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Chi Square Formula

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Understanding Chi-Squared (χ²) Tests: Formulas and Applications

Comprehensive Definition, Description, Examples & Rules 

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Introduction

Chi-square is an analysis of data based on some observations of random variables. It is represented as x².   

Provide an introduction to chi-squared tests and their importance in statistics.

Chi-square has invaluable significance in statistics. It can be used for comparing two data. This test was coined by Karl Pearson in the year 1900.

Chi-square has wide importance that includes: 

  • Chi-square is a concept that is useful in testing the nature of two variables, whether they are connected or independent. 
  • It is helpful in getting correct results due to data observation. 

Explain the relevance of chi-squared tests in different  sectors.

The chi-square concept is widely used in daily life: 

  • It is useful in the healthcare sector, where you can find risk factors and health outcomes through this concept. 
  • In social science, chi-square can be used for analyzing voting behaviour.
  • The Chi square concept is beneficial from an environmental and educational perspective. 

Understanding Chi-Squared (χ²) Test?

The Chi-square test is a data analysis of a set of random variables. It is useful in comparing two data.

What is the chi-squared test and its objective? 

The sample distribution of test statistics is referred to as the chi-square distribution. You can find the normal and observed frequency through this concept. It is referred to as X². 

How chi-squared tests are used in analysing categorical data?

You can find out  if two categorical variables are independent or related to each other. 

  • You can analyze the data through chi-square by making hypotheses. There is a null hypothesis and another one is the alternative. The null hypothesis, referred to as (HO) reflects the independent variable, and the alternative variable reflects dependence among variables. 
  • You need to make a table for better analyzing the data. 
  • It becomes easy to calculate chi square through a table by using its formula. 
  • You will find the degree of freedom through the cross-tabulation dimension. 
  • You can get the results by comparing your chi-square data. If you will get x² more than the critical value, then the hypothesis will be rejected. 
  • You can get the conclusion about the dependent and independent data. 

Chi Squared Formula

There are two types of Chi-square formulas that are based on the test types. You will get to know about these formulas in a further section. 

Present the chi-squared formula used to calculate the chi-squared statistic.

The Chi-squared formula is different as per the type of test. There is two chi-square formula that is : 

Chi-square test of independence and chi-square goodness of fit test. 

χ2 = ∑(observed -expected value) ²/ expected value 

You can also refer it is χ2 = ∑(Oi – Ei)2/Ei 

The goodness of fit test formula is : 

= (r – 1)(c – 1)

Provide the mathematical expression for the chi-squared formula 

In the formula

  • X² is chi-square statistics
  • O is the observed frequency
  • E is the expected value 

What is the importance of the chi-squared in statistic ?

  • It is helpful in finding out the nature of variables.
  • You can make a comparison between two data by the chi-square test. 
  • You can calculate the chi-square through expected frequency and critical value. 

Degrees of Freedom Chi Square Tests:

The degree of freedom is the value in the final calculation of chi-square. You will get to know about its formula and usage in the next section. 

Define degrees of freedom in the context of chi-squared tests.

The degree of freedom is a measurement of the number of values in statistics that may vary without having any influence on statistics results. 

Explain how degrees of freedom impact the chi-squared distribution.

  • Chi-square distribution becomes downward with the increasing degree of freedom. 
  • The mean of the chi-square distribution is related to the degree of freedom. 
  • With a large degree of freedom, the distribution becomes normal. 

Provide formulas for calculating degrees of freedom in different chi-squared test scenarios.

Chi-square independence test : 

You can refer it is x² = ∑(Oi – Ei)²/Ei 

  • The goodness of fit test formula is : 

DF = (r – 1)(c – 1)

  • Here, R is a row of table
  • And C is the column of the table 
  • The degree of freedom formula is 

K-1 (Where k is the category) 

To compare two observations, you can use the formula: 

(R-1)*(k-1) 

  • R is population
  • K is cells 

Calculating Chi Square Statistic

You need to use the chi-square formula according to the question requirements to get the correct answer. You will get to know about the basic steps to calculate chi-square statistics. 

Offer step-by-step instructions on how to calculate the chi-squared statistic.

Here are some steps that must be followed by the students: 

  • The first step contains the contingency table that includes rows and columns. 
  • You need to use the formula: 

 Row total * columns total/total 

This formula is useful in calculating expected frequency. 

You need to use this formula for each cell. 

x² = ∑(Oi – Ei)²/E

  • After applying this formula, you need to find the degree of freedom through the formula: 

DF = (r – 1)(c – 1)

  • You need to consider the critical value by using your cross-tabulation, and after that, you need to draw a conclusion by comparing chi-square statistics and critical value. 

Include examples of chi-squared calculations for contingency tables and goodness-of-fit tests.

X

30

20

Y

40

60

  • You need to calculate the expected value for each cell that is

 30+20*30+40/100 =25

30+20*20+60/100=25

40+60*30+40/100=35

40+60*20+60/100=35

  • You need to use the chi-square test formula for each cell: 

(30-25)²/25= 1

(20-25)²/25= 1

(40-35)²/35 = 0.71

(60-35)²/35 = 52.74

  • After the sum, you need to obtain the sum that is 1+1+0.71+52.74= 55.42
  • After that, you need to determine the degree of freedom through (2-1)*(2-1)=1

Through using the table, you need to find out the critical value, and then you need to make decisions. The critical value in this case is 3.8, and you have obtained x² as 55.428. 

Here, your chi-square is more than the critical value, so the null hypothesis is rejected in this case. 

