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Correlation coefficient

Variance & correlation

Japanese school year: Math I

What you learn

You will learn how to quantify the linear relationship between two variables using scatter plots and the correlation coefficient. It is widely used to analyze whether two factors, like height and weight, are related. Prior knowledge of calculating variance and standard deviation will help you follow this topic smoothly.

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Key points

Use this formula to examine whether two variables xx and yy tend to increase or decrease together (covariance sxys_{xy}). It averages the products of deviations from their respective means xˉ\bar{x} and yˉ\bar{y} across nn pairs.

sxy=1n∑i=1n(xi−xˉ)(yi−yˉ)s_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})

Use this formula to measure the strength of the linear relationship between two variables (correlation coefficient rr). Dividing covariance sxys_{xy} by the standard deviations sxs_x and sys_y removes units for fair comparison.

r=sxysxsyr = \frac{s_{xy}}{s_x s_y}

This inequality shows the range of the correlation coefficient rr, which always lies between -1 and 1. Values near 1 indicate strong positive correlation, while values near -1 indicate strong negative correlation.

−1≤r≤1-1 \le r \le 1

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