Category: Maths Notes PPT

Quadratic Equation: Definition

The simple definition of a quadratic equation is the polynomial equation whose highest order is two. It commonly gets expressed as ax² + bx + c = 0. In which, x is the unknown variable, and a, b, c are the constant terms. Also, note that ‘a’ is never equal to 0; otherwise, the equation becomes a linear one. 

Since the quadratic has a single unknown term or variable, it also gets referred to as a univariate. The power of the variable x always has to be a non-negative integer. Thus, it becomes a polynomial equation with the highest degree as two. 

The solution for the equation is the values of x, and they get called as zeros. They are the final solutions which satisfy the equation. When it comes to quadratics, there are two zeros or roots of the equation. When you insert the value of x in the L.H.S. of the equation, then you get a zero. That’s why they get called as zero.    

There are two fundamental concepts to solve a quadratic equation: 1. Formula method, and 2. Factorization method. These are the quickest methods to solve any quadratic equation example. You can learn both of these methods, as follows. 

Solving Quadratics by Formula

By learning the quadratic equation formula, you can solve any quadratic equation quickly. If the quadratic equation looks like ax² + bx + c = 0, then below is the formula you need to apply.

The signs (+/–) in the formula indicate that you obtain two values/solutions for the x. 

Examples of the Quadratic Equations :

Below are the examples for quadratic equations of the form: ax² + bx + c = 0.

• x² –x – 7 = 0

• 4x² – 2x – 9 = 0

• –x² + 2x + = 0

Below are the examples for quadratic equations where ‘c’ or a constant term is absent. 

• -x² – 6x = 0

• x² + 4x = 0

• -14x² + 9x = 0

Below are the examples for quadratic equations where ‘bx’ or a linear coefficient is absent.

• x² – 14 = 0

• 5x² + 54 = 0

• -x² – 7 = 0

Below are the examples for quadratic equations in factored form. 

• (x – 6)(x + 1) = 0 (after solving, you get x² – 5x – 6 = 0)

• (2x+3)(3x – 2) = 0 (after solving, you get 6x² + 5x – 60)

• (x – 4)(x + 2) = 0 (after solving, you get x² – 2x – 8 = 0)

Solving Quadratics by Factoring

Apart from the quadratic equation formula, factorization is another method of obtaining solutions for quadratic equations. Below are steps to find the solution of the quadratics by factoring.

• You begin with an equation in the form of ax² + bx + c = 0. 

• Then, you factor the L.H.S. of the equation while assuming zero on the R.H.S. of the equation. 

• By assigning each factor to zero, you can solve the equation to obtain the values of x.

When the main coefficient is not equal to zero, then have to arrange the factors in a way as below. 

Consider an equation: 2x² – x – 6 = 0

(2x + 3) (x – 2) = 0

2x + 3 = 0

X = -[frac{3}{2}]

In the end, you get: X = 2. 

Solved Examples

Using quadratic equations, you can solve word problems, typical equations, which involve determining the speed, area, etc. Below you can find solved quadratic equation example to help you understand the topic even better.

Question 1: Find the value of x: 27x² − 12 = 0

 A) 2/3 B) ± 2/3 C) Ambiguous  

Answer : Here, a = 27, b = 0 and c = -12. 

Now, by putting the values in the quadratic equation formula, you get:

x = [frac{-0 pm sqrt{0^{2} – 4(27)(-12)}}{2(27)}]

x = ± [sqrt{frac{4}{9}}]

Finally, x = ± [frac{2}{3}]. So, the correct option is B. 

Question 2: The area of a rectangle is 336 cm². Its length is four more than twice its width. Find its actual width.

Answer: Consider the width of a rectangle as ‘x.’ 

From the given data, length = (2x + 4) cm

As you know, Area of rectangle = Length x Width

Now, we get the equation as x(2x + 4) = 336

By further solving, 2x² + 4x – 336 = 0

x² + 2x – 168 = 0

x² + 14x – 12x – 168 = 0

x (x + 14) – 12 (x + 14) = 0

(x + 14) (x – 12) = 0

x = -14, x = 12

Since a measurement cannot be negative; the width of the rectangle is, x = 12 cm.

