# Overview of Chi-Square

Population Variance Test

The Chi-Square Population Variance Test is a hypothesis test is used to determine if the variance of a normally-distributed population has changed. One common use of this test is to determine whether an adjustment made to a production line causes a change in variance at some measurement point on the production line.

The Chi-Square Variance Test can be performed as a one-sample or a two-sample test.

A one-sample test usually involved using a single sample taken from a normally-distributed population to determine whether the variance of that population has changed from a known variance measured in the past. The production line example just mentioned is the most common use of the one-sample Chi-Square Variance Test. In this case the test is most accurate when the benchmark population standard deviation has been calculated from a stable process over a long period of time.

A two-sample test is used to determine whether two normally-distributed populations have the same variance. This is known as the F Test.

As with most hypothesis tests, the Chi-Square Population variance Test can be conducted as a one-tailed test or a two-tailed test. When this hypothesis test is conducted as a one-tailed test, it is used to determine whether the population variance has moved in one direction, i.e., the test is being used to determine only whether the population variance has increased or the test is being used to determine only whether the population variance has decreased.

When this hypothesis test is being conducted as a two-tailed test, it is being used to determine whether the population variance has changed in any direction (a one-sample test) or whether two populations have the same variance (a two-sample test). The two-tailed test is more stringent than the one-tailed test; the two-tailed test requires more change to reject the Null Hypothesis than a one-tailed test of the same alpha level. The Null Hypothesis states either that a single population variance has not changed (a one-sample test) or that two populations have the same variance (an F Test, which is a two-sample test).

## One-Sample Chi-Square Population

Variance Test

The one-sample Chi-Square Population Variance Test is used to determine if a normally-distributed population’s variance has changed. A sample of size n is taken from the population and its variance is measured. A test statistic called the Chi-Square Statistics is then created from n (sample size), s^{2} (sample variance), and σ^{2} (population variance) as follows:

Chi-Square Statistic = (n-1)* s^{2 }/ σ^{2}

This test statistic is called the Chi-Square Statistic because the distribution of this test statistic can be approximated by the Chi-Square distribution with n-1 degrees of freedom if the population is normally distributed.

Critical Chi-Square Values are then calculated based upon the degrees of freedom (n-1), alpha, and the number of tails in the hypothesis test as follows:

**Two-tailed test**

Left Critical Value = CHISQ.INV(α/2,df)

Right Critical Value = CHISQ.INV(1 – α/2,df)

**One-tailed test – Right tail**

Critical Value = CHISQ.INV(1 – α,df)

**One-tailed test – Left tail**

Critical Value = CHISQ.INV(α,df)

The Null Hypothesis of the One-Sample Chi-Square Population Variance Test states that population variance has not changed. This Null Hypothesis is rejected if the Chi-Square Statistic is outside of a Critical Value on the same side of the Chi-Square PDF curve. The Null Hypothesis is not rejected of the Chi-Square Statistic falls inside of the Critical Value on that side of the curve.

## Example of 1-Sample, 2-Tailed, Chi-

Square Population Variance Test in

Excel

A specific measurement is taken on each unit that is completed from a production line over a long period of time. These measurements have been determined to be normally distributed with a population variance = σ^{2} = 0.09.

An adjustment was made to the production line that may have affected the variance of the measurement taken on each completed unit. A random sample of 150 units from the newly-adjusted production line had the measurement taken. The sample variance of this 150-unit sample is s = 0.32. Determine with 95 percent certainty whether the population variance has **changed in any direction** as a result of the adjustment.

### Problem Information

Sample size = n = 150

Degrees of Freedom = n – 1 = 149

Sample Standard Deviation = s = 0.32

Sample Variance = s^{2} = 0.1024

Long-term, Benchmark Population Variance = 0.09

Alpha = 1 – Required Level of Certainty = 1 – 0.95 = 0.05

### Requirement of Population Normality

Before performing this test, it is important to verify that the population of data measurements is normally distributed. If the data measurements are not normally distributed, this statistical test could produce *totally* invalid results.

