## Inferential Statistical Analysis

### Defining inferential statistics

In quantitative research, a sample is used to obtain information relating to the total population (Stephen & Ruth,1999). Inferential statistics make conclusions and predictions, about the whole population, basing these on the information obtained from the sample (Mason, 1999). This means that whatever is concluded for a sample, applies to the population that is represented by such a sample. Similar sample information such as trends can be used to predict the outcome of an element in the population.

Inferential statistics may be used to look into differences between groups or even cause and effect. The technique used to carry out the inferential statistics varies depending on levels of measurement in question say nominal, interval, ordinal, or ratio. The techniques include chi-square and other non-parametric statistics, measures of central tendency and dispersion, ANOVA, and ANCOVA Correlation, regression, and discriminant analysis.

### Example of an inferential test: Chi-Square Test

- Chi-square is an inferential statistic test that is widely used to test the level at which two sets of data differ in terms of their distribution.
- Chi-square uses numerical as well as categorical data derived from numerical variables and categorical variables respectively.
- It derives its criteria of analysis from the difference that occurs between the observation made and the expected values of information.

### Steps to the chi-square test

- Identify the expected values of a given situation which form the question at hand (E).
- Collect the data as they are presented by the study (O).
- Compute the difference between the observed and the expected values (O -E).
- Find the square of these differences (O -E)
^{ 2} - Divide the squared differences; each category by its expected value [(O -E)
^{ 2}]/E. - Compute the sum of [(O -E)
^{ 2}]/E; this is the chi-square of the problem.

### Illustration

A researcher conducted research in which he did a fertilizer trial, on a group of plants and he observed that the plants planted with fertilizers A, B, C, D, E, F, G, grew to different heights (in inches). It was expected that the plants would grow to a height of 15 inches, regardless of the fertilizer applied. The researcher’s study resulted in the following Data as presented in the table (Level of significance= 0.05).

### Test the hypothesis

- Ho: height of the plant is dependent on the type of fertilizer.
- Ha: height of the plant is independent of the type of fertilizer.

### Statistical Data

## Table analysis

### Chi square formula

X^{2 }= ∑(O –E )^{2}/E

X^{2}=3.94

Degrees of freedom= (rows-1) (columns-1)

6×1= 6d.f

Tabulated value of chi-square at 5% level of significance and 6d.f is 12.59

### Decision rule

Generally, when computed x^{2} is more than the critical value at a given probability level, the null hypothesis is rejected. Since the calculated x^{2} at a 5% significant level, is less than the tabulated, we do not reject the null hypothesis that the observed height of the plant is dependent on fertilizer.

Conclusion – the height of the plant is dependent on the type of fertilizer used.

### How inferential statistical analysis increases understanding of the data

- Inferential statistical analysis provides a more detailed view of data and therefore a better decision about the population can be made.
- This type of analysis is known to answer the cause and effect problems. This consequently enables the researchers to avoid guess work.
- Inferential statistics can give the researcher a better view of the relationships existing between variables.
- Inferential statistical analysis brings out some meaning on data collected with theoretical information that requires proof. Raw data does not tell much but when this type of analysis is conducted on it, an informed conclusion about allusion made about a population is made.
- It evaluates quantitatively. Inferential statistics is defined by several techniques that can deal with different categories of data. Quantitative evaluation paints a better picture of the situation at hand than the raw categorical information.

## References

- Mason R. D. (1999). Statistical techniques in business and economics. Phoenix: Phoenix University Press.
- Stephen, B. & Ruth, B. (1999). Elements of statistics II: inferential statistics. New York: McGraw-Hill.