https://www.youtube.com/watch?v=htXELssWGEk&list=PLsri7w6p16vsOVj5cL8U4z0m4RdMZfhme&index=2 ================================================================================ $$$H_0$$$: no difference, $$$\bar{x}_A = \bar{x}_B = \bar{x}_C$$$ $$$H_1$$$: there is difference, other cases ================================================================================ ================================================================================ Sum of squares total (SST) $$$SST = \sum\sum (x_{ij}-\bar{x})^2$$$ $$$(1-3.434)^2+(4-3.434)^2+\cdots+ (3-3.434)^2 = 13.973$$$ ================================================================================ Sum of squares between samples (SSB) $$$SSB = \sum n_i (\bar{x}_j - \bar{x})^2$$$ $$$ 7(2.857-3.434)^2 + 8(4.000-3.434)^2 + 9(3.444-3.434)^2 = 4.894$$$ ================================================================================ Sum of squares within samples (SSW) $$$SSW = \sum\sum (x_{ij} - \bar{x}_i)^2$$$ $$$SSW_A=(1-2.857)^2+(4-2.857)^2+(3-2.857)^2=4.857$$$ $$$SSW_B=(4-4.000)^2+(4-4.000)^2+(4-4.000)^2=2.000$$$ $$$SSW_C=(4-3.444)^2+(3-3.444)^2+(3-3.444)^2=2.222$$$ $$$SSW = 4.857+2.000+2.222 = 9.079$$$ ================================================================================ ================================================================================ When you compare some values, basic values are mean value Degree of freedom of SST: n-1 Degree of freedom of SSB: i-1 Degree of freedom of SSW: n-i ================================================================================ Mean square of between samples (MBS) $$$MBS = \frac{\text{SSB}}{i-1}$$$ Mean square of within samples (MSW) $$$MSW = \frac{\text{SSW}}{n-i}$$$ ================================================================================ Ratio of variance (F) $$$F \\ = \dfrac{\text{change between classes}}{\text{changes within classes}} \\ = \dfrac{MSB}{MSW} \\ = \dfrac{\frac{SSB}{i-1}}{\frac{SSW}{n-i}}$$$ ================================================================================ $$$MSB = \dfrac{4.894}{2} = 2.447$$$ $$$MSW = \dfrac{9.079}{21} = 0.432$$$ $$$F = \dfrac{2.447}{0.432} = 5.664$$$ ================================================================================ One-way ANOVA table report ================================================================================ $$$H_0$$$ is rejected