# C Appendix: Symbols used for statistics and parameters

Almost all confidence intervals have the form

$\text{statistic} \pm ( \text{multiplier} \times \text{s.e.}(\text{statistic})).$ Notes:

• The multiplier is approximately 2 for an approximate 95% CI (based on the 68–95–99.7 rule).
• $$\text{multiplier} \times \text{s.e.}(\text{statistic})$$ is called the margin of error.
• Odds ratios have a slight complication, so this formula does not apply for odds ratio (for which the test statistic is $$\chi^2$$ and not $$t$$). For the same reason, a standard error for ORs is not given.

For hypothesis testing, $$t$$-scores all have the form:

$t = \frac{\text{statistic} - \text{parameter}}{\text{s.e.}(\text{statistic})}$ Notes:

• Odds ratios have a slight complication, so this formula does not apply for odds ratio (for which the test statistic is $$\chi^2$$ and not $$t$$). For the same reason, a standard error for ORs is not given.
• The $$\chi^2$$ statistic is approximately like a $$z$$-score of (where $$\text{df}$$ is the ‘degrees of freedom’ given in the software output):

$\sqrt{\frac{\chi^2}{\text{df}}}.$

TABLE C.1: Some sample statistics used to estimate population parameters. Empty table cells means that these are not studied. The asterisk means that no formula is given in this text.
Parameter Statistic Standard error S.E. formula reference
Proportion $$p$$ $$\hat{p}$$ $$\displaystyle\text{s.e.}(\hat{p}) = \sqrt{\frac{ \hat{p} \times (1 - \hat{p})}{n}}$$ Def. 20.2
Mean $$\mu$$ $$\bar{x}$$ $$\displaystyle\text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}}$$ Def. 22.1
Standard deviation $$\sigma$$ $$s$$
Mean difference $$\mu_d$$ $$\bar{d}$$ $$\displaystyle\text{s.e.}(\bar{d}) = \frac{s_d}{\sqrt{n}}$$ Def. 23.2
Diff. between mean $$\mu_1 - \mu_2$$ $$\bar{x}_1 - \bar{x}_2$$ $$\displaystyle\text{s.e.}(\bar{x}_1 - \bar{x}_2)$$
Odds ratio Pop. OR Sample OR $$\displaystyle\text{s.e.}(\text{sample OR})$$
Correlation $$\rho$$ $$r$$
Slope of regression line $$\beta_1$$ $$b_1$$ $$\text{s.e.}(b_1)$$
Intercept of regression line $$\beta_0$$ $$b_0$$ $$\text{s.e.}(b_0)$$
R-squared $$R^2$$
TABLE C.2: Some other symbols used
Symbol Meaning Reference
$$H_0$$ Null hypothesis Sect. 28.2
$$H_1$$ Alternative hypothesis Sect. 28.2
df Degrees of freedom Sect. 31.4
CI Confidence interval Chap. 21
s.e. Standard error Def. 18.2
$$n$$ Sample size
$$\chi^2$$ The chi-squared test statistic Sect. 31.4