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What is statistical testing and statistical significance?
Significance testing involves using a range of statistical methods to identify what are termed 'statistically significant' differences or trends in data.
A significant difference is a difference which is assumed to be real, rather than due to (what is termed) 'sampling error'.
Sampling error simply refers to the fact that when you select people to do surveys, it is often the case that you may not have selected or 'sampled' in a way, which represents the true population of people you are looking at.
For instance, while a result of 80% for group 1 and 85% for group 2 may look like a real difference, it’s quite possible that this 5% difference is just due to what is called 'sampling error'.
To assess whether differences in data are likely to be real, statistical significance testing can be quite useful. By applying a range of statistical methods, we can test whether that 5% difference is statistically significant (or just due to sampling error).
This produces a probability figure represented as 'p'.
A probability less than 0.05 is typically used as an indication of statistical significance. It is also the research industry standard.
This indicates that the theoretical chance of the result occurring by chance is less than 5%.
As such, a result of p<.001 (implying that the probability figure was so small and the result is very unlikely as a result of chance) means that the difference is very statistically significant.
Conversely, figures greater than a probability or 'p' value of 0.05 are generally considered to be nonsignificant or not statistically significant.
Note that < implies 'less than'.
Different statistical tests can be used depending on the structure of the data. For instance, ttests are used for continuous measures (like a rating scale), while chi square and regular logistic regression are often used for categorical measures (e.g., the proportion of people in category A v B).
Understanding statistical significance in social and market research
Some examples of results to assist with the interpretation of statistical significance are below:

ttests  Group 1 Mean=5.3, Group 2 Mean=5.5 (p<.05) – implies that Group 2 was rated significantly higher than Group 1 (ie. The 0.2 difference in mean ratings was 'statistically significant')

Pearson correlations – Between age and attitude  r=.86 (p<.05) – implies that there was a statistically significant relationship between age and attitude (note that the .86 implies that, as age increased, attitudinal ratings became significantly higher – in contrast, .86 would have implied the reverse – i.e., As age increased, attitudinal ratings decreased)

Chi square  p>.07 – this implied that the probability of obtaining the chi square test result was more than 0.05 (0.07), so this result is not statistically significant

Logistic regression  Odds ratios can tell us that, compared to a reference group, another group is significantly four times more likely to suffer anxiety (OR=4.0, p<.001)
Note that there are many statistical tests which can be used to identify trends in data and to assess whether a difference is 'statistically significant'.
It should also be noted a result of p<.05 does not necessarily mean that the result is PRACTICALLY SIGNIFICANT.
Indeed, while the difference is significant, the group difference may only be very small, implying that groups were fairly similarly overall in spite of the significant trend.