An instructor in a statistics course set a final examination...

An instructor in a statistics course set a final examination and also required the students to do a data analysis project. For a random sample of 10 students, the scores obtained are shown in the table. Find the sample correlation between the examination and project scores. $$ \begin{array}{lcccccccccc} \hline \text { Examination } & 81 & 62 & 74 & 78 & 93 & 69 & 72 & 83 & 90 & 84 \\ \hline \text { Project } & 76 & 71 & 69 & 76 & 87 & 62 & 80 & 75 & 92 & 79 \\ \hline \end{array} $$

An instructor in a statistics course set a final examination and also required the students to do a data analysis project. For a random sample of 10 students, the scores obtained are shown in the table. Find the sample correlation between the examination and project scores.
$$
\begin{array}{lcccccccccc}
\hline \text { Examination } & 81 & 62 & 74 & 78 & 93 & 69 & 72 & 83 & 90 & 84 \\
\hline \text { Project } & 76 & 71 & 69 & 76 & 87 & 62 & 80 & 75 & 92 & 79 \\
\hline
\end{array}
$$

Key Concepts

Bivariate Data Analysis

Bivariate data analysis involves the examination of two variables simultaneously to understand relationships, associations, or correlations between them. Techniques in bivariate analysis, such as calculating the Pearson correlation coefficient, help in determining whether a linear relationship exists and in assessing the strength and direction of that relationship.

Standard Deviation

Standard deviation is a measure of the dispersion or spread of a set of values around their mean. It quantifies the amount of variation in a dataset, and when used in the formula for correlation, it serves to normalize the covariance, ensuring that the resulting correlation coefficient is dimensionless and lies between -1 and 1.

Pearson Correlation Coefficient

This is a statistical measure that quantifies the strength and direction of the linear relationship between two continuous variables in a sample. It is calculated as the ratio of the covariance of the variables to the product of their standard deviations. Its value ranges between -1 and 1, where values closer to 1 or -1 indicate a stronger linear relationship and values near 0 indicate a weak or no linear relationship.

Covariance

Covariance measures how two variables change together. A positive covariance indicates that as one variable increases, the other tends to increase as well, while a negative covariance indicates that as one variable increases, the other tends to decrease. In the calculation of correlation, the covariance is normalized by the product of the standard deviations of the two variables, which allows for comparison across different scales.

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