Quadratic regression calculator: definition, formula and calculation Patient survey software

Quadratic regression calculator: definition, formula and calculation

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What is quadratic regression?

Quadratic regression is the process of finding the equation of a parabola, that best fits your dataset. 

You can identify a quadratic regression by the plotting of your scatterplots. If the scatterplots are in a shape looking like a “U” (concave up), or the scatterplot are plotted in a shape like an up-side down U like “∩” (concave down), then you can say that you have a Quadratic regression at your hand which is best fitting your data. The shape of the scatterplots are not always complete. So you might see a half U or just a 3/4th of it. 

Quadratic regression calculator: definition, formula and calculation Patient survey software

As being an extension of simple linear regression, we can say that the major difference between the both is that in linear regression, a straight line can be drawn using only two points too. But when it comes to Quadratic regression, as it is a parabolic curve, you need as much points as possible to plot a perfect curve. Due to this disadvantage of Quadratic regression, it is generally more costly than simple linear regression.

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Quadratic regression formula

y = ax2 + bx + c

a = { [ Σ x2 y * Σ xx ] – [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2 x2] – [Σ xx2 ]2 }

b = { [ Σ xy * Σ x2 x2 ] – [Σ x2y * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] – [Σ xx2 ]2 }

c = [ Σ y / n ] – { b * [ Σ x / n ] } – { a * [ Σ x 2 / n ] }

Where, 

a, b, c are the coefficients of Quadratic regression.

Σ x x = [ Σ x 2 ] – [ ( Σ x )2 / n ]

Σ x y = [ Σ x y ] – [ ( Σ x * Σ y ) / n ]

Σ x x2 = [ Σ x 3 ] – [ ( Σ x 2 * Σ x ) / n ]

Σ x2 y = [ Σ x 2 y] – [ ( Σ x 2 * Σ y ) / n ]

Σ x2 x2 = [ Σ x 4 ] – [ ( Σ x 2 )2 / n ] 

a, b, c are the coefficients of Quadratic regression.

How to calculate Quadratic regression?

Let’s take an example to understand how the formula works around the give data.

Example: draw a second degree polynomial with polynomial regression for the given dataset.

x values

y values

5

3

6

2

4

4

Step 1: Count the total given numbers.

In our case, n=3.

Step 2: Find all the values needed for our formula.

x

y

x2

x3

x4

xy

x2y

5

3

25

125

625

15

75

6

2

36

216

1296

12

72

4

4

16

64

256

16

64

∑x

∑y

∑x2

∑x3

∑x4

∑xy

∑x2y

15

9

77

405

2177

43

211

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Step 3: Substitute the values in the formula

Σ x x = [ Σ x 2 ] – [ ( Σ x )2 / n ]

∑xx = [77] – [(15)2 / 3]

∑xx = 77 – [225/3]

∑xx = 77 – 75

∑xx = 2

Σ x y = [ Σ x y ] – [ ( Σ x * Σ y ) / n ]

∑xy = [43] – [(15 x 9) /3]

∑xy = 43 – [135/3]

∑xy = 43 – 45

∑xy = -2

Σ x x2 = [ Σ x 3 ] – [ ( Σ x 2 * Σ x ) / n ]

∑xx2 = [405] – [(77 x 15) / 3]

∑xx2 = 405 – [1155 / 3]

∑xx2 = 405 – 385 

∑xx2 = 20

Σ x2 y = [ Σ x 2 y] – [ ( Σ x 2 * Σ y ) / n ]

∑x2y = [211] – [(77 x 9) / 3]

∑x2y = 211 – [693 / 3]

∑x2y = 211 – 231

∑x2y = -20

Σ x2 x2 = [ Σ x 4 ] – [ ( Σ x 2 )2 / n ] 

∑x2x2 = [2177] – [(77)2 / 3]

∑x2x2 = 2177 – [5929 / 3]

∑x2x2 = 2177 – 1976

∑x2x2 = 201

Step 4: Calculate a

a = { [ Σ x2 y * Σ xx ] – [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2 x2] – [Σ xx2 ]2 }

a = {[-20 * 2] – [-2 * 20]} / {[2 * 201] – [20]2}

a = {(-40) – (40)} / {(402) – (400)}

a = -80 / 2

a = -40

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Step 5: Calculate b

b = { [ Σ xy * Σ x2 x2 ] – [Σ x2y * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] – [Σ xx2 ]2 }

b = {[-2 * 201] – [-20 * 20]} / {[2 * 201] – [20]2}

b = {[-402] – [-400]} / {(402) – (400)}

b = -2 / 2

b = -1

Step 6: Calculate c

c = [ Σ y / n ] – { b * [ Σ x / n ] } – { a * [ Σ x 2 / n ] }

c = [9 / 3] – {-1 * (15 / 3)} – {-40 * (77 / 3)}

c = 3 – [-1 * 5] – [-40 * 25.66]

c = 3 – (-5) – (-1026.4)

c = 1034.4

Step 7: Substitute the value of a, b, c in the Quadratic regression equation

y = ax2 + bx + c

y = -40x2 + (-1x) + 1034.4

y = -40x2 – x + 1034.4

Hence, the Quadratic regression equation of your parabola is y = -40x2 – x + 1034.4 

Apart from this, there are various online Quadratic regression calculators that make your task easy and save all these steps and give the Quadratic regression equation straight away. 

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Hindol Basu 
GM, Voxco Intelligence

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