The Range

In mathematics, a range is a difference between the lowest and highest values of a numeral. In {7, 15, 4, 6, 9} the lowest value is 4, and the highest is 15, thus the range is 15 − 4 = 11. The range can also imply all the values of the output of a function. Moreover, when you start studying functions in mathematics, you’ll encounter a second definition of range. To better understand range, it aids to think of functions as tiny math machines.

Range of a Function

Talking about the range of a function definition, it is the set of outputs the function accomplishes when it is pertained to its whole set of outputs. In the function machine metaphor, the range is the set of items that arise out of the machine when you insert in all the inputs.

For instance, when we apply the function notation f: R→R, we imply that f is a function →from the real numbers →to the real numbers. By this notation, we are aware that the domain (set of all inputs) of ‘f’ is the set of all possible inputs (the codomain) and as well the set of all real numbers.

But, without having to know the function f, we will be unable to identify what its outputs are further cannot even determine what its range is. All we know is that the range should be a subset of the codomain, so the range should be a subset (likely to be the whole set) of the real numbers. It is possible objects are available in the subset of codomain for which there are no inputs and for which the function will output that object.

For instance, we could describe a function f: R→R as f(x) =x2. Seeing that f(x) will invariably be non-negative, the number −3 is in the codomain set of f, but it is not in the range, since there is no input of x for which f(x) =−3. For this f, the codomain is the set of all real numbers whereas the range is the set of non-negative real numbers.

Domain and Codomain in Range

The set of values we can insert into the math machine are known as the domain (another very important concept in the range). The set of possible outcomes, once we crank those values via the math machine, is known as the ​co domain​. And the set of actual outputs or outcomes we obtain is called the range​.

Interquartile Range

The Interquartile Range also known as IQR, defines the mid ( 50%) of values when arranged from lowest to greatest in the data set. In order to determine the interquartile range (IQR), we need to ​first find the median (middle value) of the lower and upper half of the set of data. These values are assigned as quartile 1 (Q1) and quartile 3 (Q3). The IQR is thus the difference between Q3 and Q1.

Solved Examples

Example:​

Think that you happen to view your math’s teacher’s notebook, and you snuck the peek so far that you saw the students’ grade percentages in class are {91, 84, 37, 53, 52, 88, 46, 62}. Now, you need to find out the range of this data set or we can say the range of the students’ grades?

Solution:

First, we need to determine the highest as well as the lowest value of the data set i.e

The highest data point = 91

The lowest data point= 37

Next, subtract the lowest value from the highest value determined:

91 – 37 = 54

Thus, the range of this specific data set is 54 percentage points.

Example:

Mr Alex drove through 8 southern states on his summer vacation. Fuel prices varied from state to state he travelled. Calculate the range of fuel prices?

Rs. 2.79, Rs. 0.61, Rs. 2.96, Rs. 3.09, Rs. 1.64, Rs. 2.25, Rs. 3.73, Rs. 1.67

Solution: 

Arranging the data from least to greatest, we obtain,

0.61, 1.64, 1.67, 2.25, 2.96, 2.79, 3.09, 3.73

highest – lowest = 3.73 – 0.61 = ＄0.48

Answer: The range of fuel prices is Rs. 3.12

Fun Facts

While finding the range, curly brackets are commonly used to enclose a set of data, so you are aware everything inside the curly brackets belongs together.

Reciprocal and Division of fractions are different from each other. When two terms, such that the second is the inverse of first, gives a product of 1 upon multiplication, then the fractions are called reciprocal or multiplicative inverse of each other. It can be achieved by interchanging the numerator and denominator of the fraction. If a term is x/y, then its reciprocal will be y/x. A fraction is a numerical quantity that represents a part of the whole number.

