One-Sample Sign / One-Sample Wilcoxon

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About this lesson

The One-Sample Sign Test and the One-Sample Wilcoxon Test accomplish the same purpose, but each has strengths and weaknesses. When the data is not normal, or it is not known that it definitely is normal, these tests can be used to determine if the data set statistics meets or exceeds a target value. The application of this test using Minitab is illustrated in this lesson.

Quick reference

One-Sample Sign / One-Sample Wilcoxon

There are hypothesis tests that can be used when the data is non-normal.  One-Sample Sign Test and One-Sample Wilcoxon Test are non-normal hypothesis tests used when there is only one data sample.

When to use

Both tests compare the data sample median to a target median.  The One-Sample Wilcoxon Test is more sensitive and therefore more accurate, but it only works with symmetric data.  The One-Sample Sign Test can be used with any non-normal data set, but since it is less sensitive it is more likely to fail to reject the Null hypothesis.

Instructions

Data is often non-normal.  Fortunately, there are many non-normal hypothesis tests that can be used with non-normal data.  In some cases, non-normal data may be transformed into normal data.  If using Minitab, I would not transform but rather use the non-normal hypothesis test.  However Excel does not have non-normal hypothesis tests in its Data Analysis menu, so when using Excel attempt to transform the data.  I suggest trying Box-Cox transformations which were discussed in a previous lesson.

The non-normal data hypothesis tests are often “tuned” to a particular type of non-normality.  This will be discussed with each test.  The table below shows the non-normal data hypothesis test and its normal data test equivalent.

The One-Sample Sign Test compares the median of the non-normal sample data to a target median.  It is similar to the One-Sample T Test which compares the sample mean to a target mean.  This test is relatively insensitive.  It works with any type of non-normality.

  • Minitab:
    1. Stat > Non-parametric – 1 Sample Sign
    2. Enter the column containing the data
    3. Set the relationship level (equal, less than, greater than)

The One-Sample Wilcoxon Test compares the median of the non-normal sample data to a target median.  The One-Sample Wilcoxon Test is designed for use with symmetric data.  When the data is symmetric it is more sensitive and provides a better test than the One-Sample Sign Test.  It should not be used with data that is heavily skewed.  In all other respects it is similar to the One-Sample Sign Test.

  • Minitab:
    1. Stat > Non-parametric > 1 Sample Wilcoxon
    2. Enter the column containing the data
    3. Set the relationship level (equal, less than, greater than)

