are very important since they directly determine the error with which we work. So, The more dispersion we collect in a sample, the larger size we will need to work with the same error.

On the other hand, these measures help us determine if our data is too far from the central value. With this, they give us information about whether this central value is adequate to represent the study population. This is very useful for comparing distributions and understanding risks in decision making (1).

These measures are very useful for comparing distributions and understanding risks in decision making. The greater the dispersion, the less representative the central value is.. These are the most used:

Range or range. The mean deviation. Variance. The typical or standard deviation. The coefficient of variation.

Functions of each of the dispersion measures

Range

First of all, The range is recommended for primary comparison. Thus, consider only the two extreme observations. That is why it is recommended only for small samples (1). It is defined as the difference between the last value of the variable and the first (3).

This measurement is easily calculated. However, its disadvantage is that it does not really express the concentration of the data; There are cases in which exaggerated intervals are obtained when in reality the series has a great concentration, but its extreme values differ greatly from the rest of the values of the series.

Statistical deviation

For its part, The mean deviation indicates where the data would be concentrated if they were all the same distance from the arithmetic mean (1). We consider the deviation of a value of the variable as the difference in absolute value between that value of the variable and the arithmetic mean of the series. Thus, it is considered as the arithmetic mean of the deviations (3).

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Variance

Variance is an algebraic function of all values, appropriate for inferential statistics tasks (1). It can be defined as the squared deviations (3).

Some observations on variance

The variance, like the mean, It is an index very sensitive to extreme scoresIn cases where the mean cannot be found, it will not be possible to find the variance either. The variance is not expressed in the same units as the data, since the deviations are squared.

Standard or typical deviation

For samples drawn from the same population, the standard deviation is the most used (1). It is the square root of the variance (3).

It is the measure of dispersion that best provides the variation of the data with respect to the arithmetic mean.. Its value is directly related to the dispersion of the data, the greater the dispersion of the data, the greater the standard deviation, and the less dispersion the less the standard deviation.

Observations on standard deviation

The standard deviation, like the mean and variance, It is an index very sensitive to extreme scoresIn cases where the mean cannot be found, it will not be possible to find the standard deviation either. The smaller the standard deviation, the greater the concentration of data around the mean.

Coefficient of variation

This is a measure used mainly to compare the variation between two sets of data measured in different units. For example, height and body weight of the students in a sample. Thus, it is used to determine in which distribution the data are most grouped and the mean is most representative (1).

The coefficient of variation is a more representative measure of dispersion than the previous ones, because it is an abstract number. That is, it is independent of the units in which the values of the variable appear. In general, this coefficient of variation is usually expressed as a percentage (3).

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Causes of data dispersion in a sample

The variability of the data will depend on the phenomenon being studied and the measurement instruments used..

For example, in the physical scienceswhere the phenomena are usually stable, the dispersion may be due to random measurement errors. In these cases, instrument measurements are usually not completely precise, that is, reproducible. Additionally, there is additional variability between raters in interpreting and reporting measured results.

For its part, in the biological and social sciences, what is measured is rarely immutable and stable. Hence the observed variation may also be intrinsic to the phenomenon. In this case, the variation is usually due to:

Interindividual variability: when different members of a population differ from each other. Or it can be associated with intra-individual variability: the same subject differs in the tests taken, either at different times or under different conditions.

Final thoughts

Thus, these dispersion measures will indicate, on the one hand, the degree of variability in the sample. On the other hand, they will indicate the representativeness of the central value, since If a small value is obtained, it will mean that the values are concentrated around that center.

This would mean that there is little variability in the data and the center represents everyone well. On the other hand, if a large value is obtained, it will mean that the values are not concentrated, but dispersed. Therefore, there will be a lot of variability and the center will not be very representative.

On the other hand, when making inferences we will need a larger sample size if we want to reduce the error, increased precisely by the increase in variability.