Statistical Techniques an their Overview Assumptions Ranges Uses

Statistical Techniques
Introduction
Statistics is defined as a methodology of gathering data, analyzing it, understanding the data and forming appropriate conclusions from the analysis. Not many subjects are as broad or have as many important applications as statistics (Wooldridge & Jeffrey, 2013). Mathematicians, scientists and researchers in general rely on statistics to interpret the information they collect or encounter in their specific fields of work. In general, nearly everything that covers the collection, interpretation, manipulation and presentation of data is considered statistics (Armitage, Berry, 2011). This paper discusses various statistical instruments, their assumptions and their uses. In the end, a conclusion is given.
Pearson’s Correlation Coefficient
Pearson’s correlation coefficient, r, is a statistical instrument that looks at the strength and the direction of the relationship between any two variables. It is more accurately known as the Person product-moment correlation coefficient (Taylor, Jeremy 2010). Simply put, it is a statistical instrument meant to weigh the relationship between two related variables. For example, it can be used to understand the link between the time one spends revising for a test and his or her eventual performance. It could also be utilized to look at whether there is a link between duration of unemployment and depression.
One can also think of the coefficient as the line of best fit between the collected for two variables.
Assumptions
Assumption 1:
When using the coefficient, the assumption that the data is normally distributed is made. This is because one cannot evaluate the important or statistical significance of the correlation between two variables without bivariate normality. However, this assumption is not easy to evaluate so a more appropriate and practical method is often used. The method entails evaluating the normality of the variables individually. This can be done by utilizing SPSS Statistics to conduct the Shapiro-Wilk test of normality.
Assumption 2:
For Pearson’s correlation coefficient to be accurate the data from the two variables has to be measured at ratio/ interval level. In essence, they have to be continuous (Taylor, Jeremy 2010). Some examples of this kind of data include weight measured in pounds or kilograms, test scores measure between 0 and 100, intelligence measured utilizing the IQ score, study time measured in minutes or hours and so on.
Range
Pearson’s coefficient for ranges from -1 to +1
Examples of uses
It is thought that Pearson’s coefficient can be applied to many different types of investigations and analyses. For instance, Chinese scientists have used it to investigate the genetic divergence between different types of rice in the country (Armitage, Berry 2011)
The objective of this particular objective was to determine the evolutionary propensity of the different types of rice. When the correlation was measured, it revealed a positive correlation of between 0.78 and 089. They rightfully concluded that the evolutionary potential was very high.
ANOVA
Overview
ANOVA is a statistical instrument that measures whether the results of a test, experiment or survey are significant. Whether they are important. Simply put, the ANOVA test is meant to help you decide whether or not to accept your null hypothesis (Armitage, Berry, 2011). Essentially, it tests different groups of data two see whether they are similar or different. There are several instances when the ANOVA test may come in handy (Wooldridge & Jeffrey, 2013). For example, when different types of therapies such as biofeedback, medication and counseling are tried for a group of psychiatric patients, the test can be used to see which therapy is more effective. Another example where the ANOVA test can be utilized is when a manufacturer used two different processes to make the same type of product (Taylor, Jeremy 2010). The test can be utilized in this case to find out which of the two products works better. Also, if university students from different campuses took the same test, examiners can use the ANOVA test to determine which campus outdid the rest.
Assumptions
· The groups being tested should have the same sample sizes
· The data is normally distributed or near normal distribution
· Population variances should be the same
· Samples should be independent
Range/uses
As mentioned before, the ANOVA test looks at whether the results of a particular study or test are significant. When doing the test, scientists often aim to answer two questions: (1) whether the variance/ difference between the groups is significant? (2) If yes, then how much variance is there between the groups?
Many tests often compare group SDs (Standard deviations), group medians, and group means (Armitage, Berry 2011). For group means, several types of measures are available including the Tukey HSD procedure. This procedure can reveal group differences by looking at plotting the means and pinpointing the cases where there is no overlap.
Example of uses
The ANOVA test is utilized to measure whether there is a significant difference between factors in response to some sort of stimuli (Wooldridge & Jeffrey, 2013). This has dozens of important applications. For example, the when a manufacturer uses two distinct procedures to make a bulb, he or she can use the test to determine which bulb is better. Another example, is when students from different countries do the same exam, ANOVA can be utilized to tell which group actually outperformed the other.
T-tests
Overview
The T-test is also a measure of difference/ variance between two groups. The t score is the ratio of the difference within a group and between the group and another group. The more the score, the bigger the variance between the two groups, while the smaller it is, the more similar the two groups are (Taylor, Jeremy 2010). For example, a T-test score of two is interpreted as meaning that the groups are two times more difference from each other than they are on their own. Furthermore, the higher the t value, the more likelihood that the results gathered can be repeated.
Assumptions
Several assumptions are made when conducting a T-test. The number one assumption is that repeated observations/ measures on the same sample do not qualify as independent measurements. For instance, if there were ten cases of malaria in a population sample of three, then we only have three independent observations (Taylor, Jeremy 2010).
The second assumption is that the data distribution is normal and that the data/ information are quantitative and lastly that the observations are independent from each other.
Uses
1. To measure the significance of multiple and partial correlation coefficient
2. To analyze the line of regression
3. To measure the significance between the population rank coefficient and the hypothetical rank coefficient
4. To measure the difference between the pre and post-stimuli mean of any sample
5. To measure the significance between the average of two different samples
6. To measure the significance between a population means an a hypothetical mean of a sample
OLS REGRESSION
Overview
OLS is an abbreviation for Ordinary Least Squares. So OLS Regression is the Ordinary Least Squares Regression (Taylor, Jeremy 2010). It is frequently used to measure linear regression. For a P number of variables, the Ordinary Least Squares Regression model is:
Y = ?0 + ?j=1..p ?jXj + ?
Where:
?0= Model intercept
Y = Dependent variable
X j = The jthexplanatory variable of the model (j= 1 to p)
e = The random error with expectation 0 and variance ?²
If there is n number of observations, the measure estimated value of the variable Y (dependent variable) for the ith observation can be arrived at by utilizing the formula below:
yi = ?0 + ?j=1..p ?jXij
The OLS regression model is equal to minimizing the total square differences between predicted and observed data. This minimization gives these estimators for the model:
[? = (X’DX)-1 X’ Dy ?² = 1/(W –p*) ?i=1..n wi(yi - yi)]

