# Significance Tests on Trend Plots

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Trends that are significantly increasing or decreasing are color-coded (these colors can be modified using Table Styles).

Where a Date is used as the Date/Filter question of a trend chart, significance is computed by splitting the time periods into two halves, where ties go with the left half (see Comparing Two Independent Groups). For example, if the Aggregation of the question is Months, and there are five months, January, February, March, April and May, significance is computed by comparing the pooled data from January and February with the data from April and May (March is excluded).

Where a Pick One question is used as the Date/Filter question the trend is shown with a column chart and significance is shown via the color-coding of the columns.

For reasons of computational speed, a shortcut is used in computing significance tests for trend charts involving averages. Each bar is compared as a complex sample two sample t-test. The shortcut is that the second group’s standard error is computed as a function of the existing standard errors. First, all the standard errors are combined sequentially, whereby the first two groups’ standard errors are combined, then this combined standard error is combined with the third group, and so on, using the following formula to combine pairs of standard errors:

${\displaystyle e_{a+b}^{2}={\frac {[n_{a}(n_{a}-1)e_{a}^{2}+n_{b}(n_{b}-1)e_{b}^{2}+(n_{a}n_{b})/(n_{a}+n_{b})(m_{a}-m_{b})^{2}]}{(n_{a}+n_{b})(n_{a}+n_{b}-1)}}}$

where ${\displaystyle n_{a}}$ is the effective sample size of the first group, ${\displaystyle e_{a}^{2}}$ is the squared standard error of the first group, ${\displaystyle m_{a}}$ is the relevant statistic (proportion, mean or coefficient, depending upon the data) of the first group and the second group’s terms are denoted with the subscript b (see: Baker, R. W. R. and J. A. Nissim (1963), "Expressions for Combining Standard Errors of Two Groups and for Sequential Standard Error," Nature, 198 (8 June 1963).