# What is meant by non-zero correlation

(Chapter 7 - page 2/3)

### Product-moment correlation

We are now looking for a measure that not only tells us how close the connection (or how high the proportion of common variation) is between two features, but which also tells us something about the direction of the connection.

In addition, this dimension should be independent of the selected scale or the measuring scale; i.e. we want a measure that tells us something about the relationship between two variables that are scaled differently (such as willingness to take risks / extraversion or motivation to achieve / openness).

Furthermore, the measure should be independent of the sample size so that we can compare measures of association from different studies.

This measure is the so-called "product-moment-correlation coefficient". It was developed by PEARSON and BRAVAIS. Whenever correlation is mentioned in the following, the product-moment correlation coefficient is always meant, unless otherwise noted.

The formula of the correlation was constructed in such a way that the numerical size of the coefficient can never be greater than +1 and never less than -1 (unless one miscalculates!).

The higher the value of the correlation (the closer it is to +1 or -1), the closer the relationship between the variables under consideration.

The sign of the correlation says nothing about the closeness of the connection, only something about the direction of the connection. A missing correlation is expressed by a correlation close to zero.

Usually correlations are interpreted as follows:

.00 = no connection

.00 to .25 = lower "

.25 to .50 = medium "

.50 to .75 = higher "

.75 to 1.0 = more complete "

These values represent approximate benchmarks and not exact limit values.

For example, a correlation of .08 should not be interpreted as a "low correlation" but as "no correlation".

However, the phrase "connection between two characteristics" must not be misunderstood. It is not meant that there is a causal relationship between two features (x and y). The correlation coefficient itself does not permit any statement about a cause-effect relationship, for example in the sense that the feature x causes the feature y.

For example, a high positive correlation between intelligence and school performance cannot simply be interpreted in such a way that school performance is based on intelligence. Such a coefficient initially only says that a student with high (low) intelligence will usually also show good (poor) school performance. The creation of this connection could, however, be explained by a number of alternative causes, e.g. a third variable may be responsible (e.g. socio-economic status of the parents).

Now, within the framework of science that is as precise as possible, we cannot be satisfied with estimates, but must try to calculate exact values. Therefore we cannot do without the formula for calculating the coefficient of the product-moment correlation.

*The correlation is the sum of the deviation products of all x _{i} and y_{i} from the respective mean, divided by the sum of the squared deviations of all x_{i} times that of all y_{i}. *

Application rules:

- The measured values of the test person with regard to feature 1 (x values) and feature 2 (y values) are entered in a table in such a way that the two measured values for each test person are side by side.
- The x and y values are squared and (in the case of marginal calculation) entered in the table.
- The product xy is formed for each pair of measured values.
- The values in the x, y, x columns
^{2}, y^{2}and xy are added. - The values obtained are entered in the formula.

We want to apply this rule to the example shown at the beginning:

x = intelligence quotient (IQ) / y = computing test performance (RT)

Vp | x | y | x^{2} | y^{2} | x * y |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 120 118 100 102 96 90 112 115 116 104 95 108 111 119 101 | 10 7 4 4 1 3 6 8 8 5 4 6 7 10 5 | 14400 13924 10000 10404 9216 8100 12544 13225 13456 10816 9025 11664 12321 14161 10201 | 100 49 16 16 1 9 36 64 64 25 16 36 49 100 25 | 1200 826 400 408 96 270 672 920 928 520 380 648 777 1190 505 |

In this (fictitious) example there is a very high positive correlation of .92 (the zero in front of the decimal point is usually left out and one reads: "Point ninety-two").

In order to be able to calculate the product-moment correlation coefficient, the following requirements must be met:

- The data to be correlated must have been measured at the interval scale level.
- The relationship between the two features x and y must be linear.
- Therefore the features a) unimodal and b) must be distributed symmetrically.
- Conditions 2) and 3) are always met if the characteristics are normally distributed.

What is meant by normal distribution should not be described in more detail here.

Just this much:

The normal distribution looks like a bell and means that values which deviate only slightly from the mean value are observed very frequently, while extreme values, on the other hand, are very rare. The greater the deviation from the mean, the lower the probability of actually measuring such an extreme measured value.

Example:

People over 2.00 meters tall are rare, as are those under 1.40 meters. Most people are average height (around 1.70 meters).

Furthermore, linearity means that the swarm of points in a bivariate distribution can best be represented by a straight line and not by a curve-linear function (parabola, hyperbola, etc.).

- Adderall restricts creative thinking
- What's your reading list for 2016
- Why did Anil Ambanis Rcom fail
- What conflicts did Queen face?
- What is Heat Resistant Concrete
- Is Shayari a relevant page for Shayaris
- What advantages does Postgres have over MySQL
- Is the NDA exam in writing or online
- What happens when narcissists turn into psychopaths
- How do I say Duraznos in English
- What makes a country really secular?
- Who founded the social news company NowThis
- What can crowdfunding be used for?
- Does a brand need a logo
- Where do you go to relax
- Why is a pressurized gas cold
- The latest engine technologies will make engines smaller
- Who is the most inspiring Blackpink member?
- What are some social delusions
- Where does Jakarta's soil of society live?
- What's the point of point
- Does the universe have a certain volume
- Is it unhealthy not to think positively
- Should I do an internship for money