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Below is a worked statistics example covering ANOVA. Hopefully it will provide some much needed statistics help! For solutions tailored to your particular class, please contact us providing details of how you would like us to help you (in a few sentences), and we will respond promptly with a quote.

ANalysis Of VAriance—ANOVA—is a statistical method that compares the difference between the means of two or more samples. It uses the f-statistic, which is a ratio comparing the variance within each sample to the overall variance of the data among all the samples. The motivation for ANOVA is that Student’s t-test can only compare two means at a time, with an accepted error rate of 5% (from α = 0.05). Thus, as the number of means being compared increases linearly, the number of t-tests needed increases exponentially (round-robin testing) with the 5% error rate guaranteeing that as the number of t-tests increases the probability of erroneous results increases at a rate of 1 false result for every 20 tests. ANOVA allows comparison of ALL the means SIMULTANEOUSLY while maintaining an overall error-rate of 5%. That is the power of ANOVA! Below is a simple example of a one-factor one-way ANOVA.

__ANOVA example:__

Three different bacteria, X, Y and Z, were grown in a lactose broth in order to ascertain which of the three could best utilize lactose as a growth substrate. Four replicate flasks of lactose broth were used for each bacterium and after 24h the biomass in each flask was recorded as shown in the table below. Use a one-factor ANOVA to determine if there was any significant difference in the mean biomass for each of the three bacteria, X, Y, and Z.

Biomass of bacterium |
|||

Flask |
X |
Y |
Z |

1 |
37 |
10 |
22 |

2 |
43 |
15 |
18 |

3 |
45 |
16 |
20 |

4 |
41 |
12 |
23 |

- The f-statistic is based on the formula where MS is the Mean Square value, which contrary to what its name would suggest is a measure of variance, and not the mean. The “between” variables apply to differences between the three bacteria, while the “residual” variables represent random effects associated with all the measurements. Adding “between” and “residual” variances gives the total variances.
- where SS
*between*is the Sum of Squares between the different bacteria and can be found using the formula where*n*is the number of measurements for each bacterium (*n =*4) and*N*is the total number of measurements (*N*= 12). df*between*represents the Degrees of Freedom between the three bacteria, and can be found using the formula where*u*is the number of bacteria (i.e. the number of conditions being compared) - Once the f-statistic is calculated, the final step involves comparing the value to that listed in a table of the f-distribution for dfbetween and dfresidual where p = 0.05. Should the value calculated exceed that found in the table, there is a significant difference between the different conditions. As shown for the bacterial experiment, the value of the f-statistic was far higher than that required for significance (p = 0.05), high significance (p = 0.01) or even very high significance (p = 0.001) which tells us that the difference between the mean biomass for bacteria X, Y and Z was very highly significant. It does NOT tell us what the individual significances between X and Y, X and Z, and Y and Z were. Also, it does NOT tell us what the size of the effect between the three bacteria was. Gaining this information would require further analysis.

The values for all the calculations above are shown in the table below:

__f-distribution: table of values for P = 0.05, P = 0.01, P = 0.001__