If the normality of the population cannot be confirmed, the normality of the sample must be confirmed. Large sample size (n > 30) does not waive the normality requirement as occurs with t Tests.

Common ways to confirm normality of sample data are the following:

An Excel histogram of the sample data in Excel

A normal probability plot of the sample data in Excel

The Kolmogorov-Smirnov test for normality of the sample data in Excel

The Anderson-Darling test for normality of the sample data in Excel

The Shapiro-Wilk test for normality of the sample data in Excel

**The above tests are all performed in several articles in this blog.**

### Non-Parametric Alternatives to 1-Sample Chi-Square Population Variance Test

When population normality cannot be confirmed, nonparametric alternatives for the one-sample Chi-Square Population Variance Test include Levene’s Test and the Brown-Forsythe Test.

Levene’s Test and the Brown-Forsythe Test are nonparametric tests that are used to compare variances of two samples when the F Test’s normality requirement cannot be met. These two nonparametric tests can also be used in place of the one-sample, Chi-Square Population Variance Test.

Since both of these nonparametric tests are used to compare variances between two samples and require that two data samples be taken for comparison.

The one-sample Chi-Square Population Test must be changed slightly in order to meet the requirement of two samples to compare. A “Before” sample must now be taken in place of the known population standard deviation data. The one-sample Chi-Square Population Variance Test compares an “After” sample with known population variance data. The nonparametric tests requires that a “Before” sample be taken to compare with the “After” sample. The “Before” sample is taken before to the adjustment is made to, for example, a production line. The “After” sample is taken after the adjustment is made.

**Levene’s Test and the Brown-Forsythe Test are performed in several articles in this blog.**

### Null and Alternative Hypotheses

The Null Hypothesis is created as follows:

H_{o}: σ^{2} = 0.09

σ^{2 }equals the variance of the new population of measurements taken after the adjustment was made to the production line,

This Null Hypothesis states that the new population variance, σ^{2}, is the same as the old, long-term population variance of measurements taken before the adjustment was made.

The Alternative Hypothesis is created as follows:

H_{1}: σ^{2} ≠ 0.09

The non-directional operator, ≠, indicates that this hypothesis test is a two-tailed test. A directional operation (< or >) would indicate that this hypothesis test is a one-tailed test.

### Chi-Square Statistic and Chi-Square Critical Values

The Chi-Square Statistic and Critical Values for this two-tailed test are calculated as follows:

*(Click On Image To See a Larger Version)*

These left and right Critical Values are shown in this Chi-Square PDF distribution curve for 149 degrees of freedom as follows:

*(Click On Image To See a Larger Version)*

The Chi-Square Statistic (169.5) falls inside of the Critical Values (117, 184) and into the red Region of Acceptance. The Null Hypothesis is therefore not rejected. It cannot be stated with 95 percent certainty that the variance of the measurement taken from the completed production unit has changed as a result of the adjustment made to production line.

## Example of 1-Sample, 1-Tailed,

Right Tail, Chi-Square Population

Variance Test in Excel

A specific measurement is taken on each unit that is completed from a production line over a long period of time. These measurements have been determined to be normally distributed with a population variance = σ^{2} = 0.09.

An adjustment was made to the production line that may have affected the variance of the measurement taken on each completed unit. A random sample of 150 units from the newly-adjusted production line had the measurement taken. The sample variance of this 150-unit sample is s = 0.33. Determine with 95 percent certainty whether the population variance has **increased** as a result of the adjustment.

### Problem Information

Sample size = n = 150

Degrees of Freedom = n – 1 = 149

Sample Standard Deviation = s = 0.33

Sample Variance = s^{2} = 0.1089

Long-term Population Variance = 0.09

Alpha = 1 – Required Level of Certainty = 1 – 0.95 = 0.05

### Requirement of Population Normality

Before performing this test, it is important to verify that the population of data measurements is normally distributed. If the data measurements are not normally distributed, this statistical test could produce *totally* invalid results.

