variation ratio

Examples of variation ratio in the following topics:

• Measures of Variability of Qualitative and Ranked Data

  • The variation ratio is a simple measure of statistical dispersion in nominal distributions.
  • Just as with the range or standard deviation, the larger the variation ratio, the more differentiated or dispersed the data are; and the smaller the variation ratio, the more concentrated and similar the data are.
  • This group is more dispersed in terms of gender than a group which is 95% female and has a variation ratio of only 0.05.
  • Similarly, a group which is 25% Catholic (where Catholic is the modal religious preference) has a variation ratio of 0.75.
  • This group is much more dispersed, religiously, than a group which is 85% Catholic and has a variation ratio of only 0.15.

• Overview of How to Assess Stand-Alone Risk

  • 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.
  • The coefficient of variation, an example of which is plotted in this graph, can be used to measure the ratio of volatility to expected return.

• Mean Squares and the F-Ratio

  • There are two sets of degrees of freedom for the $F$-ratio: one for the numerator and one for the denominator.
  • To calculate the $F$-ratio, two estimates of the variance are made:
  • The variance is also called variation due to treatment or explained variation.
  • The variance is also called the variation due to error or unexplained variation.
  • Then, the F-ratio will be larger than one.

• The F-Distribution and the F Ratio

  • The F statistic is a ratio (a fraction).
  • To calculate the F ratio, two estimates of the variance are made.
  • The variance is also called variation due to treatment or explained variation.
  • Then the F-ratio will be larger than 1.
  • If the groups are the same size, the calculations simplify somewhat and the F ratio can be written as: F-Ratio Formula when the groups are the same size

• Direct Variation

  • When two variables change proportionally to each other, they are said to be in direct variation.
  • No matter how many toothbrushes purchased, the ratio will always remain: $2 per toothbrush.
  • Direct variation is easily illustrated using a linear graph.
  • Graph of direct variation with the linear equation y=0.8x.
  • The line y=kx is an example of direct variation between variables x and y.

• Comparisons Within an Industry

  • When comparing past and present financial information, one will want to look for variations such as higher or lower earnings.
  • Ratios of risk such as the current ratio, the interest coverage, and the equity percentage have no theoretical benchmarks.
  • Ratios must be compared with other firms in the same industry to see if they are in line .
  • Ratio analyses can be used to compare between companies within the same industry.
  • For example, comparing the ratios of BP and Exxon Mobil would be appropriate, whereas comparing the ratios of BP and General Mills would be inappropriate.

• Test of Two Variances

  • For instance, college administrators would like two college professors grading exams to have the same variation in their grading.
  • In order for a lid to fit a container, the variation in the lid and the container should be the same.
  • Since we are interested in comparing the two sample variances, we use the F ratio
  • NOTE: The F ratio could also be $\frac{(s_1)^2}{(s_2)^2}$.
  • Two college instructors are interested in whether or not there is any variation in the way they grade math exams.

• Risk Adjusting the Discount Rate

  • Individuals and firms use methods for adjusting discount rates for risk, include adding risk premiums, Sharpe Ratios, rNPV, and Monte Carlo evaluation.
  • Risk will also increase due to the amount of variation in the possible outcomes.
  • The Sharpe Ratio is a measure of risk premium per unit of deviation in an investment asset.
  • The Sharpe Ratio is defined as: where Ra is the asset return, Rb is the return on a benchmark asset, such as the risk free rate of return or an index such as the S & P 500.
  • The Sharpe ratio characterizes how well the return of an asset compensates the investor for the risk taken.

• Total Debt to Total Assets

  • The debt ratio is expressed as Total debt / Total assets.
  • Financial ratios are categorized according to the financial aspect of the business which the ratio measures.
  • Debt ratios measure the firm's ability to repay long-term debt.
  • The higher the ratio, the greater risk will be associated with the firm's operation.
  • Like all financial ratios, a company's debt ratio should be compared with their industry average or other competing firms.

• Comparing Two Population Variances

  • In order for a lid to fit a container, the variation in the lid and the container should be the same.
  • Since we are interested in comparing the two sample variances, we use the $F$ ratio:
  • If the null hypothesis is $\sigma_1^2 = \sigma_2^2$, then the $F$ ratio becomes:
  • Note that the $F$ ratio could also be $\frac { { s }_{ 2 }^{ 2 } }{ { s }_{ 1 }^{ 2 } }$.
  • Two college instructors are interested in whether or not there is any variation in the way they grade math exams.