The one sample chi-square test is used to test the hypothesis that the population from which a sample is drawn has
a specified variance or standard deviation. The F test compares the variances of two samples. Other tests, such as
Levene's test and Bartlett's test, are used to compare the variances of multiple samples. They are covered in their
own section on testing equality of variances.
The One Sample Chi-Square Test
The one sample chi-square test is used to test the hypothesis that a sample comes from a population with a
specified variance or standard deviation. The test is based on the assumption that the sample is randomly selected
from the population, and that the population itself follows a normal distribution. If either of these assumptions is
violated, the reliability of the chiSquare test may be compromised.
The null hypothesis is always that the population underlying the sample has a variance that is equal to the
proposed variance. The alternative hypothesis depends on whether a one or two-tailed test is performed.
For the one-tailed test, the alternative hypothesis is that the population from which the sample was drawn has a
variance that is either less than (lower tailed) or greater than (upper tailed) the proposed variance. For the
two-tailed version, the alternative hypothesis is that the variance of the population does not equal the the proposed
variance.
This test should not be confused with the chi-square goodness-of-fit test, which is used to verify that a sample
comes from a specified distribution.
The one sample chi-square test is implemented by the OneSampleChiSquareTest class. It has five constructors. The
first constructor takes no arguments. The data and conditions for the test must be specified by setting properties of
the OneSampleChiSquareTest object.
The remaining four constructors can be divided into two pairs.
The second and third constructors take 2 or 3 arguments. The first argument is a NumericalVariable that specifies the sample. The second argument is
the proposed variance. The third, optional argument is a HypothesisType value that specifies whether the test is one or
two-tailed. The default value is HypothesisType.TwoTailed.
The last two constructors take 3 or 4 arguments. The first argument is an integer that specifies the size of the
sample. The second argument specifies the variance of the sample. The third parameter specifies the variance to test
again. The optional fourth argument is a HypothesisType
value that specifies whether the test is one or two-tailed. The default value is
HypothesisType.TwoTailed.
Example
The test scores of a class on a national test are as follows:
61, 77, 61, 90, 72, 51, 75, 83, 53, 82, 82, 66, 68, 57, 61, 61, 78, 69, 65.
We want to investigate if the variance of the scores is greater than 5. The following code performs the test:
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double[] group1Data = new double[]
{62, 77, 61, 94, 75, 82, 86, 83, 64, 84,
68, 82, 72, 71, 85, 66, 61, 79, 81, 73};
NumericalVariable results = new NumericalVariable("Class 1", group1Data);
OneSampleChiSquareTest chiSquareTest = new OneSampleChiSquareTest();
chiSquareTest.Sample = results;
chiSquareTest.Variance = 5;
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic);
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
chiSquareTest.Reject() ? "yes" : "no");
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Dim group1Data As Double() = New Double() _
{62, 77, 61, 94, 75, 82, 86, 83, 64, 84,
68, 82, 72, 71, 85, 66, 61, 79, 81, 73}
Dim group1Results As NumericalVariable = _
New NumericalVariable("Class 1", group1Data)
Dim chiSquareTest As OneSampleChiSquareTest = _
New OneSampleChiSquareTest()
chiSquareTest.Sample = results
chiSquareTest.Variance = 5
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic)
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue)
Console.WriteLine("Reject null hypothesis? {0}", _
IIf(chiSquareTest.Reject(), "yes", "no"))
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The value of the chi square statistic turns out to be -2.4505 giving a p-value of 0.0143. As a result, the
hypothesis that on average, the students in this class score no different than the national average is rejected at
the 0.05 level.