wi = ith observation weight
W = wi weights total
D = matrix diagonal whose weight is wi 
? = the ?iparameters estimators vector
X = the matrix of the explanatory variables whenpreceded by a vector of 1s
y = n values (observed) of the dependent variable vector
p* = number of explanatory variables (1 is added if the intercept is unfixed)
Assumptions
The Ordinary Least Square regression has two main assumptions.
The OLS regression assumption 1:
The first assumption is that the error terms follow a normal distribution. Basically the assumption implies that the errors not only follow a normal distribution but depend on the on the independent variables (Wooldridge & Jeffrey, 2013). Even though this assumption is not necessarily needed for the OLS regression model to be valid, it is important in cases where a scientist, mathematician or researcher would like to define extra finite-sample qualities. And while it is important for the error terms to follow a normal distribution, the Y dependent variable should not be normally distributed. This is the only way one can best define finite-sample qualities with the OLS model.
The OLS regression assumption 2:
Perhaps the most important assumption when using the Ordinary Least Square (OLS) regression is that the conditional mean is zero (Armitage, Berry 2011). That means that the mean of all the error terms in the OLS regression should have the value zero when taking into account the values of independent observations.
It can be explained statistically as Eleft( { varepsilon } right) =0E(?)=0 or more commonly as Eleft( { varepsilon }|{ X } right) =0E(??X)=0.
Others have explained this assumption in much more simpler terms by stating that the error terms are distribution has a mean of zero and is not dependent on the variables X'sX?s (independent variables). And therefore, there must be no relationship between the independent variables and the error terms.
Uses
OLS is often utilized to investigate process or analyze multiple real-life situations. For instance, OLS regression model can be used to investigate the case of a multinational company looking to identify the variables that may affect the number of products it sales (Armitage, Berry 2011). The OLS model can tell which factors are the most important. Organizations can also use the model internally for the same purpose. In the subject of econometrics, the OLS regression model is often utilized to approximate the parameter of a liner regression model.
Conclusion
Statistics is a broad discipline that requires whose main goal is to investigate sample populations and extrapolate to the wider population using that information (Taylor, Jeremy 2010). There are many different types of statistical instruments out there on the market that can be used to investigate data so as to interpret it in the best way possible. Some of the instruments discussed in this paper include Pearson’s correlation coefficient, ANOVA, the T-test, and the OLS regression model. The models investigate the extent of variance/ significance/ certainty in the data collected from sample populations. The significance can then tell us how much we should trust the conclusions or the interpretations made (Wooldridge & Jeffrey, 2013). However, for the conclusions we make from the instruments to be valid, there are some assumptions we have to make. The paper has listed some of the important assumptions. All these instruments/ methods of analysis have important uses in real-life situations.












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