What are variability and covariability? - Chapter 3

What is variability?

Variability (also called variance) is the difference within a set of test scores or between the values ​​of a psychological trait. Inter-individual differences are differences that occur between people. Intra-individual differences are differences within one person at different times. Individual differences are very important in psychological tests. Reliability and validity of tests depend on the ability of a test to quantify differences between people. All research in psychology and all scientific applications of psychology depend on the ability of a test to measure individual differences. It is important to know that every area of ​​scientific psychology depends on the existence and quantification of individual differences.

You can quantitatively display scores of a group of people or scores of one person at different times in a so-called distribution of scores. A distribution of scores is quantitative, because the differences between scores are expressed in figures. The difference between scores within a distribution is called the variability.

Calculating variability

To display a distribution of scores, you first have to calculate a few things. First of all, we calculate the average, since it is the most used, in addition to the mode and the median. Secondly, we calculate the variability, which consists of steps. These steps are as follows:

1. Calculate the mean by dividing the sum X by the total number of N people or things.
2. Calculate the deviation by displaying the difference between X and the average of X.
3. Calculate the squared deviation by squaring the deviation.
4. Calculate variance s² by dividing the sum of squared deviations by the total N.
5. Standard deviation √s² = calculate s by taking root of the variance.

Interpretation of the variance and standard deviation

• Is never less / lower than zero, or s² ≥ 0 and s ≥ 0.
• You can never interpret a single score as a large or small value.
• Comparison is only possible if two or more scores are based on the same measuring instrument / variable, for example IQ. After this you can also determine whether it is a large or small value.
• The variance and the standard deviation can only be used in certain concepts, for example in correlations or when you measure the reliability of scores.

Normal distribution

Distribution forms and normal distributions are qualitative because they represent the scores in a graphical way. The variable is placed on the x-axis, for example IQ scores from low to high. The proportions of the number of people who have achieved a certain score are displayed on the y-axis. A figure emerges from this: the normal distribution. This rarely (almost never) has a mirror shape. Usually the figures are either crooked to the right or crooked to the left. Skewed to the right means that there are more people who score low. Skewed to the left means that there are more people who score high.

What is covariability?

With a variance, the difference is calculated within one set of scores. With covariability, also called covariance, the difference of one set of scores is compared with the difference of another set of scores. In other words: with a covariance, the relationship between two variables is searched for, for example IQ and GPA. With a variance, one variable is used.

There are important characteristics with a covariance that clearly show the relationship between the two variables. The direction and strength of the relationship, but also the consistency between the two variables is important.

Direction and strength

The direction of the relationship between the two variables can have a positive or negative relationship. There is a positive (or direct) relationship when high scores on the first variable and high scores on the second variable occur at a time. A negative relationship exists when high scores for the first variable and low scores for the second variable occur. This can also be reversed, so low scores on the first variable and high scores on the second variable.

The strength of a connection is difficult to interpret.

Consistency

A strong relationship (positive or negative) between two variables shows that there is a high degree of consistency between them. If there is no clear relationship between two variables, then individual differences on one variable are inconsistent with individual differences on the other variable.

Variance is the variability of a single distribution of scores. Covariance is the variability of two distributions of scores. We have discussed the distribution of scores of a variance for this. We will calculate the distribution of covariance scores as follows:

1. Calculating deviations from variable X and from variable Y. You do this by calculating the difference between X and the average of X. You also calculate the difference between Y and the average of Y.
2. Calculate cross-products by multiplying the deviation of X and the deviation of Y by each other. A positive cross product can come out here, which means that the relationship between the variables is consistent. Or when the scores of an individual on both variables are consistent with each other, the individual scores either above the average on both variables or just below the average on both variables. A negative cross product can also be the result. This means that the coherence is uneven and therefore inconsistent. In other words, the individual scores below the average for one variable (so a negative deviation results from this), but on the other variable above the average (this results in a positive deviation).
3. Calculate covariance using a formula: Cxy = ∑ of the cross-products (multiplying deviation X by deviation Y) by the total N number of people or things.

The covariance provides clear information about the direction of the relationship but not about the strength of the relationship. Correlation coefficients provide clear information about the direction and strength of the relationship.

Variance-covariance matrix

The variance-covariance matrix is ​​always structured in a certain way, with a number of standard features:

1. Each variable has a row and a column.
2. The variances of the variables are shown in a diagonal line from top left to bottom right.
3. All other cells contain covarities between sets of variables.
4. The covariations are symmetrical. All values ​​below the diagonal are identical to the values ​​above the diagonal.