Dividing fractions requires inverting the divisor (reciprocal of the divisor) and then follow the steps of multiplication. The term x/y, when divided by a non-zero fraction p/q, then the expression looks like:

x/y ÷ p/q = x/y ÷ q/p

Division of fractions involves multiple steps.

Parts of Fraction

The parts of a fraction are:

• Numerator: The numerator is the number which is on the top of the line. It shows how many equal parts of the whole or collection is taken.

• Denominator: The number below the line is the denominator. It shows the total divisible number of equal parts or the total number of equal parts which they are in a collection.

Types of Fraction

There are three different types of fractions:

Example: 4/5, 1/6, 7/9, 3/7, etc.

Example: 5/2, 7/3, 8/5, 5,3, etc.

Example: 4²/₃, 3¹/₂, etc.

Reciprocal of Fractions

Interchanging or swapping the numerator or denominator with each other gives us the reciprocal of the fraction. Like for example, the reciprocal of 1/2 is 2/1, or that of 4 is 1/4.

In order to obtain a reciprocal from a mixed fraction, it must be converted to an improper fraction, and then the numerator and denominator must be swapped. For example, to find the reciprocal of 4²/₃, it is first converted into an improper fraction.

4²/₃ =14/3[improper fraction]

14/3 = 3/14

Therefore, the reciprocal of 4²/₃ is 3/14.

The product of a fraction and its reciprocal is always 1, as it is nothing but its inverse.

Division of Fractions

Division of fractions involves certain rules and follows multiple steps. To perform Division, we have to multiply the first fraction with the reciprocal of the second. Division involves some steps to be followed:

• Step 1: Changing the division sign (÷) to the multiplication sign (×).

• Step 2: If we change the sign to multiplication, we also have to write the reciprocal of the second term or fraction.

• Step 3: Multiplying the fractions and simplifying the result.

Here, is an example of Division of fractions:

15³/₇ ÷ 1²³/₄₉

First, we have to change the mixed  to an improper fraction.

= 108/ 7 ÷ 72/49.

Step 1: Changing the sign to multiplication from division and writing the reciprocal of the second term [72/49 = 49/72]

 = 108/7 × 49/72

Step 2: Multiplying the first with the reciprocal of the second fraction.

= (108 × 49)/ (7 × 72)

= (3 × 7)/ (1 × 2)

= 21/2

Step 3: Getting the simplified result of the expression.

Division of fractions is the multiplication of fractions by just changing the second fraction to its reciprocal.

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Solved Example:

1. 5/9 ÷ 2/3

[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 2/3 becomes 3/2.]

= 5/9 × 3/2 

 [Step II: Multiplying the first fraction by the reciprocal of the second: (3 × 5)/(2 × 9)]

= 5/6 

[Step III: It is the simplified expression, hence no further simplifications].

2. 4/9 ÷ 2/3

[Step I: Turning over the second fraction upside-down (it becomes a reciprocal): 2/3 becomes 3/2.]

= 4/9 × 3/2

[Step II: Multiplying the first fraction by the reciprocal of the second]

= (4 × 3)/ (9 × 2)

= (2 × 1)/(3 × 1)

= 2/3

[Step III: It is the simplified expression, hence no further simplifications]

In Geometry, a Polygon is a closed two-dimensional figure, which is made up of straight lines. A Polygon is a two-dimensional geometric figure that has a finite number of sides. Generally, from the name of the Polygon, we can easily identify the number of sides of the shape of a polygon. For example, a triangle is a Polygon having three sides. There are different types of Polygons here we will learn about Regular Polygons. In this article, we will learn about Regular and irregular Polygons, properties of Regular Polygons along with solved examples of them.

In other words, a Polygon is a geometric shape with a finite number of sides in two dimensions. A Polygon’s sides are made up of straight line segments that are joined end to end. As a result, a Polygon’s line segments are referred to as sides or edges. The intersection of two line segments is known as the vertex or corner, and an angle is generated as a result. A triangle with three sides is an example of a Polygon. Although a circle is a planar figure, it is not regarded as a Polygon since it is curved and lacks sides and angles. As a result, we may claim that all Polygons are two-dimensional forms, but not all two-dimensional figures are Polygons.

