Dispersion measures are fundamental, because unlike mathematics, statistics work with random variables. For example, the big media worked with deterministic values, such as candidate A has 45% and B with 30%. For mathematics, A is clearly ahead, but for statistics, I don’t know. Today they present it in a statistical way, as they talk about the “margin of error”. If the “margin of error” is 15%, the two are technically tied.

Example 1.

Dispersion measure

As motivation, consider two one-piece production lines. The average piece length measurement is 100cm and both lines are producing pieces with averages close to this value. In this case, do we consider that the parts produced by both lines are suitable?

Example 1.

The less dispersion between production lines, the more accurate (in terms of position) and more reliable the production line. More specifically, we have a better chance of hitting the target.

Example 1.

Definition (Measure of dispersion): are statistics that represent a series of data and guide us regarding the dispersion (variation, variability) of the data distribution.

The most important dispersion measures are:

  • amplitude;
  • variance;
  • standard deviation;
  • coefficient of variation.

Amplitude

Amplitude is the most basic dispersion measure and is applied in most applications because it is simple to calculate. In everyday life, audio platforms use amplitude as a concept a lot.

Example 1.

Definition (Amplitude): is the difference between the largest and smallest value in the data set.

$$\\{\text{menor valor}, \dots, \text{maior valor}\\}$$$$X_{(1)} \leq X_{(2)} \leq X_{(3)} \leq \dots \leq X_{(n−1)} \leq X_{(n)}$$

The amplitude R of the data is given by:

$$R = X_{(n)}−X_{(1)}$$

Note: Below are some important notations:

  • When an index is in parentheses, it means that the numbers are ordered.
    • $X_{(1)}$ is the smallest value
    • $X_{1}$ is not necessarily the smallest value
  • From the concept of inequalities we have that $X_{(1)} \leq X_{(2)}$:
    • $X_{(1)}$ is less than or equal to $X_{(2)}$
    • $X_{(1)} \leq X_{(2)} \leq X_{(3)} \leq \dots \leq X_{(n−1)} \leq X_{(n)}$ are ordered from smallest to smallest value.
  • The amplitude is given by the symbol $R$, which comes from the English Range.

Example (Amplitude) for frog jump

Be the height of the frog’s jump and the measurements are given below:

Example 1.
Jump Height (cm)
1 0
2 3
3 10
4 3
5 0

Therefore, the maximum and minimum heights are given below:

  • $\max \\{\text{alturas}\\} = 10\text{ cm}$
  • $\min \\{\text{alturas}\\} = 0\text{ cm}$
  • $\text{Amplitude} = R = \max \\{\text{alturas}\\} - \min \\{\text{alturas}\\} = 10 - 0 = 10\text{ cm}$

Variance

The next measure of dispersion described is variance.

There are two types of variance:

  • Population variance: is the calculation of the dispersion value for all population information.
  • Sampling variance: is the calculation of the dispersion value for all information in the sample.
Example 1.

When we model the population by a probability distribution. The value of the population variance 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 variance of 10 measurements is equal to 2. Therefore, the estimate of the sampling variance is 2.

Population variance

Population variance is a little more complicated than calculating other descriptive measures. It is calculated based on the deviations from the average.

$$d_{i} = \text{desvio} = (\text{medida} - \text{média}) = (X_{i} - \overline{X})$$

An important property is that the sum of the deviations is equal to zero. Suppose we have 5 measurements.

$$d_1 + d_2 + d_3 + d_4 + d_5 = 0$$

In a population of N elements, the population variance is given by:

$$ \begin{aligned} \sigma^2 &= \dfrac{d^2_1+\ldots+d^2_N}{N} \\ \sigma^2 &= \sum^N_{i=1}\dfrac{(X_i-\mu)^2}{N} \\ \end{aligned} $$

where: ​

  • N: population size or number of population elements;
  • $\mu$: parameter referring to the population average;
  • $x_1, \dots, x_N$: population elements.
  • $d_i = (X_i - \mu)$: is the deviation of measurement i, for $i=1,\dots, N$.
  • $\sum^N_{i=1}$: is a sum symbol, in this case the sum goes from $1$ to $N$.
Example 1.

Note: the parameters are generally given by Greek letters, in the case of variance it is given by the Greek letter $\sigma^2$.