Hints & tips

  • Check your data to see if it is normal.  If it is not, use a non-normal hypothesis test.
  • When working with one sample set of data, check your non-normal data to see if it is symmetric.  This will determine which test to use.
  • Non-normal data uses the median instead of the mean for the measure so central tendency.  This reduces the impact or skewed data and outliers.
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  • 00:04 Hi, I'm Ray Sheen.
  • 00:06 Now, so far we've been focusing on what to do when the data is normal.
  • 00:10 But sometimes your data is not normal.
  • 00:13 So now let's start to look at this situation, and
  • 00:15 we'll begin by using the One-Sample Sign and
  • 00:18 the One-Sample Wilcoxon tests. >> But
  • 00:21 we are moving to a different section of a hypothesis testing decision tree.
  • 00:26 We now take the path for non normal data and move toward the bottom of the tree.
  • 00:31 When there is one data sample involved, the decision is to use either
  • 00:35 the One-Sample Sign or the One-Sample Wilcoxon.
  • 00:38 We will look at both of these in this lesson.
  • 00:40 But let's talk for a few moments, first, discussing non normal tests.
  • 00:45 Up until now, we've focused on tests for normal data.
  • 00:48 But many real world situations are not normal.
  • 00:51 That doesn't mean there are special causes present, just that the physical attributes
  • 00:55 that are being studied don't act with the bell shaped Gaussian distribution.
  • 01:00 Don't become so focus on normality that you fail to do an appropriate test.
  • 01:05 There are plenty of non-formal hypothesis tests that are very effective.
  • 01:09 With the aid of the computer, the math is not overwhelming.
  • 01:12 Many of the normal tests are somewhat forgiving for non-normal data.
  • 01:17 We mention that in particular when discussing ANOVA.
  • 01:19 So if it is a minor non-normality, you may still be okay.
  • 01:23 But even if it is a major non-normality you can still transform the data into
  • 01:27 normal data if you believe it's necessary.
  • 01:31 One legitimate concern with respect to non-normal data
  • 01:33 is that Excel does not have any non-normal tests in its analysis tool pack
  • 01:38 that is accessed through the data analysis menu on the data ribbon.
  • 01:42 So you will likely need a statistical analysis application like Minitab
  • 01:46 to do these tests.
  • 01:48 Fortunately, Minitab has all these tests, and even some additional ones that I won't
  • 01:52 be discussing they could be used for hypothesis testing.
  • 01:56 Let me talk for a moment about non-normal test selection.
  • 01:59 The hypothesis decision theory often includes several options for
  • 02:03 non-normal test.
  • 02:04 This is do impart to the fact that non-normal tests are often optimized for
  • 02:08 particular data set conditions.
  • 02:10 Non-normal tests or as they are called in Minitab, non-parametric tests
  • 02:16 do not work with a mean and standard deviation of a distribution.
  • 02:19 Because those statistical parameters are often whacked out
  • 02:22 by the nature of the non-normality.
  • 02:24 Instead, they are optimized for data distributions that are not bell-shaped.
  • 02:29 One advantage of these tests is that they generally can be used
  • 02:32 with either attribute or ordinal data.
  • 02:34 Again, the shape of the curve is more important than the smoothness of
  • 02:38 the distribution curve.
  • 02:40 Which of course, means that they work with both discrete and continuous data types.
  • 02:44 Let's look at this table for a moment, it helps to draw a comparison between some of
  • 02:48 the common hypothesis tests used with normal data, and
  • 02:52 those used with non normal data.
  • 02:53 When there's one set of sample data being compared to a target
  • 02:57 we use a 1-Sample t-Test for normal data, and
  • 03:00 either 1-Sample Sign or 1-Sample Wilcoxon with normal data.
  • 03:04 More about both of those in the next few slides.
  • 03:07 With 2 Samples that are normal, we use the F Test and the sample t-Test.
  • 03:12 When they are not normal we use Levene's Test/Mann-Whitney Test.
  • 03:16 Finally, for multiple sample test with normal data we use ANOVA.
  • 03:21 With non--normal data we use Mood's Median, Kruskal-Wallis and the Friedman.
  • 03:25 Depending upon the shape of the data.
  • 03:28 More about all those on the other lesson.
  • 03:31 So, let's consider the One-Sample Sign Test first.
  • 03:34 This test is similar to the One-Sample T-Test for Normal data.
  • 03:37 But it is used when the data is not normal such as, when it's skewed, truncated,
  • 03:42 or non-symmetric.
  • 03:43 Since the data is not normal, the Median is used instead of the Mean.
  • 03:48 The Median is a better measure of central tendency for non normal data.
  • 03:51 The hypothesis is normally structured such that the null hypothesis states that
  • 03:56 the sample Median equals the Target Median.
  • 03:59 The alternative hypothesis is normally stating that the sample Median
  • 04:03 will be either above or below the target
  • 04:05 depending upon the underlying question of the hypothesis test.
  • 04:09 Conducting this test in Minitab is very simple, start with the Stat pull down
  • 04:14 menu, select Nonparametric which is near the bottom of that menu.
  • 04:18 And then, select 1-Sample Sign which is at the top of the Nonparametric menu.
  • 04:22 Select your data column the same way we've done on other tests, and
  • 04:26 then enter the target median in the field that is labeled test median.
  • 04:30 Finally, set the operator for your alternate hypothesis at less than,
  • 04:34 greater than, or not equal to.
  • 04:36 The P value result will be shown in the session window.
  • 04:40 The One-Sample Wilcoxon works almost exactly the same as
  • 04:43 the One-Sample Sign Test.
  • 04:46 It's also similar to the One-Sample T-Test for Normal data.
  • 04:49 However, a difference between this test and the One-Sample Sign Test
  • 04:53 is that the One-Sample Wilcoxon requires the data to be symmetric.
  • 04:57 It can be uniform of even bath tub shape, but it should be symmetric.
  • 05:02 Since the data is not normal,
  • 05:04 the Median is used instead of the Mean as the better measure of central tendency.
  • 05:08 The null and alternative hypotheses are the same as the One-Sample Sign Test.
  • 05:13 The null is that the Median equals the Target.
  • 05:16 And the alternative hypothesis is that the Median is greater than or
  • 05:19 less than the Target.
  • 05:20 The Minitab actions are the same as 1-Sample Sign, except that,
  • 05:24 of course, 1-Sample Wilcoxon is selected from the Nonparametric menu.
  • 05:28 The data column selection, entering the target, and setting the operator are done
  • 05:33 in the same manner as the one sample sign. >> One-Sample Sign and
  • 05:38 One-Sample Wilcoxon, are great tests to use when checking to see if
  • 05:42 your solution is able to meet a desired standard.

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