If the normality of the population cannot be confirmed, the normality of the sample must be confirmed. Large sample size (n > 30) does not waive the normality requirement as occurs with t Tests.

Common ways to confirm normality of sample data are the following:

An Excel histogram of the sample data in Excel

A normal probability plot of the sample data in Excel

The Kolmogorov-Smirnov test for normality of the sample data in Excel

The Anderson-Darling test for normality of the sample data in Excel

The Shapiro-Wilk test for normality of the sample data in Excel

**The above tests are all performed on the sample data in several articles in this blog.**

### Non-Parametric Alternatives to the One-Sample Chi-Square Population Variance Test

When population normality cannot be confirmed, nonparametric alternatives for the one-sample Chi-Square Population Variance Test include Levene’s Test and the Brown-Forsythe Test.

Levene’s Test and the Brown-Forsythe Test are nonparametric tests that are used to compare variances of two samples when the F Test’s normality requirement cannot be met. These two nonparametric tests can also be used in place of the one-sample, Chi-Square Population Variance Test.

Since both of these nonparametric tests are used to compare variances between two samples and require that two data samples be taken for comparison.

The one-sample Chi-Square Population Test must be changed slightly in order to meet the requirement of two samples to compare. A “Before” sample must now be taken in place of the known population standard deviation data. The one-sample Chi-Square Population Variance Test compares an “After” sample with known population variance data. The nonparametric tests requires that a “Before” sample be taken to compare with the “After” sample. The “Before” sample is taken before to the adjustment is made to, for example, a production line. The “After” sample is taken after the adjustment is made.

**Levene’s Test and the Brown-Forsythe Test are performed on two samples in several articles in this blog**.

### Null and Alternative Hypotheses

The Null Hypothesis is created as follows:

H_{o}: σ^{2} = 0.09

σ^{2 }equals the variance of the new population of measurements taken after the adjustment was made to the production line,

This Null Hypothesis states that the new population variance, σ^{2}, is the same as the old, long-term population variance of measurements taken before the adjustment was made.

The Alternative Hypothesis is created as follows:

H_{1}: σ^{2} > 0.09

The directional operator, >, indicates that this hypothesis test is a one-tailed test in the right tail. A non-directional operator (≠) would indicate that this hypothesis test is a two-tailed test. The directional operation, <, would indicate that the hypothesis test is a one-tailed test in the left tail.

### Chi-Square Statistic and Chi-Square Critical Values

The Chi-Square Statistic and Critical Values for this two-tailed test are calculated as follows:

*(Click On Image To See a Larger Version)*

These left and right Critical Values are shown in this Chi-Square PDF distribution curve for 149 degrees of freedom as follows:

*(Click On Image To See a Larger Version)*

The Chi-Square Statistic (180.3) falls outside of the Critical Value (178) and into the blue Region of Rejection. The Null Hypothesis is therefore rejected. It can be stated with 95 percent certainty that the variance of the measurement taken from the completed production unit has increased as a result of the adjustment made to production line.

It should be noted that the Null Hypothesis would not be rejected if this were a two-tailed test. The Chi-Square Statistic, 180.3, falls inside of the Chi-Square Critical Values of the two-tailed test (117, 184). The two-tailed test is more stringent than the one-tailed test. This is the case with nearly every type of hypothesis test.

### Example of 1-Sample, 1-Tailed, Left Tail, Chi-Square Population Variance Test in Excel

A specific measurement is taken on each unit that is completed from a production line over a long period of time. These measurements have been determined to be normally distributed with a population variance = σ^{2} = 0.09.

An adjustment was made to the production line that may have affected the variance of the measurement taken on each completed unit. A random sample of 150 units from the newly-adjusted production line had the measurement taken. The sample variance of this 150-unit sample is s = 0.33. Determine with 95 percent certainty whether the population variance has ** decreased** as a result of the adjustment.

