# Measures of central tendency

The measures of central tendency are statistical parameters that inform about the center of the distribution of the sample or statistical population.

Sometimes we deal with a large amount of information. Variables that present a lot of data and very disparate. Data with many decimal places, of different sign or length. In these cases, it is always preferable to calculate measures that provide us with summary information about said variable. For example, measurements that tell us which is the value that is repeated the most.

Notwithstanding the above, you don't have to go so far. If we look at the following table that shows the salary received by each of the workers of a company that manufactures cardboard boxes, we will have the following:

Worker | Salary |

1 | € 1.235 |

2 | € 1.002 |

3 | € 859 |

4 | € 486 |

5 | € 1.536 |

6 | € 1.248 |

7 | € 1.621 |

8 | € 978 |

9 | € 1.125 |

10 | € 768 |

Someone might wonder, how much does the average worker in this company earn? In that case, central tendency measures could help us. Specifically, the average. However, a priori, the only thing we know is that the number will be between the minimum and the maximum.

## What are measures of central tendency good for?

Central tendency measures, obviously, pursue a series of objectives that justify their existence.

In the first place, the measures of central tendency serve to know where the average element, or typical of the group, is located. Let's imagine that we want to know which music group is the favorite of the class. For this, we can use fashion.

Likewise, the measures of central tendency serve to compare, as well as to interpret the results obtained in relation to the different observed values. Let's imagine that the average grade of students in a class is 7, while there are students who are 3.

Also, measures of central tendency serve to compare and interpret the value of the same variable on different occasions. Let's imagine that the mean value of a variable is not representative, so we can complement with the median value to extract a true image.

Finally, these types of measures are used to compare the results with other groups, taking into account these same measures of central tendency. Let's imagine that we want to compare the average grade between the different classes of a school. The average allows us to compare them and to know which class gets better grades.

## Measures of central tendency

Next, let's see the main measures of central tendency, as well as the different formulas that allow us to calculate these measures in any case.

These measurements are the mean, the mode, and the median.

### Half

The mean is the average value of a set of numeric data, calculated as the sum of the set of values divided by the total number of values. Below is the formula for the arithmetic mean:

Arithmetic mean formula Consult explanation and example of the meanAs explained in the article linked above, there are many types of media. The choice of each type of average has to do, mainly with the type of data on which it is calculated.

### Median

The median is a central position statistic that splits the distribution in two, that is, it leaves the same number of values on one side as on the other. The proposed formulas will not give us the value of the median, what they will give us will be the position in which it is within the data set. The formulas that indicate the position of the median in the series are the following:

- When the number of observations is even:

Median = (n + 1) / 2 → Mean of the observed positions

- When the number of observations is odd:

Median = (n + 1) / 2 → Observation value

Consult explanation and example of the median

### fashion

The mode is the value that occurs the most in a statistical sample or population. It has no formula in itself. What should be done is the sum of the repetitions of each value. For example, what is the mode of the following table of wages?

Worker | Salary |

1 | € 1.236 |

2 | € 1.236 |

3 | € 859 |

4 | € 486 |

5 | € 1.536 |

6 | € 1.536 |

7 | € 1.621 |

8 | € 978 |

9 | € 1.236 |

10 | € 768 |

The mode would be € 1,236. If we look at the wages of the 10 workers, we would see that € 1,236 is repeated three times.

## Criticism of measures of central tendency

The measures of central position are helpful in summary form but are not categorical. As a summary, they can give us information on what, on average, one would expect. But they are not always accurate.

To better analyze these measures, it is advisable to combine measures of central tendency with measures of dispersion. Dispersion measures are not infallible either, but they offer us information about the variability of a certain variable. Thus, suppose following the example of wages, that there are two companies A and B. In company A the average salary is $ 3,100, while company B is also $ 3,100. This could lead us to make the mistake that wages are the same or very similar. But this is not necessarily the case.

It can happen that company A has a standard deviation of $ 400, while company B has a standard deviation of $ 1,000. This indicates that there is greater inequality, for whatever reason, in the wages of company B than in those of company A.

## Examples of measures of central tendency

To finish, let's look at some examples of the different measures of central tendency discussed previously:

Average example: Let's imagine that we have obtained 4 different grades in 4 exams, with our final grade being the average grade obtained. Let's imagine that these ratings have been 7, 6, 8, and 5.

To find the average grade, we will add the grades and divide the result by the number of values we have.

(7+6+8+5) / 4 = 6,5.

A process that would culminate with an average rating of 6.5.

Example of median: Let's imagine that we have thrown a data 10 times and we have obtained the following results (ordered from least to greatest): 1, 2, 2, 3, 4, 5, 5, 5, 6, 6.

Performing the calculation, applying the formula, we obtain the following: Median = 10 + 1/2 = 5.5.

Next, we calculate the average of the values that occupy position 5 and 6, that is, 4 and 5:

5 + 4 / 2= 4,5.

In this case, the median would be 4.5.

Fashion example: Let's imagine that we have rolled a die among a group of 8 friends, and we want to know the fashion.

The results in the throws have been (ordered from lowest to highest): 2, 3, 3, 3, 3, 4, 5, 5.

Thus, since the mode does not have a formula, but is the observed value that is repeated the most, the mode in the following distribution is 3. Since 3 is the observed value that is repeated the most times (x4).

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