The measures of central tendency are statistical tools, whose calculations generate numerical results related to the centralized location of the data that are studied. For their part, measures of central tendency serve to summarize and interpret a set of data or study variables, with the aim of analyzing their behavior and making generalizations about an entire population.

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## Role of measures of central tendency

The measures of central tendency serve to summarize and order the information collected, thus allowing a better, quick and easy understanding of the data being studied; All of this makes it possible to know the sample or population studied, in addition, the measures of central tendency serve as support for the calculation of other statistical parameters.

Among the benefits to be gained from measures of central tendency are the following:

### Improve performance

The measures of central tendency allow to interpret a set of data or variables, in order to obtain certain information about the phenomenon under study. In this sense, a common example of the use of measures of central tendency is the interpretation of academic grades or grades.

With the application of measures of central tendency, it is possible to analyze, decipher and understand the meaning and scope of these data for a specific situation or event; in this particular case, they serve to clarify how a student’s academic performance has been and where the trend is heading, whether positive or negative.

### Summary

The measures of central tendency, in addition to allowing the interpretation of a series of data, also serve to summarize, simplify or synthesize all the data collected from the sample or population studied; in order to obtain knowledge about it in less time and incurring fewer costs.

### Estimate

Another benefit provided by the use of central tendency measures is that through these it is possible to make estimates about the behavior of all the data and variables studied, in this way, it is possible to know how the descriptive statistical elements are distributed.

### Location

Finally, the measures of central tendency allow to analyze, study and know the location of the data in studies; In this way, with the calculation of these measurements it is possible to make graphs, through which the location and distribution of the data in a certain space is observed, as well as the frequency with which they appear; In short, it is used to determine the past of the data and the future of the same, in a certain timeline.

## Formulas for measures of central tendency

There are three measures of central tendency: mean, median and mode, each one of them shows how the data are grouped in the central zone and what their behavior is. In this sense, these measures can be used to study grouped or non-grouped data; To carry out these studies, some formulas must be applied, which are described below:

### Arithmetic average

It is a measure of central tendency used only for quantitative variables (which can be measured, counted or quantified), and consists of the sum of all values, divided by the total number of values that are under study. The formula to calculate the arithmetic mean is

### Weighted average

The mean is also known as the average, and as regards the weighted mean, it is the one in which the data studied have a different degree of importance, for which, each of them is assigned a weight relative to the importance they have in the study. To calculate it, the following formula should be used:

### Median (Me)

The median is a numerical value that divides the data series in half, having the same number of data higher and lower than this; To calculate this measure of central tendency, the number of data being studied must be taken into account, in order to know if the numerical series is of an even or odd type.

In both cases, the data must be ordered from smallest to largest, or from largest to smallest; Once this operation has been carried out, the median is determined, in the case of odd series, the median will be the central value of the series. On the other hand, in the case of even series, the median will be the arithmetic mean of the two central values.

### Moda (Mo)

The mode is a measure of central tendency, which refers to the data or value that occurs most frequently in the series of data analyzed; It is price to mention that a series can contain several fashions, in this sense, unimodal series are known as those that have a single mode and bimodal those that have two modes.

## Measures of Central Tendency for Pooled Data

Grouped data are called a series of study variables, whose size is greater than 20, therefore, to be analyzed, they must be classified, grouped and ordered according to the characteristics they share. In this sense, the measures of central tendency for grouped data are those mentioned above (mean, median and mode), however, their calculation formula is different.

### Media

Like the arithmetic mean for non-grouped data, this represents the average of the data studied; In this case, it multiplies the class marks and the absolute frequency of the data, later, it is divided by the total number of data that are the object of study. The formula for the mean for grouped data is:

### Median

The median divides the grouped data set in half, resulting in two subsets with the same amount of data, of which one is less than the mean and the other is greater than this same measure of central tendency. To calculate the median of the grouped data, the following formula is used:

### Fashion

Finally, the mode represents the value or values that occur the most in the grouped data series; It should be noted that, as in the mode for non-grouped data, a study series may have more than one mode or data trend. To calculate this statistical parameter, the formula is used:

## Measures of dispersion

The measures of dispersion or variability are a type of statistical parameters and indicators, which serve to know and analyze the distribution of the data around the arithmetic mean; Through them, it is determined how far or close the data is to the arithmetic mean. The main measures of dispersion are range, variance, and standard deviation.

### Range (R)

The range denoted by the letter “R” is a dispersion measure that evaluates the distance or path that exists between the limit values of the data series under study, that is, it observes the distance between the minimum value and the maximum value. . This measure is calculated with the difference between the maximum value and the minimum value.

### Standard deviation

The standard deviation of a population or a sample is known as the statistical parameter that represents and analyzes the dispersion of the data under study around the arithmetic mean. In this sense, the higher the standard deviation, the more dispersed the data in relation to the mean; its calculation formula is:

- Sample standard deviation (samples: small representative group of the population):
- Population standard deviation (population: total set of study elements):

### Variance

For its part, variance is a measure of dispersion that expresses the variability between the data series, in relation to its arithmetic mean. It is also called variance, the mathematical expression of the standard deviation squared, to calculate it the following formula is used:

## Conclusion: Are these measures of central tendency really useful?

The measures of central tendency are statistical parameters that serve to study the behavior of the variables, in other words, by calculating these statistics it is possible to know the location of the data under study and reflect them in graphs of different types. Likewise, the measures of central tendency allow summarizing the information provided by the collected data, with the aim of facilitating and improving their interpretation and understanding.

Dr. Samantha Robson ( CRN: 0510146-5) is a nutritionist and website content reviewer related to her area of expertise. With a postgraduate degree in Nutrition from The University of Arizona, she is a specialist in Sports Nutrition from Oxford University and is also a member of the International Society of Sports Nutrition.