Dundas Chart for Windows Forms
Z Test Formula
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Formula Reference > Statistical Formulas > Z Test Formula



Glossary Item Box

Overview

This formula performs a Z Test using Normal distribution.

Applying the Formula

All statistical formulas are calculated using the Statistics class, and the following table describes how to use its ZTest method to perform a Z Test.

 

Value/Description Example
Formula Name: Z Test  Statistics.ZTest (0.2, 2.5, 4.5, 0.05, "Series1", "Series2")
Parameters:
  1. hypothesizedMeanDifference: the hypothesized difference between the means of the data groups.
  2. varianceFirstGroup: the variance within the first group of data.
  3. varianceSecondGroup: the variance within the second group of data.
  4. probability: the alpha value (probability that the hypothesis is rejected).
  5. firstInputSeriesName: The name of the Series object that stores the first group of data.
  6. secondInputSeriesName: The name of the Series object that stores the second group of data.
Statistics.ZTest(0.2, 2.5, 4.5, 0.05, "Series1", "Series2")
Return:

A ZTestResult object, which has the following members:

  • FirstSeriesMean
  • SecondSeriesMean
  • FirstSeriesVariance
  • SecondSeriesVariance
  • ZValue
  • ProbabilityZOneTail
  • ZCriticalValueOneTail
  • ProbabilityZTwoTail
  • ZCriticalValueTwoTail

-

 

Note
Make sure that all data points have their XValue property set, and that their series'  XValueIndexed property has been set to false.

Statistical Interpretation

Suppose that you have two different populations whose means you want to compare. Assume that the random variables  (mean , variance ) and  (mean , variance ) have approximately normal distributions. For sample sizes of  and , then the sample means  and  are normal random variables.

If we want to test hypotheses about the difference between the population means , our null hypothesis might be that the population means are equal:

 or .

If the population variances  and  are known, we can form the test statistic Z:

This statistic will have a standard normal distribution if the null hypothesis is true.

Calculation

  1. Calculate  and .
  2. Calculate the Z statistic:

     

  3. Use the Normal Inverse Distribution function to test the hypothesis.

Example

This example demonstrates how to perform a Z Test, using Series1 and Series2 for the input series.

Visual Basic Copy Code
Imports Dundas.Charting.WinControl
  ...
  
' Perform the Z test.
Dim result As ZTestResult = Chart1.DataManipulator.Statistics.ZTest(0.2,2.5,4.5,0.05,"Series1","Series2")
C# Copy Code
using Dundas.Charting.WinControl;
  ...
  
// Perform the Z test.
ZTestResult result = Chart1.DataManipulator.Statistics.ZTest(0.2,2.5,4.5,0.05,"Series1","Series2");

Example

This example demonstrates how to perform a Z Test, using Series1 and Series2 for the input series. The results are returned in an ZTestResult object. The object values are then added as titles to a separate chart. We assume series data was added at design-time. Further, we assume a "DundasBlue" template was applied for appearance purposes.

 

Figure 1: Two Charts; One containing Series data (left), and the other containing the ZTestResult object (right).

 

Visual Basic Copy Code
Imports Dundas.Charting.WinControl
  ...
  
ZTestResult ztestresult = chart1.DataManipulator.Statistics.ZTest(0.2, 
        chart1.DataManipulator.Statistics.Variance("Series1",True), 
        chart1.DataManipulator.Statistics.Variance("Series2",True),
        0.05, 
        "Series1", 
        "Series2"
        )
 
' Inital Titles are added.
chart2.Titles.Add("ZTestResult Object")
chart2.Titles.Add("Hypothesized Mean Difference: 0.2 | Probability: 5%\n")
 
' Change Appearance properties of first title.
chart2.Titles(0).BackColor = Color.FromArgb(255, 65, 140, 240)
chart2.Titles(0).Font = New Font("Trebuchet", 12, FontStyle.Bold)
chart2.Titles(0).Color = Color.White
chart2.Titles(0).Style = TextStyle.Shadow
 
' Change Appearance properties of second title.
chart2.Titles(1).BackColor = Color.FromArgb(255, 224, 64, 10)
chart2.Titles(1).Font = New Font("Trebuchet", 8, FontStyle.Bold)
chart2.Titles(1).Color = Color.White

' ZTestObject properties are created into titles using the ToString() method.
chart2.Titles.Add("First Series Mean: " + ztestresult.FirstSeriesMean.ToString() + 
" | Second Series Mean: " + ztestresult.SecondSeriesMean.ToString())
chart2.Titles.Add("First Series Variance: " + ztestresult.FirstSeriesVariance.ToString() + 
" | Second Series Variance: " + ztestresult.SecondSeriesVariance.ToString())
chart2.Titles.Add("Probability Z One Tail: " + ztestresult.ProbabilityZOneTail.ToString())
chart2.Titles.Add("Probability Z Two Tail: " + ztestresult.ProbabilityZTwoTail.ToString())
chart2.Titles.Add("Z Critical Value One Tail: " + ztestresult.ZCriticalValueOneTail.ToString())
chart2.Titles.Add("Z Critical Value Two Tail: " + ztestresult.ZCriticalValueTwoTail.ToString())
chart2.Titles.Add("ZValue: " + ztestresult.ZValue.ToString())
C# Copy Code
using Dundas.Charting.WinControl;
  ...
  
ZTestResult ztestresult = chart1.DataManipulator.Statistics.ZTest(0.2, 
        chart1.DataManipulator.Statistics.Variance("Series1",true), 
        chart1.DataManipulator.Statistics.Variance("Series2",true),
        0.05, 
        "Series1", 
        "Series2"
        );

// Inital Titles are added.
chart2.Titles.Add("ZTestResult Object");
chart2.Titles.Add("Hypothesized Mean Difference: 0.2 | Probability: 5%\n");

// Change Appearance properties of first title.
chart2.Titles[0].BackColor = Color.FromArgb(255, 65, 140, 240);
chart2.Titles[0].Font = new Font("Trebuchet", 12, FontStyle.Bold);
chart2.Titles[0].Color = Color.White;
chart2.Titles[0].Style = TextStyle.Shadow;

// Change Appearance properties of second title.
chart2.Titles[1].BackColor = Color.FromArgb(255, 224, 64, 10);
chart2.Titles[1].Font = new Font("Trebuchet", 8, FontStyle.Bold);
chart2.Titles[1].Color = Color.White;

// ZTestObject properties are created into titles using the ToString() method.
chart2.Titles.Add("First Series Mean: " + ztestresult.FirstSeriesMean.ToString() + 
" | Second Series Mean: " + ztestresult.SecondSeriesMean.ToString());
chart2.Titles.Add("First Series Variance: " + ztestresult.FirstSeriesVariance.ToString() + 
" | Second Series Variance: " + ztestresult.SecondSeriesVariance.ToString());
chart2.Titles.Add("Probability Z One Tail: " + ztestresult.ProbabilityZOneTail.ToString());
chart2.Titles.Add("Probability Z Two Tail: " + ztestresult.ProbabilityZTwoTail.ToString());
chart2.Titles.Add("Z Critical Value One Tail: " + ztestresult.ZCriticalValueOneTail.ToString());
chart2.Titles.Add("Z Critical Value Two Tail: " + ztestresult.ZCriticalValueTwoTail.ToString());
chart2.Titles.Add("ZValue: " + ztestresult.ZValue.ToString());

 

See Also

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