*Root mean square is the square root of a mean square or it can be defined as the arithmetic mean of squares of group values. This is a special case of quadratic mean whose exponent value is two. Root mean square is also taken as the variable function based on the integral of squares of values that are instantaneous in a cycle.*

RMS is the square of function that defines the waveform in continuity. Also, the RMS of a periodic function is generally equal to RMS of a single function. RMS of a continuous function is generally calculated in approximate values by computing RMS of a sequence of equally spaced entities. Also, The RMS can be computed for different waveforms without using the calculus.

The other popular term associated with Root Mean Square is RMSE (Root Mean Square Error) predicted through an estimator or you could take help of a mode. It will tell the differences between the actual value and the calculated value.

\[\large X_{rms}=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}….x_{n}^{2}}{n}}\]

Where,

$x_{1},\; x_{2},\; x_{3}$ are observations

*n* is the total number of observations

**Question:** Calculate the root mean square of the following observations; 5, 4, 8, 1 ?

**Solution:**

Using the Root Menu Square formula: \[\large X_{rms}=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}….x_{n}^{2}}{n}}\]

\[\large Root\;Mean\;Square=\sqrt{\frac{5^{2}+4^{2}+8^{2}+1^{2}}{4}}\]

Root mean square = 5.14

Chi-square is a common term in non-parametric statistics.it is used to analyze data consist of people distributed across various categories. It is necessary to check either distribution is performed as per the expected values or it is different. There is extremely small chi-square test in statistics that will check either observed data suits your expectations well or not. If the value is small then data fits your expectations well otherwise in case of large value you need to reject the null hypothesis.

Chi-square formula is the perfect way how can you show the relationship between two categorical variables. The two types of categoric variables are numerical variables and the non-numeric variables. And the final values will be the difference between observed frequency and the expected frequency. The common applications of chi square distribution are –

The chi-square test is associated with the non-parametric test and it can be utilized for nominal data too. This is used in advanced mathematics for tabular association and analysis. One major application is the hypothesis that will check either two populations are same or different in terms of characteristics and behavior.

Also, you can study two random samples in terms of different parameters. It can also be used to analyze a particular distribution by comparing the observed data with expected values. One of the drawbacks of chi-square test is that it does not allow the calculation of confidence intervals so the size of a sample is not available readily.

\[\LARGE X^{2}=\sum \frac{(O-E)^{2}}{E}\]

Where,

**O** = Observed frequency

**E** = Expected frequency

Σ = Summation

X^{2} = Chi Square value

In elementary mathematics, the completing a square is generally applied for the computation of quadrilateral polynomials. Completing the square formula is also used to derive the quadratic formulas.

\[\LARGE ax^{2}+bx+c\Rightarrow (x+p)^{2}+constant\]

Since the degree of this polynomial is two, so it has two roots or solutions. There are a plenty of methods to solve the roots of a quadratic equation and one of these popular techniques is completing the square method.