Different sorts of Polygons may be seen in our daily lives, and we may use them deliberately or unknowingly. In this article, you will learn the concept and definition of a Polygon, as well as the many types of Polygons, as well as real-life examples of Polygon forms, their characteristics, and corresponding formulae.

Definition of Regular Polygon

A Polygon having all the sides equal are all Regular Polygons. Since all the sides of the Polygon are equal therefore all the angles of a Regular Polygon are equal.

The most common examples of Regular Polygons are square, rhombus, equilateral triangle etc.

To form a closed figure, a minimum of three line segments must be connected end to end. As a result, a triangle is a Polygon having at least three sides, often known as a 3-gon. The term n-gon refers to a Polygon with n sides.

The below diagram represents a Regular Polygon.

Regular Polygon Examples

• A square has all its sides equal, and all the angles are equal to 90°.

• An equilateral triangle has all three sides equal and the measure of each angle is equal to 60°.

• A Regular pentagon has five equal sides and all the interior angles of the pentagon are equal to 108⁰.

Properties of Regular Polygons

Following are the properties of Regular Polygons:

• All the sides and interior angles of a Regular Polygon are all equal.

• The bisectors of the interior angles of a Regular Polygon meet at its centre.

• The perpendiculars drawn from the centre of a Regular Polygon to its sides are all equal.

• The lines joining the centre of a Regular Polygon to its vertices are all equal.

• The centre of a Regular Polygon is the centre of both the inscribed and circumscribed circles.

• Straight lines drawn from the centre to the vertices of a Regular Polygon divide it into as many equal isosceles triangles as there are sides in it.

• Formula to calculate the angle of a Regular Polygon of n side is                                      =(2n−4n)×90o

Regular and Irregular Polygons

A polygon whose sides are not of the same length and angles are not of the same measure is called Irregular Polygons. In simple words, polygons that do not fulfil the properties of a Regular Polygon are known as Irregular Polygons.

Different Diagram of an Irregular Polygon

Types Of Irregular Polygon

Concave or Convex Polygon

• There are no angles pointing inwards in a convex Polygon. No internal angle can be more than 180 degrees.

• The polygon is concave if any internal angle is bigger than 180°. (Concave has the word “cave” in it.

Simple or Complex Polygon

A simple Polygon has only one border that does not cross over. When a complicated Polygon interacts with itself, it creates a new Polygon! When it comes to Polygons, many rules don’t apply when the situation is complicated.

Parts of Regular Polygons

Following are the five basic parts of a Regular Polygon:

• Vertices

• Sides

• Interior Angles

• Exterior Angles

• Diagonals

The Diagram Shown Below Represent Basic Parts of a Regular Polygon 

Based on the number of sides there are different types of Regular Polygons few of them are listed below

Regular Polygons List

Regular Polygon Angles

Exterior Angles Of A Regular Polygon

Exterior angles of every simple Polygon add up to 360o, because a trip around the Polygon completes a rotation, or return to your starting place. For example in a hexagon where sides meet, they form vertices, so the hexagon has six vertices.

Interior Angles of a Regular Polygon

Inside the hexagon’s sides, where the interior angles are, is called the hexagon’s interior. Outside its sides is the hexagon’s exterior. This becomes important when we consider complex Polygons, like a star-shape (such as pentagram).

Formulas of Regular Polygon

The formula to find the sum of interior angles of a regular Polygon when the value of n is given

The sum of an interior angle = (n-2) x 180⁰

Where n is the number of sides of the Polygon 

The formula to calculate each interior angle of a regular Polygon 

 Interior angle – (n – 2)×180°/n

The formula to calculate each exterior angle of a regular Polygon 

Since all exterior angles sum up to 360°.