With a correlation, the value is easier to interpret than with a covariance. Correlation is always between -1 and +1. If the value is below zero, then the relationship between the two variables is negative. If the value falls above zero, it is positive.

Zero means that there is no correlation. The closer to zero the correlation falls, the poorer the relationship between the variables. You can also say that the coherence is inconsistent. The farther away from zero, the better / stronger the coherence between the variables. You can also say that the coherence is consistent. Correlation = Rxy = Cxy / SxSy.

Variance for compound variables / items is used when a psychological test contains a large number of variables / items. You first calculate the variances s² of the number of variables / items separately and add them together. You then calculate the correlation between the scores of the number of variables / items. You multiply this two times. In turn, you multiply this by the standard deviations of the number of variables / items (which you first calculated separately). In the end you add everything.

The total test score variance depends solely on item variability and the correlation between the item pairs. This is an important part of the dependency theory, which is discussed in a later chapter.

There are dichotomous reactions with binary items. This means that when answering a question you can choose from two answers (yes/no, agree/disagree, or 0/1). For example, we ask people to answer a yes or no to a question or we ask people whether they agree or disagree with the statement. It is also possible that certain scores are considered right or wrong. Whether we look at whether or not a certain disorder occurs. We often indicate this with codes, namely code 0 for a negative response (no, disagree, wrong, not true) and code 1 for a positive response (yes, agree, good, true). Code 1 is indicated by p = ∑X / N. Code 0 is indicated by q = 1-p.

You can also calculate the variance using p and q, namely: s² = p x q or p x (1-p). The maximum variance that you can get is 0.25; if p = q = 0.50 then s² = 0.50 x 0.50.

What should you pay attention to when interpreting test scores?

Problems arise when interpreting test scores. As an example:

• What is a high score and what is a low score?
• What does it mean if you score high or if you score low?

Consider for example a score of 35 on neuroticism. Is this a high score? And if this is a high score, what does that mean? Am I neurotic or not at all?

A frame of reference ensures that the figures and percentages are easy to interpret. You look at whether the scores fall above or below or even on the average score. You also view how many above or below the average score they fall (think of standard deviations). With this data you can calculate the so-called z-scores. A z-score shows how far above or below the average test score a score falls. Z-scores are easy to compare, even when two completely different variables / units of measurement are used within a test score. For example weight and optimism. Z = (X minus the average of X) / Sx (s = standard deviation of X). The z-score is expressed in the "number of standard deviations". An example, z = 0.5 or -0.5 this means that the score 0.5 standard deviations is above or below the average. This is very close. Another example, z = 2 or -2, this means that the score of 2 standard deviations is above or below the average. This is further away. A z-score distribution includes the following distribution: Z (0; 1) where 0 is the average and 1 is the standard deviation. Z scores say something about a score in relation to the rest of the group. It says how good or bad your score is compared to the average person but says nothing about your abilities in general. Correlation between variables using z-scores: Rxy = ∑ZxZy / N.

Z scores may be easy to compare, they are more difficult to interpret because many people are not familiar with concepts such as "standard deviations" or "distance to the average". Therefore, T-scores (standardized scores) are used with T (50; 10) where 50 is the average and 10 is the standard deviation. T = (z) times (s) + (average of X). Either T = z (10) + 50. Other means / standard deviations may also be given. Another way to interpret scores is in percentiles. An example: an individual has achieved a score of 194. The total number of people taking part in this test is 75. Only 52 people score lower than 194. So: (52/75) x 100 = 69%. You can interpret this as if the score of the individual falls in the 69th percentile and that this person scores higher than 69% of the other people who took the test.

What are normalized scores?

It is often assumed that a psychological trait is normally distributed, but this is not always the case and then a problem arises. It may be thought that a property (such as intelligence) is normally distributed, but the test data (IQ test) is not normally distributed. Researchers then assume that their theory is correct and that the test data (IQ scores) do not accurately reflect the distribution of the construct. Researchers have tried to solve this problem with the help of normalization transformations / area transformations. The scores are then converted to T-scores.

Variability (also called variance) is the difference within a set of test scores or between the values ​​of a psychological trait. Inter-individual differences are differences that occur between people. Intra-individual differences are differences within one person at different times. Individual differences are very important in psychological tests. Reliability and validity of tests depend on the ability of a test to quantify differences between people. All research in psychology and all scientific applications of psychology depend on the ability of a test to measure individual differences. It is important to know that every area of ​​scientific psychology depends on the existence and quantification of individual differences.

You can quantitatively display scores of a group of people or scores of one person at different times in a so-called distribution of scores. A distribution of scores is quantitative, because the differences between scores are expressed in figures. The difference between scores within a distribution is called the variability.