negative correlation

Examples of negative correlation in the following topics:

• The Correlation Coefficient r

  • If r = 1, there is perfect positive correlation.
  • If r = − 1, there is perfect negative correlation.
  • A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation).
  • We say "correlation does not imply causation."
  • (a) A scatter plot showing data with a positive correlation. 0 < r < 1 (b) A scatter plot showing data with a negative correlation. − 1 < r < 0 (c) A scatter plot showing data with zero correlation. r=0

• Correlational Research

  • The strength, or degree, of a correlation ranges from -1 to +1 and therefore will be positive, negative, or zero.
  • Direction refers to whether the correlation is positive or negative.
  • For example, two correlations of .78 and -.78 have the exact same strength but differ in their directions (.78 is positive and -.78 is negative).
  • A negative correlation, such as -.8, would mean that one variable increases as the other increases.
  • The same is true for panels (c) and (d)—the strong negative linear pattern more closely resembles a straight line than does the weak negative pattern.

• Correlation and Causation

  • A correlation can be positive/direct or negative/inverse.
  • A negative correlation is just the opposite; as one variable increases (e.g., socioeconomic status), the other variable decreases (e.g., infant mortality rates).
  • Ice cream consumption is positively correlated with incidents of crime.
  • This diagram illustrates the difference between correlation and causation, as ice cream consumption is correlated with crime, but both are dependent on temperature.
  • Thus, the correlation between ice cream consumption and crime is spurious.

• Exercises

  • Make up a data set with 10 numbers that has a negative correlation.
  • Is this a positive or negative association?
  • Is this a positive or negative association?
  • Just from looking at these scores, do you think these variables are positively or negatively correlated?
  • (AM) Would you expect the correlation between the Anger-Out and Control-Out scores to be positive or negative?

• Coefficient of Correlation

  • The most common coefficient of correlation is known as the Pearson product-moment correlation coefficient, or Pearson's $r$.
  • Pearson's correlation coefficient when applied to a sample is commonly represented by the letter $r$ and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient.
  • A negative value of $r$ means that when $x$ increases, $y$ tends to decrease and when $x$ decreases, $y$ tends to increase (negative correlation).
  • If $r=-1$, there is perfect negative correlation.
  • Put the summary statistics into the correlation coefficient formula and solve for $r$, the correlation coefficient.

• Properties of Pearson's r

  • State the relationship between the correlation of Y with X and the correlation of X with Y
  • A correlation of -1 means a perfect negative linear relationship, a correlation of 0 means no linear relationship, and a correlation of 1 means a perfect positive linear relationship.
  • Pearson's correlation is symmetric in the sense that the correlation of X with Y is the same as the correlation of Y with X.
  • For example, the correlation of Weight with Height is the same as the correlation of Height with Weight.
  • For instance, the correlation of Weight and Height does not depend on whether Height is measured in inches, feet, or even miles.

• Describing linear relationships with correlation

  • We denote the correlation by R.
  • If it is strong and negative, it will be near -1.
  • Sample scatterplots and their correlations.
  • The second row shows variables with a negative trend, where a large value in one variable is associated with a low value in the other.
  • Sample scatterplots and their correlations.

• Values of the Pearson Correlation

  • Give the symbols for Pearson's correlation in the sample and in the population
  • The Pearson product-moment correlation coefficient is a measure of the strength of the linear relationship between two variables.
  • It is referred to as Pearson's correlation or simply as the correlation coefficient.
  • The symbol for Pearson's correlation is "$\rho$" when it is measured in the population and "r" when it is measured in a sample.
  • An r of -1 indicates a perfect negative linear relationship between variables, an r of 0 indicates no linear relationship between variables, and an r of 1 indicates a perfect positive linear relationship between variables.

• Rank Correlation

  • If as the one variable increases the other decreases, the rank correlation coefficients will be negative.
  • To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers $(x, y)$:
  • This means that we have a perfect rank correlation and both Spearman's correlation coefficient and Kendall's correlation coefficient are 1.
  • This graph shows a Spearman rank correlation of 1 and a Pearson correlation coefficient of 0.88.
  • Define rank correlation and illustrate how it differs from linear correlation.

• Hypothesis Tests with the Pearson Correlation

  • We decide this based on the sample correlation coefficient $r$ and the sample size $n$.
  • If the test concludes that the correlation coefficient is significantly different from 0, we say that the correlation coefficient is "significant."
  • If the test concludes that the correlation coefficient is not significantly different from 0 (it is close to 0), we say that correlation coefficient is "not significant. "
  • If $r$ is not between the positive and negative critical values, then the correlation coefficient is significant.
  • If $r$ is less than the negative critical value or $r$ is greater than the positive critical value, then $r$ is significant.