For a completely randomised design with a total of k treatments. let x_ij : i^th observation corresponding to of j^th treatment, i = 1, 2, ..... n_j, j = 1, 2, ......, k T : total of all observations, x̅_j : mean of i^th treatment, x̅ : \(\rm \frac{T}{N}\) where \(\rm N=\Sigma_{j=1}^kn_j\). The sum of squares due to treatments (SST) can be calculated by using the formula:

\(\rm \Sigma_{j=1}^k n_j(\bar x_{j}- \bar x)^2 \). 
 Key Points

• The sum of squares due to treatments (SST) in a completely randomized design can be calculated using the formula: \(\rm \Sigma_{j=1}^k n_j(\bar x_{j}- \bar x)^2 \)
  • Σk represents the sum of the k treatments. It indicates that we consider each treatment separately.
  • Σj=1nj represents the sum of the observations within each treatment. It implies that we calculate the sum of squares for each treatment individually.
  • x̅j is the mean of the jth treatment, which represents the average value of the response variable for that particular treatment.
  • x̅ is the grand mean, which represents the overall average value of the response variable across all treatments and observations.
• To calculate the SST, we take the difference between each treatment mean (x̅j) and the grand mean (x̅). We square this difference to eliminate any negative signs and emphasize the magnitude of the difference. We then sum up these squared differences across all treatments and observations. 
• The resulting SST value reflects the total variation in the response variable that can be attributed to the differences between the treatment means. It represents the variability between treatments and is used to assess the significance and effectiveness of the different treatments in the study.
•  By comparing the SST to other sums of squares, such as the sum of squares due to error (SSE) or the total sum of squares (SSTotal), we can further analyze and interpret the effects of treatments on the response variable.

Hence, the required formula is \(\rm \Sigma_{j=1}^k n_j(\bar x_{j}- \bar x)^2 \)