Exterior angles = 360⁰/n

To find the sum of  interior and exterior angle of a regular Polygon 

Since each exterior angle is adjacent to the respective interior angle in a regular Polygon and their sum is 180°.

Sum of interior angles + exterior angle = 180⁰

Formula to Find the Number of Diagonals of a Polygon

For an ‘n’-sided Polygon, the number of diagonals can be calculated using the given formula

Number of diagonals = n(n−3)/2n(n−3)2

Area of a Regular Polygon

To calculate the area of a regular Polygon use the below  formula

Area=l2 × n4tan (πn)

Where

l  is the length of any side

n  is the number of sides

tan  is the tangent function calculated in degrees

Formula to Find the Perimeter of a Regular Polygon

The perimeter of a regular Polygon can be calculated with given below formula

Perimeter = ns

where n is the number of sides of the Polygon, s is the measure of one side of the Polygon.

Fun Facts About Polygons

• A Polygon is a two-dimensional geometric shape with at least three straight sides and angles.

• The sides of a regular Polygon are all the same length, and the angles are all the same. The equilateral triangle and square are examples of regular Polygons.

• A triangle is a three-sided Polygon with 180-degree inner angles.

• A square is a four-sided Polygon with 360-degree internal angles. A square is a quadrilateral as well.

• A pentagon is a five-sided Polygon with 540 degrees of inner angles.

• Regular pentagons have equal-length sides and 108-degree interior angles.

• Okra is a pentagonal-shaped dietary plant.

• Hexagonal cells make up beehive cells.

• Curved heptagons are used on the British 50 and 20 penny coins.

Solved Examples

1. Calculate the Area of 5 Sided Regular Polygon Having a Side Length of 5 cm.

Ans: The given parameters are,

l = 4 cm and n = 5

The formula for finding the area is, Area=l2 × n4tan (πn)

A=l2×n4tan(πn)

A=52×54tan(πn)

A=52×54tan(πn)

A=1255×0.726

A=1255×0.726

A=1253.63

A=1253.63

A = 34.43 cm2

Hence the area of a Polygon is 34.43 cm2

2. Calculate the Value of the Exterior Angle of a Regular Hexagon?

Ans:  Here we will use the formula of an exterior angle

Exterior angle = [frac{360^{0}}{n}]

As we a hexagon has 6 sides, so n = 6

Put the value of n in the above equation

Exterior angle =[frac{360^{0}}{6}]

                          =60o

Hence exterior angle regular hexagon = 60o 

What are Plane Figures and Their Properties?

A plane figure is a closed, two-dimensional flat figure. It has only length and breadth but no thickness at all. Since it has two measurements only, it is two-dimensional. Plain figures only have vertices and sides, and owing to their two-dimensional properties, only their perimeter and area can be measured. You cannot measure their volume, which is reserved for 3-dimensional figures.

Properties of a Rhomboid

1. The Opposite Sides of a Rhomboid are Parallel. 

Let’s take a figure, where ABCD as a rhomboid. So hence, AB is parallel to DC, and AD is parallel to DC. 

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2. The Opposite Sides of the Rhombus are Congruent as Well. 

In the figure below, quadrilateral DABC is a rhomboid. In the rhomboid, DA = CB and DC = AB. So, opposite sides of the rhomboid are equal.

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3. The Diagonals of the Rhomboid Divide the Figure Into Two Congruent Triangles.

This is a property true of any parallelogram. In the figure below, ABCD is a rhomboid, with AC and BD as its diagonals meeting at point O. Hence Triangle ABC is congruent to Triangle ADC by SSS congruence test. 

To prove it:

1. AB= DC (opposite sides of a rhomboid)

2. AD=BC (opposite sides of a rhomboid)

3. AC=AC (common side)

Hence proved. 

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4. The Opposite Angles of a Rhomboid are Equal.

In the figure below, quadrilateral ADCB is a rhomboid. Here ∠ADC= ∠ABC and ∠DAB=    ∠DCB. This is because the opposite angles of a rhomboid are always equal.