Chi-Squared Test for Independence

If you want to know the dependency of the variable, then you need to follow the chi-square independence test. 

Explain the chi-squared test for independence and its application in analyzing relationships between categorical variables

There are different steps that need to be followed by students: 

  • Students need to observe the table and fund the null or alternative hypothesis. 
  • You need to use the formula: 
  •  Row total * columns total/total 
  • This formula is useful in calculating expected frequency. 
  • You need to use this formula for each cell. 

x² = ∑(Oi – Ei)²/E

  • After applying this formula, you need to find the degree of freedom through the formula: 

DF = (r – 1)(c – 1)

  • You need to consider the critical value by using your cross-tabulation, and after that, you need to draw a conclusion by comparing chi-square statistics and critical value. 

Provide an example of a chi-squared test for independence.

An example of a chi-square test for independence is similar to that of goodness of fit. 

You need to first calculate the expected value for each cell, and then you need to obtain the degree of freedom. Then, you need to take the sum of frequency. 

You will find the 55.42 after the sum. Then, you need to compare it with critical value, and you can make decisions whether the thesis is accepted or rejected. 

Chi-Squared Test for Goodness of Fit

Introduce the chi-squared test for goodness of fit and its use in comparing observed and expected frequencies.

There are various steps that need to be followed by students: 

  • Students need to observe the table and fund the null or alternative hypothesis. 
  • You need to use the formula: 

 Row total * columns total/total 

  • This formula is useful in calculating expected frequency. 

You need to use this formula for each cell. 

x² = ∑(Oi – Ei)²/E

  • You need to calculate the degree of freedom through the formula (k-1) 
  • After that, you need to make comparisons between the critical and chi-square values. 

Include an example illustrating a chi-squared goodness-of-fit test.

  • Assume you have 5 candies 180,250,120,225,225 from 200. After considering the thesis you will calculate expected frequency for each candies and then take a sum of it.
  • 2+12.5+32+3.12+3.12=52.75
  • The, you need to find degree of freedom by 5-1=4
  • Suppose if standard deviation is  as 0.05 and 4 degree of freedom is 9.488
  • Then you will find the conclusion of rejecting the null hypothesis because 52.75 is larger than the critical value 9.488.

Calculate Chi Square P Value

P is the probability that is related to the chi-square. You will get to know about its importance through the next section. 

Describe how to calculate the p-value associated with the chi-squared statistic

  • You need to make a null or alternative hypothesis. 
  • Then, you will be expected to calculate chi-square (x²). 
  • After the calculation of the chi-square, you need to calculate the p-value. 
  • Then, you need to make decisions whether the hypothesis is accepted or not. 

Include examples of p-value calculations.

chi square test p value: 

Monday 8

Tuesday 6

Wednesday 10

Thursday 12

Friday 13

Saturday 6

Sunday 15

  • Compute the value of (O-E)2 / E for each day.

Weekday Number of customers visiting

Monday (8 – 10)² / 10 = 0.4

Tuesday (6 – 10)² / 10 = 1.6

Wednesday (10 – 10)² / 10 = 0

Thursday (12 – 10)²/ 10 = 0.4

Friday (13 – 10)² / 10 = 0.9

Saturday (6 – 10)²/ 10 = 1.6

Sunday (15 – 10)²/ 10 = 2.5

Now, let’s calculate the p-value of the test statistic. The q is equal to 7.4, and df is equal to 6

Hence, the p-value associated with X2 = 7.4 and n-1 = 7-1 = 6 degrees of freedom is 0.28543311.

The p-value comes out to be equal to 0.28. Since this value is not less than 0.05, hence, it can’t be rejected. 

Chi Square Test Example

Comprehensive chi-squared 

Assume you have worked with a clothing firm and take 200 clients’ color preferences, and you need to categorise them into different colors.  

Red – 30%

Blue -25%

Silver -20%

Black -15%

White -10%

  • Do a thesis first; for the null hypothesis, the colors are consistent in the market, and for the alternative hypothesis, this color preference is different in market distribution. 
  • You need to calculate the expected frequency. 

30%*200=60

25%*200=50

20%*200=40

15%*200=30

10%*200 =20

  • You need to calculate the chi-square through  (O-E)²/E

You will observe the frequency as 

Red -65

Blue -40

Silver-50

Black-25 

White-20 

  • Then you need to use a chi-square formula like (65-60)²/60=4.16

(40-50)²/50=4.0

(50-40)²/40=12.5

(25-30)²/30=4.1

(20-20)²/20 =0

After the Chi-square, you need to take the sum of Chi square’ 4.1+4.0+12,5+4.16+0=24.83

You need to calculate the degree of freedom that is 5-1=4, and then you need to calculate the p-value. X²=24.8 and df=4, and suppose the value of p as 0.001.  You need to compare the critical value with the obtained value, which is 0.001. this value is less than 0.05; therefore, the null hypothesis is rejected in this case. 

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Key Takeaways

  1. This test was coined by Karl Pearson in the year 1900. This concept refer as the Pearson chi-square test.

  2. You can refer it is x² = ∑(Oi – Ei)²/Ei 

  3. It includes a degree of freedom that has separate formulas and conditions. 

  4. You must know the types of thesis before solving equations. 

  5. You can improve this concept through Edulyte’s worksheet.

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Frequently Asked Questions

You can use this test to compare the critical and chi-square.

You can calculate the chi-square by following some steps like hypothesis, formula, etc. 

You can consider the degree of freedom through (k-1). 

You can calculate the chi-square p values by following the steps like the degree of freedom formula and so on. 

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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