## Measuring differences

Whenever sample data are being used, we have to compensate for sampling error. Z and t tests are methods used to assess the differences among data while still taking into account the influence that sampling error may have played. The methods are employable when measurement scales are at least interval, observations are independent, only one or two groups are involved, differences between two means are involved and, in the case of the t test, where observations are drawn from a normally distributed population.

Provided that these criteria are met, we need to decide which of the two tests is the more appropriate. A Z test is used if the sample size is larger than 30. A t test is appropriate when the sample size is less than 30. If the population variance is not known, then the sample variance is assumed to equate to it and is used instead. The t test is usually employed in marketing research.

### Z test

One- and two-tailed tests The tabular value of Z at any level of significance depends on whether a one- or a two-tailed test is involved. Where we are looking for equality between two sets of data, H0: /i1 = fi2, a two-tailed test is appropriate.

Where we are looking at a situation that involves evaluating whether something is greater than something else, a one-tailed test is appropriate. The same also applies to situations where we are evaluating whether something is less than something else.

Dealing with proportions If proportions are involved rather than discrete numbers, a different formula is needed - although in other respects the procedure is the same.

The formula used is:

where p = sample proportion

P = population proportion

The procedures for using the t test are similar to those for using the Z test. However, there are different formulae that are applicable in different situations (see Kress1). For example:

s = standard deviation of the sample x = sample mean f = population mean n = sample size

When the population data do exist, but the other group is represented by a fairly small sample, this formula should be employed.

Tests with non-metric data

There are several tests that can be used, depending on the nature of the data and the comparisons made. These include:

• Mann-Whitney U test: this is used when comparing two groups, but the basis for comparison is data in ordinal form.

• Wilcoxon or signed rank test: this is appropriate for situations where two matching (non-independent) samples are being compared using ordinal (non-parametric) data.

• Kruskal-Wallis test: this is used when more than two independent samples are involved and the data are ordinal.

• Friedman two-way analysis of data: this is used when three or more related samples are involved and the data are ordinal.

We have to decide whether the difference between two or more numbers, or sets of numbers, is probably significant, and whether two numbers are causally related.

### Difference between numbers in a single array

Suppose that in a survey of 1500, results showed that 64% chose brand X, 24% chose brand Y and the remainder made no choice. Are these real differences? Assuming that this was a probability sample, the answer is yes: the sampling error for 64% is some 2.5% and for 24% it is some 2.2%.

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