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5. The Sum of the Angles in a Rhomboid is Equal to 360 Degrees. 

Owing to the angle sum property of a quadrilateral, the sum of all the angles in a rhomboid is equal to 360 degrees. In the figure below, ADCB is a rhomboid which is a quadrilateral. Hence the sum of all the angles ∠A+∠D+∠C+∠D= 360 degrees. A quadrilateral is a closed, 2-D shape which has 4 sides.

This can be verified because the Rhomboid consists of two congruent triangles whose interior angles will add up to 180 degrees each (since all interior angles of a triangle must add up to 180 degrees). Hence when you add the sum of the interior angles of both the triangles forming the quadrilateral (180+180), you can prove that the total measure is 360 degrees.

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Important formulas of a Rhomboid

1) Area of a rhomboid

Being a 2D figure, the area of the Rhomboid stands for the size of an area, i.e., space covered by the Rhomboid. To calculate the area of a Rhomboid, we first find the length of its base side and height(perpendicular).

We know that the diagonal divides the Rhomboid into two congruent triangles. 

Hence the area of a rhomboid would be = 2 x [½ x base x height]  (area of a triangle is: ½ x base x height)

I.e

Area of Rhomboid=  base x height. 

The area is expressed in unit square, eg, m2. 

Solved Examples

1. Calculate the area of a rhomboid ABCD when the base is 7 cm, and the perpendicular height is 5 cm. 

Solution:

It is given that:

Base, b=7 cm 

Height, h= 5 cm 

Area of Rhomboid=  base x height. 

Hence, 

A = b x h 

= 7 x 5 cm2

= 35 cm2

Thus, the area of rhomboid ABCD is 35 cm2

2. Find the perpendicular height of a rhomboid ABCD, where the area is 50 cm2, and the measure of the base side is 10 cm. 

Solution:    

It is given that

Area, A=  50 cm2

Base, b= 10 cm

We know that 

A = b x h 

To find h 

A= b x h

50= 10 x h 

50/10 = h 

h= 5 cm 

Hence, the perpendicular height of the rhomboid ABCD is 5 cm. 

2) The perimeter of a Rhomboid

Perimeter is basically the measure of the boundary of a figure. Thus the perimeter of the Rhomboid will be the addition of all the sides. 

Let’s assume a rhomboid to be ABCD

Hence the perimeter of the Rhomboid would be:

P= AB+BC+CD+DA

P= 2(AB+BC)                ………… ( since opposite sides of a rhomboid are congruent)

Solved Examples

1. Calculate the perimeter of a rhomboid ABCD where: 

AB= 5 cm

BC=4 cm

CD=5 cm

DA= 4 cm 

Solution:

It is given that:

AB = CD = 5 cm

Hence, 2AB= 5 x 2 = 10 cm

BC = DA = 4cm

Hence, 2BC=4 x 2 = 8 cm

Perimeter, P = 2AB + 2BC

= 2(AB+BC)

= 2(5+4) cm 

= 2(9) cm

= 18 cm.

Thus, the perimeter of the rhomboid ABCD is  18 cm.

2. If the perimeter of the Rhomboid is 60 cm, and the measure of one side is 20 cm, find the other side of the Rhomboid. 

Solution:

Let ABCD be a rhomboid. 

It is given that

Perimeter, P = 60 cm

Let, the one side be AB.

Hence, AB= 20 cm 

We know that the opposite sides of a rhomboid are congruent. 

Hence, AB = CD = 20 cm 

To find BC and DA. 

P= 2AB + 2BC 

60= 2(20) + 2 BC

60= 40 + 2BC

20 = 2 BC 

Hence, BC = 10 cm 

Since BC= DA

DA will also be 10 cm.

