Contents

# Variance|Definition & Meaning

## Definition

The **variance **is the **standard deviation **squared and represents the **spread **of a given set of data points. Mathematically, it is the **average **of squared **differences **of the given data from the mean. Since the formula involves **sums **of squared **differences **in the numerator, variance is always **positive**, unlike standard deviation.

Figure 1: Graphical representation of variance.Â

The **variance **is determined as the **sum **of the **squared deviations **from the **mean **of each **data point **divided by the number of data points. The equation looks like this:

Variance = Î£(X – Î¼)^{2} / N

Where X is a single data point, is the data set’s **average**, and N represents the total **number **of **data points **in the set.

## Importance of Variance

When analyzing **data **distribution and forecasting upcoming **data points**, **variance **can serve as a useful tool. For instance, a significant **variance **in a group of stock prices may be a sign of high market volatility, but a little variance may be a sign of stability.Â

As values that **deviate **greatly from the **mean **are likely to be viewed as outliers, **variance** can also be used to spot outliers or **abnormalities **in a data set.

## Properties of Variance

The characteristics of variance consist of the following:

- Non-Negativity:
**Variance**is never**negative**, which means it can never be less than**zero**. Since the sum of**squared**deviations from the**mean**, which is always**positive**or**zero**, is used to determine**variance**, this is the case. - Zero for constant data: If there is no
**dispersion**in the**data**and all the values in a dataset are the same, the**variance**will be**zero**. - Units: Because
**variance**is expressed as the data’s**squared**units, it might be challenging to understand. The**standard deviation**, which is the**variance’s**square root, is frequently employed to make**variance**more understandable. - Sensitivity to extreme values:
**Variance**can be sensitive to**extreme**values and can occasionally be altered by their presence. - Linear transformations: When a
**linear transformation**is done to a**dataset**, the**variance**is altered by a nominal quantity. The variance will be multiplied by a**constant**, for instance, if a**dataset**is multiplied by that constant. - Additivity: The
**variance**of two independent**datasets**added together equals the**variance**of the independent**datasets**added separately. - Independence: The total of the
**variances**of the**independent**random variables in a**linear**combination is the**variance**of the linear combination. - Consistency: The
**sample variance**approaches the**population variance**as the sample size grows.

## Uses of Variance in Different Areas

**Variance **is useful in many different **applications**, including:

- Analysis of the distribution of data:
**Variance**, which analyzes the**dispersion**between the particular values in a**dataset**, makes it possible to better comprehend the**distribution**of data. - Finding outliers: Because values that
**differ**significantly from the**mean**are likely to be regarded as**outliers**, variance can be used to find outliers or**extreme values**in a**data set**. - Significance of differences:
**Variance**can be used to examine the**significance**of differences between two**groups**or to establish whether a**relationship**between**two**variables is true or the result of chance in**hypothesis**testing. - Regression analysis:
**Regression analysis**makes forecasts of**future**values based on the regression**equation**and makes use of**variance**to assess the**strength**of the link between**variables**. Variance can be used in regression analysis to assess the strength of the link between variables and to**forecast**future values using the regression equation. - Portfolio optimization:
**Variance**is applied in**portfolio optimization**in**finance**to assess the level of**risk**in an**investment**portfolio and to choose a portfolio that is both well-diversified and has a manageable**level**of risk. - Quality control:
**Variance**can be utilized in**quality control**in**engineering**and**manufacturing**to assess a process’**consistency**and pinpoint opportunities for development. - Experimental design:
**Variance**can also be used in**experimental**design in**scientific research**to ascertain the effects of various**treatments**or conditions on a**response**variable and to draw conclusions about population parameters. - Hypothesis testing:
**Variance**can greatly be useful in**hypothesis testing**as well as other**statistical**procedures, in addition to helping us understand how**data**are distributed. For instance,**variance**can be used in**hypothesis testing**to determine the importance of**differences**between two groups or to establish if a**correlation**between two variables is**causal**or**coincidental**.Â

