- One Sample Tests
- Two Sample Tests, Independent
*K*-Sample Tests- Two Sample Tests, Dependent (matched)
- Homework

The chi-square statistic is used when the dependent variable is at the nominal level and our parametric assumptions of normality and homogeneity of variance cannot be met as discussed in the previous lesson. In this lesson we explore nonparametric tests of significance for ordinal dependent variables.

Second, the statistically more powerful
**Mann-Whitney U test**,
tests not only the median, but also the total
distribution (central tendancy and distribution).
The null hypothesis specifies that there is no
difference in the scores of the two populations sampled.
Two

When both groups are larger than 20, the sampling distribution
approaches the normal distribution and a *z*-score
can be computed from the sampling distribution mean
µ_{U} = ½*n*_{1}*n*_{2},
and standard deviation
_{U} = sqrt(
*n*_{1}*n*_{2}
(*n*_{1} + *n*_{2} +1)/12).
The *z* score is calculated in the usual way
with *z* = (*U* - µ_{U})/
_{U}.

A table of Mann-Whitney critcal *U* values is given below
for alpha=0.05 (1-tailed=directional H_{a}) and selected
sample sizes only. More extensive tables are readily
available.

n_{i}\n_{j} | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 15 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 5 |

3 | 0 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 8 | 12 |

4 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 13 | 19 |

5 | 1 | 2 | 3 | 5 | 6 | 7 | 9 | 10 | 12 | 19 | 26 |

6 | 1 | 3 | 4 | 6 | 8 | 9 | 11 | 13 | 15 | 24 | 33 |

7 | 1 | 3 | 5 | 7 | 9 | 12 | 14 | 16 | 18 | 29 | 40 |

8 | 2 | 4 | 6 | 9 | 11 | 14 | 16 | 19 | 21 | 34 | 48 |

9 | 2 | 5 | 7 | 10 | 13 | 16 | 19 | 22 | 25 | 40 | 55 |

10 | 2 | 5 | 8 | 12 | 15 | 18 | 21 | 25 | 28 | 45 | 63 |

15 | 4 | 8 | 13 | 19 | 24 | 29 | 34 | 40 | 45 | 73 | 101 |

20 | 5 | 12 | 19 | 26 | 33 | 40 | 48 | 55 | 63 | 101 | 139 |

**Tied ranks** generally have minimal effect on both
the Mann-Whitney *U* and Kruskal-Wallis *H*
and a correction factor can be applied. However,
results from either test might be questionable when
there are an excessive number of tied ranks.

In conclusion we wish to emphasize both the
role of nonnormal populations and small sample
size in the use of these statistics. Larger
samples give better statistical precision,
but sometimes a limited population or expense
preclude obtaining such.
Thus the best statistical test for a given situation
is defined more in terms of the research conditions
than in terms of parametric *vs* nonparametric.

BACK | NO HOMEWORK! | ACTIVITY | THE END |
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- e-mail: calkins@andrews.edu
- voice/mail: 269 471-6629/ BCM&S Smith Hall 106; Andrews University; Berrien Springs,
**classroom:**269 471-6646; Smith Hall 100/**FAX:**269 471-3713; MI, 49104-0140- home: 269 473-2572; 610 N. Main St.; Berrien Springs, MI 49103-1013
- URL: http://www.andrews.edu/~calkins/math/edrm611/edrm15.htm
- Copyright ©2005, Keith G. Calkins. Revised on or after Aug. 4, 2005.