# One-sample tests

Tests for whether a single sample (single column of data) comes from a population with a given, often hypothetical, mean or median. For example, are a number of oxygen isotope values from sea shells (single sample) the same as average seawater composition (given mean)? The given test value must be typed in. In addition, single-case tests are used to test whether a single value comes from the same population as the given sample.

#### One-sample t test for given mean (parametric)

Sample mean and standard deviation are estimated in the usual way. The 95% confidence interval for the difference in means is based on the standard error for the estimate of the mean, and the t distribution. Normal distribution is assumed.

The t test has null hypothesis

H0: The samples is taken from a population with the given mean.

##### Example use of results for publication:

The sample mean (μ=5, *N*=11) is not significantly different from 4 (Student's *t*=1.70, *p*=0.12).

#### One-sample Wilcoxon signed-rank test for given median M (nonparametric)

The one-sample Wilcoxon test has null hypothesis

H0: The sample is taken from a population with median M.

For large n (say n>10), the large-sample approximation to *p* can be used. This depends on the normal distribution of the test statistic *W*. For n<13, an exact *p* value is computed, by complete enumeration of all possible reassignments. This is the preferred *p* value, if available.

##### Example use of results for publication:

The sample median (*M*=5, *N*=11) is significantly different from 3 (Wilcoxon *W*=53, exact *p*<0.01).

#### Single-case tests

The single-case tests have null hypothesis

H0: The given single value *y* is taken from the same population as the given sample.

Normal distribution is assumed. A simple *z* test is often used for this purpose, and is also provided by Past. However, the *z* test is inaccurate because it assumes that the mean and standard deviations are given exactly, whereas in reality they are estimated from the sample. Therefore, Past also provides a modified *t* test (Sokal & Rohlf 1995; Crawford & Howell 1998).

##### Example use of results for publication:

The observed value of 10 is significantly different from the sample (Student's *t*=2.46, *p*=0.034).

#### Binomial proportion

Expects binary data (0 or non-zero) in the given sample. The proportion of non-zeroes in the sample is compared with the given proportion, and confidence intervals are also reported.

The same test is provided in the Single proportion module, but there a binary data column is not required, only the observed proportion.

#### References

Crawford, J.R. & Howell, D.C. 1998. Comparing an individual’s test score against norms derived from small samples. The Clinical Neuropsychologist 12:482-486.

Sokal, R.R. & Rohlf, J.F. 1995. Biometry. W.H. Freeman, San Francisco.