Using pre-calculated values for the mean and sample size, the above example would look like this:
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double mean = 75.3
int sampleSize = 20;
OneSampleChiSquareTest chiSquareTest = new OneSampleChiSquareTest(mean, sampleSize, 79.3, 7.3);
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic);
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
chiSquareTest.Reject() ? "yes" : "no");
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Dim mean As Double = 75.3
Dim sampleSize As Integer = 20
Dim chiSquareTest As OneSampleChiSquareTest = _
New OneSampleChiSquareTest(mean, sampleSize, 79.3, 7.3)
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic)
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue)
Console.WriteLine("Reject null hypothesis? {0}", _
IIf(chiSquareTest.Reject(), "yes", "no"))
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Once a OneSampleChiSquareTest object has been created, you can access other properties and methods
common to all hypothesis test classes. For instance, to obtain a 95% confidence interval around the mean, the code
would be:
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Interval meanInterval = chiSquareTest.GetConfidenceInterval();
Console.WriteLine("95% Confidence interval for the mean: {0:F1} - {1:F1}",
meanInterval.LowerBound, meanInterval.UpperBound);
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Dim meanInterval As Interval = chiSquareTest.GetConfidenceInterval()
Console.WriteLine("95% Confidence interval for the mean: {0:F1} - {1:F1}", _
meanInterval.LowerBound, meanInterval.UpperBound)
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The confidence interval for the mean is 72.1 and 78.5 at the 95% confidence level.
The F-Test
The F-test is a two sample test that is used to test the hypothesis that the variances of two populations are
equal. The test is based on the assumption that the sample is randomly selected from the population, and that the
population itself follows a normal distribution. If either of these assumptions is violated, the reliability of the
chiSquare test may be compromised.
The null hypothesis is always that the population underlying the sample has a variance that is equal to the
proposed variance. The alternative hypothesis depends on whether a one or two-tailed test is performed.
For the one-tailed test, the alternative hypothesis is that the population from which the sample was drawn has a
variance that is either less than (lower tailed) or greater than (upper tailed) the proposed variance. For the
two-tailed version, the alternative hypothesis is that the variance of the population does not equal the the proposed
variance.
The F test was named in honor of Sir Ronal Fisher who created the foundations for much of modern statistical
analysis.
The F test is implemented by the FTest class. It has five
constructors.
The first constructor takes no arguments. All test parameters must be provided by setting the properties of the
FTest object.
The remaining four constructors can be divided into two pairs. The first pair has 2 or 3 arguments. The first two
arguments are NumericalVariable objects that represent the
samples the test is to be applied to. The first constructor only has these two arguments. This creates a two-tailed
test for equality of variances. The second constructor of the pair takes a third parameter: a HypothesisType value that specifies whether the test is one or
two-tailed. One-tailed F tests are very common.
The second pair of constructors take 4 or 5 arguments. The first four arguments are, in order, the degrees of
freedom and variance of the numerator, and the degrees of freedom and variance of the denominator. The fifth
parameter, if present, is once again a HypothesisType
value that specifies whether the test is one or two-tailed.
Example
Once again, we use the same data as before. However, this time we compare the results of one group of students to
the results of a second group of students, with these test scores:
61, 80, 98, 90, 94, 65, 79, 75, 74, 86, 76, 85, 78, 72, 76, 79, 65, 92, 76, 80
We want to test if the variances of the two populations are equal. The code below performs this test:
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double[] group2Data = new double[]
{61, 80, 98, 90, 94, 65, 79, 75, 74, 86,
76, 85, 78, 72, 76, 79, 65, 92, 76, 80};
NumericalVariable group2Results = new NumericalVariable("Class 2", group2Data);
FTest fTest = new FTest(group1Results, group2Results);
Console.WriteLine("Test statistic: {0:F4}", fTest.Statistic);
Console.WriteLine("P-value: {0:F4}", fTest.Probability);
Console.WriteLine("Reject null hypothesis? {0}",
fTest.Reject() ? "yes" : "no");
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Dim group2Data As Double() = _
{61, 80, 98, 90, 94, 65, 79, 75, 74, 86, _
76, 85, 78, 72, 76, 79, 65, 92, 76, 80}
Dim group2Results As NumericalVariable = _
New NumericalVariable("Class 2", group2Data)
Dim fTest As FTest = New FTest(group1Results, group2Results)
Console.WriteLine("Test statistic: {0:F4}", fTest.Statistic)
Console.WriteLine("P-value: {0:F4}", fTest.PValue)
Console.WriteLine("Reject null hypothesis? {0}",
IIf(fTest.Reject(), "yes", "no"))
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The value of the F-statistic is 0.9573 giving a p-value of 0.5374. As a result, the hypothesis that the variance
of the scores of students from the first group is no different than that of the second group is not rejected at the
0.05 level.