# How to use the COVARIANCE.P function

**What is the COVARIANCE.P function?**

The COVARIANCE.P function calculates the covariance in two different data sets.

### Table of Contents

## 1. Introduction

**What is covariance?**

Covariance describes if two variables are connected meaning they rise together or if one decreases as the other increases. In other words, covariance measures how two random variables or datasets vary together.

Covariance is the average of the products of deviations for each pair in two different datasets. The covariance is positive if greater values in the first data set correspond to greater values in the second data set. The covariance is negative if greater values in the first data set correspond to smaller values in the second data set.

**What is the average of the products of deviations for each pair in two different datasets?**

The covariance between two datasets is computed by taking each data point, finding its deviation from its respective dataset mean by subtracting the mean, multiplying the two datasets' deviations together for each pair, and averaging these cross-products of deviations.

**What is deviation?**

In statistics, deviation is a measure of how far each value in a data set lies from the mean (the average of all values). A high deviation means that the values are spread out widely, while a low deviation means that they are clustered closely around the mean.

**What is the mean?**

It is also known as the average. It is calculated by adding up all the values in the data set and dividing by the number of values.

For example, if you have a data set of 5, 7, 9, 11, and 13, the mean is (5 + 7 + 9 + 11 + 13) / 5 = 9.

**How to interpret covariance?**

A positive covariance means that the variables tend to increase or decrease together indicating a positive linear relationship. The top left chart above shows variables with a positive covariance.

A negative covariance means that the variables tend to move in opposite directions indicating a negative linear relationship. The bottom left chart above shows variables with a negative covariance.

A zero covariance means that the variables are independent and have no linear relationship. The top and bottom right charts shows data with no linear relationship, the covariance is close to zero.

However, covariance is not a standardized measure and it depends on the scale and units of the variables. It is not easy to compare the covariances of different pairs of variables or interpret the strength of the relationship. A more common and useful measure of linear relationship is the correlation coefficient, which is the normalized version of covariance.

**How to calculate normalized version of covariance?**

To calculate the normalized version of covariance, which is also known as the correlation coefficient, you need to divide the covariance by the product of the standard deviations of the two variables. The standard deviation is a measure of how much the values in a data set deviate from the mean.

The correlation coefficient tells you also how much related the pairs are, this is not the case with the measure of covariance. The image above shows two charts, the data set in the right chart is identical to the left chart except they are ten times larger. Covariance is much larger for the right chart.

See the CORREL function for more.

## 2. COVARIANCE.P Function Syntax

COVARIANCE.P(*array1*, *array2*)

## 3. COVARIANCE.P Function Arguments

array1 |
Required. The first data set. |

array2 |
Required. The second data set. |

## 4. COVARIANCE.P Function Example 1

**You have data on the monthly closing prices of Microsoft and the monthly values of a market index (S&P 500) over a period of three years. Calculate the population covariance between the stock prices and the market index to determine if there is a relationship between the two?**

Formula in cell C18:

If the S&P500 data represents index values and the Microsoft data represents stock prices, the covariance value of 18346 would be in the units of

index value × stock price.

It's essential to understand the units of the input data to interpret the covariance value correctly.

The sign of the covariance value indicates the direction of the relationship between the two variables. A positive covariance value (18346 in this case) suggests that S&P500 and Microsoft tend to move in the same direction. When S&P500 increases (decreases), Microsoft's stock price also tends to increase (decrease).

The size of the covariance value alone is not very helpful in analyzing the strength of the relationship. A larger covariance value doesn't necessarily imply a stronger relationship. It could be influenced by the units or scales of the input data.

Use the CORREL function to analyze how strong the correlation between the S&P500 and Microsoft is. The correlation coefficient is calculated like this:

COVARIANCE.P(X, Y) / (STDEV.P(X) * STDEV.P(Y)) however, the CORREL function does this for you.

## 5. COVARIANCE.P Function not working

Text, logical or empty values are ignored, however, 0 (zeros) are included.

The COVARIANCE.P function returns

- #N/A error value if the number of data points in array1 and array2 is not equal.
- #DIV/0! error value if either array1 or array2 is empty.

## 6. How is the COVARIANCE.P Function calculated

To calculate the covariance for a population follow these steps:

- Calculate the mean of group of numbers named:

x="Temp"

y="Icecream"

For example:

Mean of X = x̄ is calculated in cell C10

Mean of Y = ȳ is calculated in cell D10 - For each data point x
_{i}and y_{i}calculate the deviations from the mean.

Deviation of x_{i}= x_{i}- x̄ are calculated in cells E3:E9

Deviation of y_{i}= y_{i}- ȳ are calculated in cells F3:F9 - Multiply the deviations between each data point pair to get their products.

For each pair: (x_{i}- x̄) * (y_{i}- ȳ) are calculated in cells G3:G9 - Sum all the deviation products.

S = Σ(x_{i}- x̄)(y_{i}- ȳ) calculated in cell G10 - The covariance is the sum of products divided by the number of samples.

The equation for COVARIANCE.P is:

COVARIANCE.P(x,y) = (Σ(x_{i} - x̄)(y_{i} - ȳ))/n

x̄ is the sample means AVERAGE(*array1*)

ȳ is the sample means AVERAGE(*array2*)

n is the sample size.

### Functions in 'Statistical' category

The COVARIANCE.P function function is one of 73 functions in the 'Statistical' category.

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