Problem Sets



Steven Holland

Stating statistical results

Statistical results should be stated succinctly, and the numerical values should always be added parenthetically or in a supporting table. Note that you should report only the significant figures that your data supports. Generally, R will report many more significant figures than are in your data. Not only are those extra digits typically not relevant and inaccurate, they also make your results harder to read.

Statistical results are commonly stated in one of two forms. When there is an emphasis on the value of the quantity in question and the uncertainty in it, confidence intervals are typically used. When there is an emphasis on statistical significance, p-values are presented, although confidence intervals can be used instead. These two forms should not be combined; do not report both a confidence interval and a p-value.

Confidence intervals

Confidence intervals can be presented in several ways, but regardless of which you choose, the confidence level is always stated. The first way is the clearest and most informative because it states the estimate in the text and the confidence interval in parentheses. With this, you know the value of what you are trying to estimate, the degree of uncertainty in that value, and whether the estimate is statistically significant. For example:

“The difference in mean length is 2.75 mm (95% C.I.: 2.33–3.17 mm)”.

“The correlation of sunlight and chlorophyll is 0.64 (95% C.I.: 0.29–0.85)”.

Notice a couple of things. First, we state the statistic (the difference in means and the correlation in these two examples). We also report the variables involved; we do not simply say “the mean” or “the correlation”. Especially in papers with many variables, not stating what variables are correlated creates confusion. Second, we provide an estimate of the statistic with its units. Third, we put the confidence interval in parentheses with the units the confidence level. Learn this succinct style.

A second but less desirable approach puts both the estimate and the confidence interval in parentheses (sometimes also with the sample size). This is not as good as the first option, as it places less emphasis on the estimate. Follow the standard order in which these are listed: estimate, confidence interval, or estimate, sample size, confidence interval. Again, note that we always state the variables and their units. When the null hypothesis has a value of zero, saying “statistically non-zero” rather than simply “statistically significant”, we convey the test results jargon-free. This approach is also more specific and avoids the word “significant” with its ambiguous meaning.

“The difference in mean length is statistically non-zero (2.75 mm; 95% C.I.: 2.33–3.17 mm)”.

“Sunlight and chlorophyll have a positive, statistically non-zero correlation (r=0.64, 95% C.I.: 0.29–0.85)”.

Even less desirable are approaches that focus on statistical significance and I recommend these unless statistical significance is specifically requested. Some editors may insist upon this approach, even though statistical significance is obvious from the confidence interval (i.e., whether the confidence interval brackets the null hypothesis). If you have stated whether the result is statistically significant, be sure to list the variables and provide the numerical support parenthetically. Again, report the variable and its units, and state the results succinctly like this:

“The difference in mean length is statistically significant (2.75 mm; 95% C.I.: 2.33–3.17 mm)”.

“The correlation between sunlight and chlorophyll is statistically significant (r=0.64, 95% C.I.: 0.29–0.85)”.

If you must state significance per se, it is better to parenthetically state what significance means. In this case, statistical significance means the correlation is not zero (the null hypothesis). Some reviewers or editors may not like this and ask you to remove it, thinking it is redundant (if they do understand statistics) or confusing (if they don’t understand statistics).

“The correlation between sunlight and chlorophyll is statistically significant (i.e., non-zero; r=0.64, 95% C.I.: 0.29–0.85)”.

Other tests follow a similar pattern of reporting: state the result in plain language, the variables involved, and follow the numerical support in parentheses. This first example compares two means by showing their confidence interval. The mean, sample size, and confidence interval are in parentheses.

“Mean phosphorous content is greater in the Atlantic Ocean (23.4, n=73, 95% CI: 22.2–24.6) than in the Pacific Ocean (17.9, n=53, 95% C.I.: 15.6–20.1).

This example from an F-test of variance follows the same approach: the parentheses include the statistic, the degrees of freedom, and the confidence interval.

“Shell concentrations are more variable in southeastern middens than midwestern middens (F=4.53, df=37 and 53, 95% C.I.: 2.64–8.21).


Confidence intervals convey more information than p-values, so confidence intervals should be generally preferred. If you report a confidence interval, do not report the p-value.

When reporting p-values is required, lead with a statement of the results and follow parenthetically with the statistic, the degrees of freedom, and the p-value, in that order. The following style is best because it puts the emphasis on the results without being formal about the null hypothesis. Notice also that you write “p=”, not the longer “p-value=” that R reports. Here are three examples:

“Mean heights of Loblolly Pine in Georgia and South Carolina are not statistically different (t=-0.37, df=29.4, p=0.72)”.

“Mean heights of Loblolly Pine in Georgia and South Carolina are statistically indistinguishable (t=-0.37, df=29.4, p=0.72)”.

“Mean heights of Loblolly Pine in Georgia and South Carolina are statistically different (t=2.23, df=152.96, p=0.027)”.

“The difference in mean heights of Loblolly Pine in Georgia and South Carolina is statistically significant (t=2.23, df=152.96, p=0.027)”.

Note the problem in all four of these examples that is so common with p-values: you have no idea of how close or different the actual measurements are, or if there is a difference, which way the difference goes. It is hard to do this without a confidence interval, but you could say something like this:

“The mean height of Loblolly Pine in Georgia is greater than in South Carolina (t=2.23, df=152.96, p=0.027)”.

“The mean height of Loblolly Pine in Georgia is greater than in South Carolina (151.3" vs. 147.7"; t=2.23, df=152.96, p=0.027)”.

Confidence intervals express this so much better, and that is another reason why they are the preferred approach.

A second way of phrasing p-values places the emphasis on the null hypothesis, making it somewhat indirect:

“The null hypothesis that the means of the two groups are equal was accepted (t=-0.37, df=29.4, p=0.72)”.

It would be better to be explicit about the two groups and what was measured, but the sentence can easily become too confusing. Many people avoid this style because it is unnecessarily formal.

Regardless of whether you use confidence intervals or p-values, you should state the conclusion succinctly and support it with the necessary statistics, usually parenthetically and always with an appropriate number of significant figures.