The Sxy value can be calculated using the formula,

**S _{xy} = Σ (X_{i }– X̄)(Y_{i} – Ȳ).**

The quantity Sxy in statistics is used to calculate the regression coefficient and correlation coefficient. In this article, we explain how to calculate Sxy and give a shortcut method for the calculation.

**Sxy Formula and Calculation:**

Suppose that you are given two values of two variables X and Y. The subscript xy in the quantity S_{xy} refers to the fact that the sum is taken over the possible values of x and y.

The quantity S_{xy} can be calculated using the below formula.

**S _{xy} = Σ (X_{i }– X̄)(Y_{i} – Ȳ).**

Here** X̄** refers to the mean of the X values and **Ȳ** refers to the mean of the Y values. We now give an example and explain how the quantity can be calculated.

**Example:**

Consider the following set of 5 data values for the variables X and Y,

X | Y |

1 | 6 |

2 | 7 |

3 | 8 |

4 | 9 |

5 | 10 |

**Step 1**: We first calculate the mean for the data values as follows,

X | Y |

1 | 6 |

2 | 7 |

3 | 8 |

4 | 9 |

5 | 10 |

ΣX_{i} = 15 | ΣY_{i} = 40 |

The mean is calculated using the formula,

X̄ = ΣX_{i}/n = 15/5 = 3.

Ȳ = ΣY_{i}/n = 40/5 = 8.

**Step 2**: We now calculate the quantity S_{xy} using the formula. Multiply the third and fourth columns and then sum up the values of the last column below,

X | Y | (X_{i }– X̄) | (Y_{i} – Ȳ) | (X_{i }– X̄)(Y_{i} – Ȳ) |

1 | 6 | 1 – 3 = -2 | 6 – 8 = -2 | -2 x -2 = 4 |

2 | 7 | -1 | -1 | 1 |

3 | 8 | 0 | 0 | 0 |

4 | 9 | 1 | 1 | 1 |

5 | 10 | 2 | 2 | 4 |

ΣX_{i} = 15 | ΣY_{i} = 40 | Σ (X_{i }– X̄)(Y_{i} – Ȳ) = 10 |

S_{xy} = Σ (X_{i }– X̄)(Y_{i} – Ȳ) = 10.

**Shortcut Method for Calculating Sxy:**

We can also calculate Sxy using a shortcut. We can use the alternative formula that is given below.

**S _{xy }= (∑ X_{i}Y_{i}) – n*X̅Ȳ.**

So in the above example, we calculate only one extra column,

X | Y | xy |

1 | 6 | 6 |

2 | 7 | 14 |

3 | 8 | 24 |

4 | 9 | 36 |

5 | 10 | 50 |

ΣX_{i} = 15 | ΣY_{i} = 40 | ∑ X_{i}Y_{i} = 130 |

S_{xy }= (∑ X_{i}Y_{i}) – n*X̅Ȳ = 130 – 5*3*8 = 130 – 120 = 10.

**Frequently Asked Questions:**

**What does Sxy Mean in Statistics and Maths?**

By Sxy, we mean the sum of the product of deviations of X and y values from their mean/average values.

We first calculate the differences between the x and y values and then multiply them. Then we sum up all of these product values in order to obtain the desired quantity.

**Is Sxy a correlation?**

Although Sxy is itself not a correlation, it can indeed be used to calculate the correlation coefficient. The correlation coefficient can be calculated using the formula,

Correlation = Sxy/√(Sxx*Syy).