What are Periodicity and Variance

In the fields of probability and statistics, the concept of “variance” refers to the expected value of the squared deviation of a random variable from its mean value. In a more colloquial sense, the term “variance” refers to an estimation of the degree to which a random group of numbers deviates from their mean value.

The square of the standard deviation, which is another important tool, is equal to the value of variance, which is another important tool.

σ², s², and var(X) are all symbols that can be used to denote variation.

Periodicity

A function is said to exhibit periodicity if it has a tendency to repeat itself in a regular pattern at intervals that are previously determined. Periodicity is a property shared by all trigonometric functions. In order for a function f(x) to be considered periodic, it is necessary for there to be a repeating pattern of values for f(x) across the entirety of the function’s domain. When it comes to determining the value of f(x), knowing the periodicity of a function is useful because, once the period of the function has been determined, one can determine the value of f(x) at any moment. Periodicity is determined by a function’s period, which is the inverse of the frequency of that function and can also be expressed as the amount of time that passes between cycles (in the context of trigonometric functions, the period is described in terms of radians per cycle).

Periodic Function

A function that iterates itself at predetermined time intervals is said to have a periodic function. The length of time during which a function is repeated is known as its period, and it is an essential quality of periodic functions that helps characterise a function. The equation for the periodic function y = f(x) that has a period of P can be written as f(X + P) = f(X).

Several Important Functions of Periodic Systems

The following is a selection of the more advanced periodic functions that there is more room to investigate.

Euler’s Formula: The cosine and sine functions are examples of periodic functions. The formula for complex numbers, eix = Coskx + iSinkx, is composed of these two functions. In this case, both of these functions are periodic, and Euler’s formula also represents a periodic function; its period is denoted by the notation 2k.

Jacobi Elliptic Functions: The graph of these functions has the shape of an ellipse, as opposed to a circle, which is the shape that is most commonly seen for trigonometric functions. These elliptical shapes appear when two factors, such as the amplitude and speed of a moving body or the temperature and viscosity of the substance, are involved simultaneously. For example, the speed and amplitude of a moving body can produce the same result. In the process of describing the swinging motion of a pendulum, these functions are frequently applied.

Fourier Series: The Fourier series can be thought of as a complex periodic function that is formed by the superposition of a number of different periodic wave function series. In most cases, it is made up of sine and cosine functions, and the summation of these wave functions is determined by allocating respective weight components to respective series. The Fourier series is useful for the modelling of heatwaves, vibration analysis, quantum mechanics, electrical engineering, signal processing, and image processing, among other areas of study and application.

Variance

The variance is a measurement that shows how different the data points are from the average. According to Layman, a variance is a measurement of how far a set of data (numbers) are spread out from their mean (average) value.

The concept of finding the expected difference of deviation from the actual value is referred to as variance. As a result, variance is determined by the standard deviation of the data set that is being used.

If the value of variance is low or at its minimum, then the data is less scattered from its mean. On the other hand, if the value of variance is high, then the data is more scattered from its mean. As a result of this, it is referred to as a measure of the dispersion of data from the mean.

The formula for variance, which can be used for figuring out answers to problems, is as follows:

Var (X) = E[( X –μ)²]

To put this another way, we may say that the expectation of the squared departure of a random set of data from its mean value constitutes the variance of the data. Here,

X = Random variable

“µ” is equal to E(X) so the above equation may also be expressed as,

Var(X) = E[(X – E(X))²]

Var(X) = E[ X² -2X E(X) +(E(X))²]

Var(X) = E(X²) -2 E(X) E(X) + (E(X))²

Var(X) = E(X²) – (E(X))²

There are situations in which the covariance of the random variable itself is used in place of the variable’s actual variance. Symbolically,

Var(X) = Cov(X, X)

Formula

As is already common knowledge, the variance can be expressed as the square of the standard deviation, or more simply put as

Variance = (Standard deviation)²= σ²

The population standard deviation, denoted by the symbol = √(Σ(X-μ )² )N

Sample standard deviation s =√(Σ(X-x)² ) / n-1

Where X, or x, Represents the Total Number of Observations

μ is the population average of all the Values.

n equals the number of individual observations made in the dataset.

x is the average value of the samples, and N is the total number of values in the population.

Properties

The following characteristics are associated with the random variable known as X’s variance, also written as var(X).

The expression Var(X + C) = Var(X) is true when C is a constant.

Var(CX) = C².

Where C is a constant, we have Var(X).

Var(aX + b) = a².

The function Var(X), where a and b are fixed values.

If X₁, X₂,……., Xn are n independent random variables, then Var(X₁+ X₂ +……+ Xn) = Var(X₁) + Var(X₂) +……..+Var (X_n).

Conclusion

In the field of statistics, the variance is a measure of dispersion that is employed to evaluate how accurately the mean summarises a complete body of information. An organisation may be more proactive in attaining their business aims with the assistance of variance analysis. This analysis also helps in detecting and reducing any potential risks, which ultimately helps to develop trust among the members of the team to achieve what was planned. For example, the greater the variance, the greater the range of values that can be found inside the collection. Data scientists can use this information to draw the conclusion that the mean may not be as accurate a representation of the set as it would be if the set had a lower variance.