Appendix C. Symbols, formulas, statistics and parameters

Section C.1 Symbols and standard errors

• Table Table C.1 lists the statistics used to estimate unknown population parameters.

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• When the sampling distribution is approximately normally distributed, under appropriate statistical validity conditions, this is indicated by ✔.

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• The value of the mean of the sampling distribution (the sampling mean) is:

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  • unknown, for confidence intervals.

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  • assumed to be the value given in the null hypothesis, for hypothesis tests.

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Section C.2 Confidence intervals

• The multiplier is approximately \(2\) to create an approximate \(95\)% CI (based on the \(68\)–\(95\)–\(99.7\) rule).

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• The quantity ’\(\text{multiplier} \times \text{s.e.}(\text{statistic})\)’ is called the margin of error.

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• Software uses exact multipliers to form exact confidence intervals.

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• When the sampling distribution for the statistic does not have an approximate normal distribution (e.g., for ORs and correlation coefficients), this formula does not apply and the CIs are taken directly from software output when available.

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Section C.3 Hypothesis testing

• If the test-statistic is a \(z\)-score, the \(P\)-value can be found using tables (Appendix Section B.3), or approximated using the \(68\)–\(95\)–\(99.7\) rule.

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• If the test-statistic is a \(t\)-score, the \(P\)-value can be approximated using tables (Appendix Section B.3), or approximated using the \(68\)–\(95\)–\(99.7\) rule (since \(t\)-scores are similar to \(z\)-scores; Sect. Section 20.4).

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• When the sampling distribution for the statistic does not have an approximate normal distribution (e.g., for ORs and correlation coefficients), this formula does not apply and \(P\)-values are taken from software when available.

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• A hypothesis test about ORs uses a \(\chi^2\) test statistic. For \(2\times 2\) tables only, the \(\chi^2\)-value is equivalent to a \(z\)-score with a value of \(\sqrt{\chi^2}\text{.}\)

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Section C.4 Sample size estimation

• To estimate the sample size needed for estimating a proportion (Sect. Section 32.3), use:

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  \begin{equation} n = \frac{1}{(\text{Margin of error})^2}.\tag{C.3} \end{equation}
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• To estimate the sample size needed for estimating a mean (Sect. Section 32.4) use:

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  \begin{equation} n = \left( \frac{2\times s}{\text{Margin of error}}\right)^2\tag{C.4} \end{equation}
  for some estimate \(s\) of the standard deviation of the data.

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• To estimate the sample size needed for estimating a mean difference (Sect. Section 32.5) use:

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  \begin{equation} n = \left( \frac{2 \times s_d}{\text{Margin of error}}\right)^2\tag{C.5} \end{equation}
  for some estimate \(s_d\) of the standard deviation of the differences.

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• To estimate the sample size needed for estimating the difference between two means (Sect. Section 32.6) use:

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  \begin{equation} n = 2\times \left( \frac{2 \times s}{\text{Margin of error}}\right)^2\tag{C.6} \end{equation}
  for each group being compared, where \(s\) is an estimate of the common standard deviation in the population for both groups. This formula assumes:

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  • the sample size for each group will be the same; and

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  • the standard deviation in each group is the same.

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• To estimate the sample size needed for estimating the difference between two proportions (Sect. Section 32.7) use:

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  \begin{equation} n = \frac{2}{(\text{Margin of error})^2}\tag{C.7} \end{equation}
  for each group being compared. This formula assumes the sample size in each group will be the same.

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• In sample size calculations, round up the sample size found from the above formulas.

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Section C.5 Other formulas

• To calculate \(z\)-scores (Sect. Section 20.4), use

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  \begin{equation} z = \frac{\text{value of variable} - \text{mean of the distribution of the variable}}{\text{standard deviation of the distribution of the variable}}.\tag{C.8} \end{equation}
  \(t\)-scores are like \(z\)-scores. When the ’variable’ is a sample estimate (such as \(\bar{x}\)), the ’standard deviation of the distribution’ is a standard error (such as \(\text{s.e.}(\bar{x})\)).

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• The unstandardising formula (Sect. Section 20.8) is \(x = \mu + (z\times \sigma)\text{.}\)

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• The interquartile range (IQR) is \(Q_3 - Q_1\text{,}\) where \(Q_1\) and \(Q_3\) are the first and third quartiles respectively (or, equivalently, the \(25\)th and \(75\)th percentiles).

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• The smallest expected value (for assessing statistical validity when forming CIs and conducting hypothesis tests with proportions or ORs) is

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  \begin{equation} \frac{(\text{Smallest row total})\times(\text{Smallest column total})}{\text{Overall total}}.\tag{C.9} \end{equation}
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• The regression equation in the sample is \(\hat{y} = b_0 + b_1 x\text{,}\) where \(b_0\) is the sample intercept and \(b_1\) is the sample slope.

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Section C.6 Other symbols and abbreviations used