So finally the measurements of the sides of the rhomboid ABCD are:

AB= CD = 20 cm 

BC = DA = 10 cm

Root mean square error or rmse is a frequently used measure of the difference between the numbers (population values and numbers) which is estimated by an estimator or mode. The root mean square is also known as root mean square deviation. The rmse details the standard deviation of the difference between the predicted and estimated values. Each of these differences is known as residuals when the calculations are completed over the data sample that was applied to determine, and known as prediction errors when estimated out of sample. The root mean square error or rmse accumulate the magnitude of the errors in estimating different times into a single measure of predictive power.

This article details about root mean square, root mean square formula, root mean square error meaning, root mean square definition, root mean square formula, etc.

Root Mean Square or rms Definition

Statistically, rmse is the square of the mean square, which is the arithmetic mean of the square of group value. Root mean square is also known as quadratic mean and is a specific situation of generalized mean whose exponent is 2. Root mean square isdefined as a varying function that relies on an integral of the square of the value which is immediate in a  cycle.

In other words, the root mean square of a group of a number is the square of the arithmetic mean or the squares of the functions which defines the constant waveform.

Root Mean Square Formula

For a group of n values including {x₁, x₂,x₃,…. xn}, the RMS is given by:

The formula for a continuous function f(x), defined for the interval [x_{1}] ≤ x ≤ [x_{2}] is given by: 

[f_{rms} = sqrt{frac{1}{x_{2} – x_{1}}} int_{x_{1}}^{x_{2}} |f(x)^{2} dx|]

[X_{rms} = sqrt{frac{a}{b}{frac{(x_{1})^{2} + (x_{2})^{2} + (x_{n})^{2}}{n}}}]

The RMS of a periodic function is always  similar to the RMS function of a single period. The continuous function’s RMS value can be estimated approximately by taking the RMS of a sequence of evenly spaced operations. Also, the RMS value of different waveforms can be calculated without calculus.

Root Mean Square Error Definition

Root mean square(rmse) is the standard deviation of the residuals ( estimated errors). Residuals are the approximation of how away from the regression line data points are. Rmse is a measure of how expanded these data are. In other words, rmse details you how intensive the data is around the line of best fit. Root means square error is primarily used in forecasting, climatology, regression analysis to verify experimental results.

Root Mean Square Error or rmse Formula 

The RMSE or root mean square deviation of an estimated model in terms of estimated value is stated as the square root of the mean square error. 

RMSE Formula = [sqrt{sum_{i=1}^{n} (X_{obs, i} – X_{model, i})^{2}}]

Here, Xobs, i is an observed value whereas Xmodel,i is known as modelled value at the time i.

What does root mean square value means?

Root means square value is defined as the square root of the mean value of a squared function. Root mean square value is primarily used as the effective d.c voltage ( or current) of an a.c. voltage ( or current). The root mean square value is further used in the computation of the average power of an AC waveform.

Solved Example

1. Find the root mean square of the following observation ; 5,4,8,1,?

Solutions: Using the root mean square formula: 

[X_{rms}  = sqrt{frac{(x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}}{n}}]

Root mean square = [sqrt{frac{5^{2} + 4^{2} + 8^{2} + 1^{2}}{4}}]

Root mean square = 5.14

2. Calculate the RMSE of the following data

Solution:

Step -1. Calculate the square of each no.

[(x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}]

= [6^{2} + 7^{2} + 8^{2} + 9^{2}]

= 36 + 49 + 64 + 81

= 230

Step 2. Calculate the mean square of each no.

[frac{1}{n} ((x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2})]

= ¼ (230)

= 27.5

[X_{rms}  = sqrt{frac{(x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}}{n}}]

= [sqrt{57.5}]

= [frac{230}{4}]

= 7.58

Quiz Time

1. RMS stands for

1. Root mean square

2. Root mean sum

3. Root maximum sum

4. Root minimum sum

2.  For a rectangular wave, the average current is ……….. Rms current.

1. Greater than

2. Less than

3. Qual to

4. Not related

3. The root mean square error is a measure of 

1. Sample size

2. Moving average period 

3. Exponential Smoothing

4. Forecast accuracy