### Problem Information

Sample size = n = 150

Degrees of Freedom = n – 1 = 149

Sample Standard Deviation = s = 0.27

Sample Variance = s^{2} = 0.0729

Long-term Population Variance = 0.09

Alpha = 1 – Required Level of Certainty = 1 – 0.95 = 0.05

### Requirement of Population Normality

Before performing this test, it is important to verify that the population of data measurements is normally distributed. If the data measurements are not normally distributed, this statistical test could produce *totally* invalid results.

If the normality of the population cannot be confirmed, the normality of the sample must be confirmed. Large sample size (n > 30) does not waive the normality requirement as occurs with t Tests.

Common ways to confirm normality of sample data are the following:

An Excel histogram of the sample data in Excel

A normal probability plot of the sample data in Excel

The Kolmogorov-Smirnov test for normality of the sample data in Excel

The Anderson-Darling test for normality of the sample data in Excel

The Shapiro-Wilk test for normality of the sample data in Excel

The above tests are all performed on the sample data in several articles in this blog.

### Non-Parametric Alternatives to 1-Sample Chi-Square Population Variance Test

When population normality cannot be confirmed, nonparametric alternatives for the one-sample Chi-Square Population Variance Test include Levene’s Test and the Brown-Forsythe Test.

Levene’s Test and the Brown-Forsythe Test are nonparametric tests that are used to compare variances of two samples when the F Test’s normality requirement cannot be met. These two nonparametric tests can also be used in place of the one-sample, Chi-Square Population Variance Test.

Since both of these nonparametric tests are used to compare variances between two samples and require that two data samples be taken for comparison.

The one-sample Chi-Square Population Test must be changed slightly in order to meet the requirement of two samples to compare. A “Before” sample must now be taken in place of the known population standard deviation data. The one-sample Chi-Square Population Variance Test compares an “After” sample with known population variance data. The nonparametric tests requires that a “Before” sample be taken to compare with the “After” sample. The “Before” sample is taken before to the adjustment is made to, for example, a production line. The “After” sample is taken after the adjustment is made.

Levene’s Test and the Brown-Forsythe Test are performed in several articles in this blog.

### Null and Alternative Hypotheses

The Null Hypothesis is created as follows:

H_{o}: σ^{2} = 0.09

σ^{2 }equals the variance of the new population of measurements taken after the adjustment was made to the production line,

This Null Hypothesis states that the new population variance, σ^{2}, is the same as the old, long-term population variance of measurements taken before the adjustment was made.

The Alternative Hypothesis is created as follows:

H_{1}: σ^{2} > 0.09

The directional operator, <, indicates that this hypothesis test is a one-tailed test in the left tail. A non-directional operator (≠) would indicate that this hypothesis test is a two-tailed test. The directional operation, >, would indicate that the hypothesis test is a one-tailed test in the right tail.

##### Chi-Square Statistic and Chi-Square Critical Values

The Chi-Square Statistic and Critical Values for this two-tailed test are calculated as follows:

*(Click On Image To See a Larger Version)*

These left and right Critical Values are shown in this Chi-Square PDF distribution curve for 149 degrees of freedom as follows:

*(Click On Image To See a Larger Version)*

The Chi-Square Statistic (121.8) falls outside of the Critical Value (120) and into the blue Region of Rejection. The Null Hypothesis is therefore rejected. It can be stated with 95 percent certainty that the variance of the measurement taken from the completed production unit has decreased as a result of the adjustment made to production line.

It should be noted that the Null Hypothesis would not be rejected if this were a two-tailed test. The Chi-Square Statistic, 121.8, falls inside of the Chi-Square Critical Values of the two-tailed test (117, 184). The two-tailed test is more stringent than the one-tailed test. This is the case with nearly every type of hypothesis test.

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