## Types of Variance

**Variance **falls into **two **categories: **sample **variance and **population **variance. In contrast, to sample variance, which is an assessment of the **variance **in a **sample **of **data **collected from a population, **population **variance is a measure of the **variance **in the **entire **population of **data**. Population variance is calculated as follows:

Var(pop) = Î£(X – Î¼)^{2} / N

while the formula for sample variance can be written as:

Var(samp) = Î£(X – xÌ„)^{2} / (N-1)

## Variance and Standard Deviation

The concept of **standard deviation**, which is the **square root of the variance**, is similarly related to variance. Given that it is given in the same units as the data points, the standard deviation is a more understandable way to assess** spread**.**Â **

Figure 2: Formulas for standard deviation and variance.

The **square root** of the **variance** is used to determine the **standard deviation**, which is represented as follows:

Standard Deviation = âˆšVariance

The **range of values** that are most inclined to lie within a particular number of standard deviations from the **mean** can be determined using **standard deviation**. For instance, a normal distribution has data that falls roughly** 68% **of the time within **one standard deviation **of the mean and **95% **of the time within **two standard deviations.**

## Sensitivity of Variance

It is crucial to remember that **variance** can be susceptible to **outliers **and **extreme values**Â and that their presence can occasionally have an **impact** on variance. This may result in **erroneous interpretations **of the **data distribution **and **skewed predictions **in statistical models.Â

In order to address this problem, researchers frequently **change **the data to lessen the impact of outliers or use alternate measurements of dispersion, for instance, **the median **or **interquartile range**.

## Summary

- As a
**statistical measure**of the variation in numbers within a**collection of data**, a**variance**is a crucial tool in many domains for interpreting data distribution and making predictions.

Figure 3: Dispersion of data from the mean value.

- Instead of employing more
**extensive**mathematical techniques like grouping the data set’s numbers into**quartiles**, statistics use variance to discover how distinct numbers**correlate**to one another. - Variance takes into account that all
**departures**from the**mean**, regardless of their direction, are the same. The**squared deviations**, however, cannot add up to 0 and show that there is absolutely no variability in the provided data set. - Finding variance has the
**drawback**of giving combined weight to**extreme results**or numbers that**deviate**greatly from the**mean**. There is a possibility that**squaring**these numbers will**distort**the available data set. - The fact that
**variation**occasionally results from sophisticated**calculations**is another**drawback**.

## Examples Explaining the Concept of Variance

### Example 1

Consider the following heights: 715, 360, 120, 220, and 175 millimeters. Find the **variance**.

### Solution

**Mean** and **Variance **are related terms. Finding the mean is the **first stage**, which is done as follows:

Mean = ( 715 + 360 + 120 + 220 + 175) / 5 = 318

Therefore, the **median **is 318 mm.

Determining the **difference** between **each term **and **the mean value **will be the next step.

715 – 318 = 397

360 – 318 = 42

120 – 318 = -198

220 – 318 = -98

175 – 318 = -143

Calculate each **individual deviation **from the **mean**, **square **it, and then find the **average **once more to determine the variance.

The **variance** in this situation is, therefore:

= (397)^{2} + (42)^{2} + (-198)^{2} + (-98)^{2} + (-143)^{2} /5 = 228630/5

The **final answer** is Variance = 45726.

### Example 2

Calculate the **population variance** of the following data:

1.5, 2.6, 3.7, 4.8

### Solution

Let us calculate the **mean **of the data in the first step:

Mean = (1.5 + 2.6 + 3.7 + 4.8) / 4 = 3.15

Population Variance = Î£(X – Î¼)^{2} / N

Now, we will find the **difference **between each term in the **data **set and the mean value:

1.5 – 3.15 = -1.65

2.6 – 3.15 = -0.55

3.7 – 3.15 =Â 0.55

4.8 – 3.15 = 1.65

The final answer will be:

= (-1.65)^{2} + (-0.55)^{2} + (0.55)^{2} + (1.65)^{2} /4

= 6.05/4

= 1.51

*All the images/mathematical drawings were created using GeoGebra.*