One-Way ANOVA
Basic statistical method is usually used for testing means and standard deviation of one or two population at the same time. However, this method cannot be used while the population is more than two groups. An Analysis of Variance, popularly known as the ANOVA test, can be used in cases where there are more than two groups.
The concept of ANOVA was born in the beginnings of agricultural era in 1918-1940. It was invented by Sir Ronald Aylmer Fisher while working at Rothamsted Agricultural Experiment Station, London, England. This method emerged because of variations, which represent inequality of a distribution of variations of certain states in a population.
There are many techniques and designs that can be used in analyzing variances. In this post, I’ll share about One-Way Analysis of Variance, the simplest version of ANOVA. A One-Way Analysis of Variance (One-Way ANOVA) is a way to test the equality of three or more means at one time by using variances. It is used for a single-factor between subjects design, i.e. for comparing two or more treatment means.
In One-Way ANOVA, we use F-test for comparing the treatment means. If the value of treatment mean is greater than the F value in certain level of confidence, we can conclude that the treatment means differ or significantly affect the response variable. The following attachment may be useful to help you in understanding the concept of One-Way ANOVA.
High-Performance Fuel Experiments
Pertamina is currently conducting a research for the formulation of an efficient and highly performance fuel. The response variable is revealed in km/hours, as top speed. The researchers wants to determine the best octane ratings (%) that produce the highest speed. They choose five levels of octane ratings that vary between 92-96% with 5 times radomized replication. The table of experiments is summarize as shown below,
|
Octane Ratings |
Observation |
||||
|
1 |
2 |
3 |
4 |
5 |
|
|
92% |
366 |
365 |
371 |
367 |
373 |
|
93% |
368 |
373 |
368 |
374 |
374 |
|
94% |
370 |
374 |
374 |
375 |
375 |
|
95% |
366 |
381 |
378 |
375 |
379 |
|
96% |
371 |
366 |
367 |
367 |
377 |
Calculation of Variation
|
Octane Ratings |
Observation |
Total y. |
||||
|
1 |
2 |
3 |
4 |
5 |
||
|
92% |
366 |
365 |
371 |
367 |
373 |
1,842 |
|
93% |
368 |
373 |
368 |
374 |
374 |
1,857 |
|
94% |
370 |
374 |
374 |
375 |
375 |
1,868 |
|
95% |
366 |
381 |
378 |
375 |
379 |
1,879 |
|
96% |
371 |
366 |
367 |
367 |
377 |
1,848 |
|
y.. |
9,294 | |||||
Total Source of Variation
SST = (366)2 + (365)2 + (371)2 + … + (367)2 + (377)2 –
= 504,56
Source of Variation : Treatment
SSTreatment = {[(1842)2 + (1857)2 + (1868)2 + (1879)2 + (1848)2] / 5 } –- {[(9294)2]/25}
= 178,96
Source of Variation : Error
SSE = SST - SSTreatment
= 577.468,70 – 577.125,10
= 325,60
ANOVA for the High-Performance Fuel
|
Source of Variation |
Sum of Square |
df |
Mean Square |
F0 |
|
| Treatment |
178.96 |
4 |
44.74 |
2.75 |
|
| Error |
325.60 |
20 |
16.28 |
||
| Total |
504.56 |
24 |
F0.1; 4; 20 = 2,25 < F0 = 2,75 ……………………(Reject H0)
Since the H0 is rejected, it can be concluded that the octane ratings in the fuel significantly affect vehicle’s top speed.
