# Sample covariance calculator:

Since there are many free programmes available online to calculate **sample covariance calculator**, you can find a **sample covariance calculator** there. To get the most accurate results, select the greatest **population covariance calculator** that includes **probability** by searching online.

## Introduction of Covariance Calculator:

The assessment of how two random variables **(X, Y)** relate to one another is known as covariance. An online covariance calculator offers a way to quickly learn and compute your values.

These variables are may be **positive** or **negative** numbers and denoted by:

$$\text{Cov(X, Y)}$$.

While the **negative** number denotes a **negative relationship**, the **positive value** denotes a **positive relationship**.

Each of the two variables tends to move in the same direction when the **covariance** value is positive, while the opposite is true when the covariance value is negative.

you can learn more about variance and its calculations from here.

## Covariance Formula to calculate Sample covariance calculator?

The **covariance formula**, used in **statistics** and **probability**, determines the sample **covariance** between two randomly varying variables, **X and Y**. The **same covariance formula** is used by a **sample covariance calculator** online to compute results. The following is the **covariance formula**:

Calculating the covariance between two variables using a formula:

$$Cov (X,Y) =$$

$$\sum_{i=1}^n (X - \overline X)(Y - \overline Y)$$

cov (X,Y) = Covariance between X and Y

x and y = components of X and Y

$$\overline x \; and \; \overline y =\;mean\; of \; X \; and \;Y $$

n = number of members

This covariance formula helps **online covariance calculator with probability** to find accurate results.

## Covariance calculator joint probability:

You may find the **covariance** between two random variables based on their **joint probability** distribution using a **covariance calculator** for **joint probability**. The joint probability distribution explains how the odds of certain outcomes are spread out across the various possible pairings of the values of the two variables.

The formula for calculate the joint probability is:

Where:

- \(X\) and \(Y\) are two random variables.

- \(\mu_x\) and \(\mu_y\) are the means (expected values) of \(X\) and \(Y\), respectively.

- \(P(X = x, Y = y)\) is the joint probability that \(X = x\) and \(Y = y\).

**Example calculation:**

- Calculate the means (\(\mu_x\) and \(\mu_y\)) of random variables \(X\) and \(Y\).
- Create a table of all possible values of \(X\) and \(Y\), along with their corresponding joint probabilities.
- For each combination of \(x\) and \(y\), calculate the product of deviations from the means: \((x - \mu_x)(y - \mu_y)\), and then multiply it by the joint probability \(P(X = x, Y = y)\).
- Sum up all the products calculated in step 3 to get the covariance.

Keep in mind that this approach is suitable for discrete random variables. For continuous random variables, you would integrate over the joint probability density function.

## Diffrence between Covariance and Correlation:

Aspect | Covariance | Correlation |
---|---|---|

Definition | Measures how two variables change together. | Determine the direction and strength of the linear relationship between two variables. |

Range | Can range from negative infinity to positive infinity. | Range between -1 and 1. |

Units | In the units of the product of the two variables. | Unitless, as it is a normalized measure. |

Interpretation | When covariance is positive, it means that variables tend to rise together, when it's negative, it means that one variable rises while the other falls, and when covariance is close to zero, it means there isn't much of a linear relationship. | Strong linear relationships are indicated by correlations that are close to 1, strong linear relationships are indicated by correlations that are close to -1, and little or no linear relationships are shown by correlations that are close to 0. |

by Scale | Sensitive to changes in scale (units) of variables. | Not affected by changes in scale; it's a relative measure. |

Standardization | Not standardized; value can vary widely based on the scale of the variables. | Standardized measure, ranging from -1 to 1, which makes it easier to compare relationships. |

Calculation | Requires calculating means of both variables. | Requires calculating means and standard deviations of both variables. |

Linearity Assumption | Measures both linear and non-linear relationships. | Primarily measures linear relationships; non-linear relationships may not be well represented. |

Purpose | Used in portfolio analysis, variance-covariance matrix, risk assessment. | Used to assess the strength and direction of linear relationships in statistics, finance, and other fields. |

## How to calculate Covariance Equation?

We will learn how to calculate **sample covariance equations** from this example. To get the covariance for this collection of four data points, let's move on to an example.

The covariance between two variables \(X\) and \(Y\) is calculated using the following formula:

**Where:**

- \( n \) is the number of data points (observations) in the dataset.
- \( x_i \) represents the \(i\)th value of variable \(X\).
- \( y_i \) represents the \(i\)th value of variable \(Y\).
- \( \bar{x} \) is the mean (average) of variable \(X\).
- \( \bar{y} \) is the mean (average) of variable \(Y\).

**To calculate the covariance:**

- Calculate the mean (\( \bar{x} \)) of variable \(X\) and the mean (\( \bar{y} \)) of variable \(Y\).
- For each data point, subtract the mean of \(X\) (\( \bar{x} \)) from the corresponding \(x\) value, and subtract the mean of \(Y\) (\( \bar{y} \)) from the corresponding \(y\) value.
- Multiply the differences obtained in step 2 for each data point: \( (x_i - \bar{x})(y_i - \bar{y}) \).
- Sum up all the products calculated in step 3.
- Divide the sum from step 4 by \( n-1 \) to get the covariance.

It's important to interpret the **covariance value** in comparison to the scales of the two variables and in the context of the problem you are analyzing.

## How to use Covariance Calculator?

The **population covariance calculator** calculates the statistical correlation and measures the difference between the sample means of the two population data sets **(x, y)**. Since their values can change, the variance of one variable is equal to the variance of the other.

Students in high school can use the **population covariance calculator** to aid them with covariance difficulties. If a learner is unable to calculate covariance on their own, they should try using our sample covariance calculator to ascertain the linear relationship between two variables.