Sample variance

In a sample of n elements, the sampling variance is given by:

$$ \begin{aligned} s^2 &= \dfrac{d^2_1+\ldots+d^2_n}{n} \\ s^2 &= \sum^n_{i=1}\dfrac{(X_i-\overline{X})^2}{n-1} \\ \end{aligned} $$

where: ​

  • n: sample size or number of sample elements;
  • $s^2$: estimator referring to sample variance;
  • $x_1, \dots, x_n$: sample elements.
  • $d_i = (X_i - \overline{X})$: is the deviation of measurement i, for $i=1,\dots, n$.
  • $\sum^n_{i=1}$: is a sum symbol, in this case the sum goes from $1$ to $n$. ​

Note: the variance estimator is given by the letter $s^2$, as it comes from the English standard deviation.

Example (Variance) for frog jump

Be the height of the frog’s jump and the measurements are given below:

Example 1.
Jump Height (cm)
1 10
2 11
3 9
4 8
5 10

In this way, we have the following calculations:

  • $\text{média} =\dfrac{10+11+9+8+10}{5} = \dfrac{48}{5}=9.6 \text{ cm}$
  • $\text{soma }\text{desvios}^2 = (10-9.6)^2 + (11-9.6)^2 + (9-9.6)^2 + (8-9.6)^2 + (10-9.6)^2 = 5.2$
  • $\text{Variância} = s^2 = \dfrac{5.2}{5-1} = 1.3 \text{ cm}^2$

Population standard deviation

The population standard deviation is given by the square root of the variance. So, what is the difference between the two?

It is merely a practical application, because if we measure a part in units of $cm$ the variance will be in $cm^2$ and the standard deviation will be in $cm$, which is what matters in practical terms. The statistician prefers variance, for mathematical simplification.

In a population of N elements, the population standard deviation is given by:

$$ \begin{aligned} \sigma &= \sqrt{ \text{variância} } \\ \sigma &= \sqrt{ \sum^N_{i=1}\dfrac{(X_i-\mu)^2}{N} } \\ \end{aligned} $$

where: ​

  • N: population size or number of population elements;
  • $\mu$: parameter referring to the population average;
  • $x_1, \dots, x_N$: population elements.
  • $d_i = (X_i - \mu)$: is the deviation of measurement i, for $i=1,\dots, N$.
  • $\sum^N_{i=1}$: is a sum symbol, in this case the sum goes from $1$ to $N$.
Example 1.

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

Sample standard deviation

In a sample of n elements, the sample standard deviation is given by:

$$ \begin{aligned} s &= \sqrt{\text{variância} } \\ s &= \sqrt{\sum^n_{i=1}\dfrac{(X_i-\overline{X})^2}{n-1} }\\ \end{aligned} $$

where: ​

  • n: sample size or number of sample elements;
  • $s^2$: estimator referring to sample variance;
  • $x_1, \dots, x_n$: sample elements.
  • $d_i = (X_i - \overline{X})$: is the deviation of measurement i, for $i=1,\dots, n$.
  • $\sum^n_{i=1}$: is a sum symbol, in this case the sum goes from $1$ to $n$. ​

Example (Standard Deviation) for Frog Jump

Be the height of the frog’s jump and the measurements are given below:

Example 1.
Jump Height (cm)
1 10
2 11
3 9
4 8
5 10

In this way, we have the following calculations:

  • $\text{média} =\dfrac{10+11+9+8+10}{5} = \dfrac{48}{5}=9.6 \text{ cm}$
  • $\text{Variância} = s^2 = \dfrac{5.2}{5-1} = 1.3 \text{ cm}$
  • $\text{Desvio-padrão} = s = \sqrt{1.3} \approx 1.14 \text{ cm}$
Example 1.

Therefore, the frog’s jump is $9.6 \text{ cm} \pm 1.14\text{ cm}$

Example (Standard deviation) for election candidates

Let the presidential election poll be for the year 2022.

Example 1.
Candidate Average vote percentage
A 35 %
B 38 %
C 20 %
D 8 %

The “margin of error” is $\pm 5$%. Which candidate is leading the electoral race?

Example 1.

With the “margin of error” we have the following scenarios:

Candidate Average vote percentage Lower Limit Upper Limit
A 35 % 30 % 40 %
B 38 % 33 % 42 %
C 20 % 15 % 25 %
D 8 % 3 % 13 %
Example 1.

As we have intersection between the ranges of A and B. Therefore, we have they are “technically” or statistically tied for first place.

This is a small introduction to dispersion measures, wait for the next posts.