A positive correlation means that there is a strong relationship between the variables.

Examples of positive correlation in the following topics:

• Correlation and Causation

  • A correlation can be positive/direct or negative/inverse.
  • A positive correlation means that as one variable increases (e.g., ice cream consumption) the other variable also increases (e.g., crime).
  • 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.

• 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.
  • In contrast, two correlations of .05 and .98 have the same direction (positive) but are very different in their strength.
  • A positive correlation, such as .8, would mean that both variables increase together.
  • Another popular example is that there is a strong positive correlation between ice cream sales and murder rates in the summer.

• The Correlation Coefficient r

  • If r = 1, there is perfect positive correlation.
  • If r = − 1, there is perfect negative correlation.
  • A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive 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

• An Intuitive Approach to Relationships

  • Correlation refers to any of a broad class of statistical relationships involving dependence.
  • These are all examples of a statistical factor known as correlation.
  • Correlation refers to any of a broad class of statistical relationships involving dependence.
  • Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring and the correlation between the demand for a product and its price.
  • This graph shows a positive correlation between world population and total carbon emissions.

• 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 positive value of $r$ means that when $x$ increases, $y$ tends to increase and when $x$ decreases, $y$ tends to decrease (positive correlation).
  • If $r=1$, there is perfect positive correlation.
  • Put the summary statistics into the correlation coefficient formula and solve for $r$, the correlation coefficient.

• Aging and Health

  • For instance, maintaining a positive attitude has been shown to be correlated with better health among the elderly.
  • Older individuals with more positive attitudes and emotions engage in less risky behavior and have lower levels of stress, both of which are correlated with better health.

• 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 the relationship is strong and positive, the correlation will be near +1.
  • Sample scatterplots and their correlations.
  • The first row shows variables with a positive relationship, represented by the trend up and to the right.
  • Sample scatterplots and their correlations.

• Exercises

  • Make up a data set with 10 numbers that has a positive 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?

• 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.

Is positive correlation strong?

What is a strong correlation between variables?

Is strong correlation positive or negative?

Is a correlation of strong or weak?