## Note on One way ANOVA.

• Note
• Things to remember

#### One way ANOVA.

It is used to test whether three or more populations have the same mean., It involves different samples subgroups that are recognised accordingly to a single factor or characteristics.

The actual mean method is usually employed to compute ANOVA is one way ANOVA.

Assuming that the k value represents populations whose measurements are randomly and independently drawn, follow a normal distribution, and have equal variances. Then the followings steps are followed for an actual mean method.

Step 1: the Null hypothesis. : H0= µ1= µ2= µ3=.....= µk That is ,there is no significance difference of k populations means.

Step 2: Alternative hypothesis:H1: µ1≠ µ2 ≠µ3≠...........≠µkThat is there is significance difference of k populations means or at least one of the mean of the populations of different from the other.

Step 3: Test statistic. Testing this hypothesis by using one-way ANOVA is accomplished by partitioning the total variance of the data into the following two variances.

a. Variance between samples (variance resulting from the treatments or columns): The following steps are required to compute the variance between samples.

1. Compute the individual means( $$\overline{X}$$1, $$\overline{X}$$2.......... $$\overline{X}$$k) of k sample subgroups. For example,

$$\overline{X_1}=\frac{∑X_1}{n_1},\overline{X_2}=\frac{∑X_2}{n_2}\,and\,so\,on$$

2. Compute the overall or grand sample mean as

$$\overline{X}=\frac{Sum\,of\,observations\,of\,all\,the\,samples}{Total\,number\,of\,observations}$$

$$=\frac{T}{N}$$

3. Compute the difference between the means of the various samples and the grand mean i,e.

$$(\overline{X_1}-\overline{X})\,(\overline{X_2}-\overline{X}).....(\overline{X_k}-\overline{X})^2$$

4. Compute the sum of squares between sample subgroup or treatments (SSB) AS:

$$SSB=n_1(\overline{X_1}-\overline{X})^2+n_2(\overline{X_2}-\overline{X})^2+..........n_k(\overline{X_k}-\overline{X})^2$$

5. Divide SSB by its degree of freedom i,e (k-1), where k is the number of samples subgroup. This gives the value of variance between samples. This is also called Mean Squares between Sample subgroups (MBS).

$$(MBS)=\frac{SSB}{K-1}$$

b. Variance within samples (Error variance or that portion of the total variance unexpected by the treatment).

1. Sample mean ( $$\overline{X}$$1, $$\overline{X}$$2........ $$\overline{X}$$has already computed of all the k samples.

2. Compute the deviations of the various items of k samples from the mean values of their correspondings samples.

3. Compute the square all the deviations obtained in step (ii) and find the total of these squared deviations. This resulting form is called Sum of Square. Within the samples subgroups (SSW) or Sum of square due to error (SSE). This means.

$$SSB=∑(\overline{X_1}-\overline{X})^2+∑_2(\overline{X_2}-\overline{X})^2+..........∑(\overline{X_k}-\overline{X})^2$$

$$Where\,X_1=the\,element\,in\,the\,first\,sample$$

$$X_2=the\,element\,in\,the\,second\,sample$$

4. Divide the SSW by its degree of freedom I,e (n-k) where n is the total number of observations. This gives the value of variance within sample subgroup. It is also called Mean or sum of the square within samples subgroups (MSW ) OR Mean Sum of squares due to Error (MSE).

$$That\,is\,MSE=\frac{SSE}{n-K}$$

To compute the value of F-ratio, divide the MSE BY MSW I,E

$$F=\frac{MSB}{MSE}$$

#### Reference

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: National Book centre, 2013.

1. To compute the test statistic of ANOVA total component of variation is split into two variations. Between samples or due to treatments and within samples or due to error.
2. H1: µ1≠ µ2 ≠µ3≠...........≠µkThat is there is significance difference of k populations means or at least one of the mean of the populations of different from the other.
3. Compute the square all the deviations obtained in step (ii) and find the total of these squared deviations. This resulting form is called Sum of Square. Within the samples subgroups (SSW) or Sum of square due to error (SSE).
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