Position Measurements

Descriptive statistics are numbers that summarize and describe all information. In this case, the data sets or simply data. But. This information will depend on what we evaluate, such as population or sample, as stated in the previous post.

Getting Started

The central theme of this post is just “describing” the data. Therefore, they do not represent more specific conclusions of the sample in relation to the population.

The technique used to extend conclusions from the sample to the population is statistical inference, a topic that does not make sense to children. An undergraduate statistics course is the most appropriate place.

Position measurement

The values of the described measure can be expressed by statistics. Thus, instead of calling it a descriptive position measure value, we will simply call it descriptive position statistics.

Definition (Position measure): are statistics that represent a series of data and guide us as to the position of the data distribution.

The most important position measurements are:

arithmetic mean
median and
fashion.

Average

The first position measurement described is the average, widely used in various areas of knowledge. For example, elementary school teachers use this position measure to grade students.

There are two types of average:

Population average: is the calculation of the position value for all population information.
Sample mean: is the calculation of the position value for all information in the sample.

When we model the population by a probability distribution. The value of the population mean is called a parameter.

Definition (Parameter): descriptive measure referring to a population.

Example: population average.

Definition (Estimator): descriptive measure referring to a sample.

Example: sample mean.

Definition (Estimate): value of the descriptive measure referring to a sample.

Example: sample mean of 10 measurements is equal to 5. Therefore, the estimate of the sample mean is 5.

Population average

The population mean is calculated by adding up all the population values and dividing the result by the total number of elements in the population. In a population of N elements, the population mean is given by:

$$\mu=\dfrac{x_1+\ldots+x_N}{N}$$

where:

N: population size or number of population elements;
$\mu$: parameter referring to the population average;
$x_1, \dots, x_N$: population elements.

Note: the parameters are generally given by Greek letters, in the case of the average it is given by the Greek letter $\mu$.

Example (Population average)

Suppose, without loss of generality, that the mass measurement of the Brazilian population is 80 kg.

$$\mu=\dfrac{40+ 120+ 86 + 31+ \ldots + 94}{220 \text{ milhões} }= 80$$

Sample mean

The sample mean, arithmetic, or simply mean, is obtained when we add the values of the sample observations and divide the result by the number of sample values. Thus, the sample mean is given by:

$$\overline{X}=\dfrac{x_1+\ldots+x_n}{n}$$

where:

n: sample size or number of sample elements;
$\overline{X}$: estimator referring to the sample mean;
$x_1, \dots, x_n$: sample elements.

Example (Sample Average) for children’s height

A sample of 5 children whose height was measured.

Child	Height
1	1.48
2	1.38
3	1.35
4	1.60
5	1.51

In this way, the average is given by

$$\overline{X}=\dfrac{x_1+x_2+x_3+x_4+x_5}{5}=\dfrac{1.48 + 1.38 + 1.35 + 1.60 + 1.51}{5} = 1.464$$

Therefore, the estimated average of 5 children is 1.464 meters.

Fashion

As the statistician does not have much creativity, it is the number that is fashionable or appears most often. More specifically, we define it as:

Definition (Mode): is the value that presents the highest frequency.

Example (Mode) for number of toys

Let the number of toys that a group of 8 children have is given by the following table:

Child	Number of toys
1	1
2	10
3	12
4	7
5	6
6	10
7	10
8	3

The mode is given by the number 10, as it appears 3 times.

Median

To calculate the median we must follow the algorithm:

Step 1: order the data from smallest to largest value;
Step 2: check the number of observations:
- if the number of observations is odd, the median will be the central observation;
- if the number of observations is even, the median will be the arithmetic mean of the two central observations.

Notation: $\widetilde{X}$

Example (Median) for math students

In an undergraduate mathematics course there are the number of students graduated per year. The observed values are presented in the following table.

Year	Number of Students
2020	13
2021	10
2022	12

The first step is to sort the data

Year	Number of Graduated Students
2021	10
2022	12
2020	13

As the number of observations is odd, the median is given by the “middle” value, which is 12. Therefore we have to:

$$\text{Mediana} = \widetilde{X} = 12$$

Example (Median) for students vaccinated for Covid-19

In a school they have the number vaccinated for COVID-19. The observed values are presented in the following table.

Class	Number of Vaccinated Students
A	15
B	7
C	3
D	9