coefficient

Examples of coefficient in the following topics:

• Calculating the Emission and Absorption Coefficients

  • We can write the emission and absorption coefficients in terms of the Einstein coefficients that we have just examined.
  • The emission coefficient $j_\nu$ has units of energy per unit time per unit volume per unit frequency per unit solid angle!
  • The Einstein coefficient $A_{21}$ gives spontaneous emission rate per atom, so dimensional analysis quickly gives
  • We can now write the absorption coefficient and the source function using the relationships between the Einstein coefficients as

• 95% Critical Values of the Sample Correlation Coefficient Table

• Rank Correlation

  • It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions.
  • However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson product-moment correlation coefficient.
  • The coefficient is inside the interval $[-1, 1]$ and assumes the value:
  • This means that we have a perfect rank correlation and both Spearman's correlation coefficient and Kendall's correlation coefficient are 1.
  • For example, for the three pairs $(1, 1)$, $(2, 3)$, $(3, 2)$, Spearman's coefficient is $\frac{1}{2}$, while Kendall's coefficient is $\frac{1}{3}$.

• A Physical Aside: Einstein coefficients

  • The Einstein coefficients seem to say something magical about the properties of atoms, electrons and photons.
  • It turns out that the relationships between Einstein coefficients (1917) are an example of Fermi's Golden Rule (late 1920s).

• 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 population is commonly represented by the Greek letter $\rho$ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient.
  • 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.
  • This fact holds for both the population and sample Pearson correlation coefficients.
  • Put the summary statistics into the correlation coefficient formula and solve for $r$, the correlation coefficient.

• Overview of How to Assess Stand-Alone Risk

  • It is able to accomplish this because the correlation coefficient, R, has been removed from Beta.
  • Another statistical measure that can be used to assess stand-alone risk is the coefficient of variation.
  • In probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution.
  • It is also known as unitized risk or the variation coefficient.
  • A lower coefficient of variation indicates a higher expected return with less risk.

• Hypothesis Tests with the Pearson Correlation

  • We need to look at both the value of the correlation coefficient $r$ and the sample size $n$, together.
  • 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. "
  • Our null hypothesis will be that the correlation coefficient IS NOT significantly different from 0.

• Economic measures

  • The Gini coefficient measures the inequality among values of a frequency distribution.
  • A Gini coefficient of zero expresses perfect equality, where all values are the same (for example, where everyone has the same income).
  • A Gini coefficient of one (or 100%) expresses maximal inequality among values (for example where only one person has all the income).
  • The Gini coefficient was originally proposed as a measure of inequality of income or wealth.
  • The global income inequality Gini coefficient in 2005, for all human beings taken together, has been estimated to be between 0.61 and 0.68.

• The Coefficient of Determination

  • r2 is called the coefficient of determination. r2 is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form. r2 has an interpretation in the context of the data:

• Testing the Significance of the Correlation Coefficient

  • The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
  • 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".
  • The test statistic t has the same sign as the correlation coefficient r.
  • Suppose you computed the following correlation coefficients.