correlation coefficients

Worksheet, Pearson coefficient of regression

Solves correlation... \[r=\frac{\sum_{}^{}(x-\bar{x})(y-\bar{y}) } { \sqrt{\sum_{}^{}(x-\bar{x})^2} \sqrt{\sum_{}^{}(y-\bar{y})^2} }\]

correlation coefficient calculator

Standard deviation calculator computes with step-by-step mean, media, mode, standard deviation, variance and coefficient of variation of sample data. The standard deviation or deviation is a concept that measures how spread out a set of numbers is from its mean.
Use correlation coefficient formula to compute correlation coefficient
correlation formula

Sample X

1 2 3 4 5

Sample Y

2 3 5 4 6

Step 1: Enter number separated by space

Enter each dataset separated by space from each sample X and Y

result

Hit the check mark to solve correlation coefficient of the data set

How The Calculator Works

Use this correlation coefficients calculator to calculate the correlation between two variables. Perfect for researchers, teachers, and others interested in correlative studies. Correlation coefficients are the measure of the strength of the linear relationship between two variables. The Pearson coefficient is the most commonly used. It ranges from -1 to 1, with -1 representing a perfect negative linear relationship, 0 representing no linear relationship and 1 representing a perfect positive linear relationship.

  • Enter datasets for variable X separated by SPACE

  • Enter datasets for variable Y separated by SPACE

  • Hit the equal sign to compute for coefficients

Understanding Pearson Correlation Coefficient

  • A positive correlation means that one variable increases as the other increases and vice versa. An example of a positive correlation is the relationship between height and weight. As a person's height increases, so does their weight. Conversely, a negative correlation means that one variable increases as the other decreases. An example of a negative correlation is the relationship between hours spent studying and grades. As hours spent studying decreases, grades tend to increase.

  • When interpreting a Pearson coefficient, it is important to consider the strength of the correlation. A correlation coefficient of -0.7 is considered to be a strong negative correlation, while a correlation coefficient of 0.2 is considered to be a weak positive correlation. Correlation coefficients closer to zero indicate a weaker relationship between the two variables.

  • It is important to note that correlation does not equal causation. Just because two variables are correlated, it doesn't necessarily mean that one causes the other. For example, a correlation between ice cream sales and the number of drownings suggests that eating ice cream causes people to drown. However, this correlation is actually caused by a third factor - warmer weather - which increases both ice cream sales and the number of people swimming.

    So, when interpreting correlation coefficients, it is important to remember to consider the strength of the correlation, as well as any potential confounding factors that may be influencing the correlation. They are a useful tool for understanding the strength of the linear relationship between two variables.

Pearson Correlation Coefficient Formula

correlation formula

How to Calculate the Pearson Correlation Coefficient

Let's say we have two variables, X and Y, and we have the following data points:

  • X: 1, 2, 3, 4, 5

  • Y: 2, 3, 5, 4, 6

To calculate the Pearson correlation coefficient, we can use the following formula:

  • r = (Σ((X - X̄)(Y - Ȳ))) / sqrt(Σ((X - X̄)^2) * Σ((Y - Ȳ)^2))

First, we need to calculate the means of X and Y:

  • X̄ = (1 + 2 + 3 + 4 + 5) / 5 = 3

  • Ȳ = (2 + 3 + 5 + 4 + 6) / 5 = 4

Next, we calculate the differences between each data point and the mean for both X and Y:

  • X - X̄: -2, -1, 0, 1, 2

  • Y - Ȳ: -2, -1, 1, 0, 2

Now, we need to calculate the sums of the squares of these differences:

  • Σ((X - X̄)^2) = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 10

  • Σ((Y - Ȳ)^2) = (-2)^2 + (-1)^2 + 1^2 + 0^2 + 2^2 = 10

Next, we calculate the products of the differences between X and X̄ and Y and Ȳ:

  • (X - X̄)(Y - Ȳ): (-2)(-2), (-1)(-1), (0)(1), (1)(0), (2)(2) = 4, 1, 0, 0, 4

Now, we sum up these products:

  • Σ((X - X̄)(Y - Ȳ)) = 4 + 1 + 0 + 0 + 4 = 9

Finally, we can substitute these values into the formula to calculate the Pearson correlation coefficient:

  • r = (9) / sqrt(10 * 10) = 9 / sqrt(100) = 9 / 10 = 0.9

Therefore, the Pearson correlation coefficient between X and Y is 0.9.

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