Examples of variation ratio in the following topics:
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- 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.
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- 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.
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- 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
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- 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.
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- 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.
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- ANOVA is a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups).
- In ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation.
- The statistical significance of the experiment is determined by a ratio of two variances.
- This ratio is independent of several possible alterations to the experimental observations, so that adding a constant to all observations, or multiplying all observations by a constant, does not alter significance.
- The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance.
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- Unequal sample size calculations are shown in the section on sources of variation.
- This ratio is named after Fisher and is called the F ratio.
- However, the F ratio is sensitive to any pattern of differences among means.
- In ANOVA, the term sum of squares (SSQ) is used to indicate variation.
- The first column shows the sources of variation, the second column shows the degrees of freedom, the third shows the sums of squares, the fourth shows the mean squares, the fifth shows the F ratio, and the last shows the probability value.
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- This question is different from earlier testing procedures since we will simultaneously consider many groups, and evaluate whether their sample means differ more than we would expect from natural variation.
- If the null hypothesis is true, any variation in the sample means is due to chance and should not be too large.
- ANOVA uses a test statistic F, which represents a standardized ratio of variability in the sample means relative to the variability within the groups.
- Then the F statistic is computed as the ratio of MSG and MSE: F = = = 1.984 ≈ 1.994.
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- r2 , when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
- 1-r2 , when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line.
- Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be ex- plained by the variation in the grades on the third exam, using the best fit regression line.
- Therefore approximately 56% of the variation (1 - 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line.
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- How might the manufacturer measure and, consequently, control the amount of variation in the car parts?
- In fact, the chi-square distribution enters all analyses of variance problems via its role in the $F$-distribution, which is the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom.