5 Appendix
5.1 Probability
5.1.1 Probability for equally likely outcomes
The probability of event \(A\) occurring is \(P(A) = \frac{x}{n}\), where \(n\) is the number of trials and \(x\) is the number of trials during which \(A\) occurred.
5.1.2 Multiplication and addition
The general case of the multiplication rule is that, for two events \(A\) and \(B\):
\(P(A \& B) = P(A) * P(B|A)\), where \(P(B|A)\) is the probability that \(B\) occurs given than \(A\) has already occurred.
The Addition Law states that, for two events \(A\) and \(B\):
\(P(A\ or\ B\ or\ A\&B) = P(A) + P(B) - P(A\&B)\)
and
\(P(A\ or\ B\ but\ not\ A\&B) = P(A) + P(B) - 2*P(A\&B)\)
and when \(A\) and \(B\) are mutually exclusive, then:
\(P(A\ or\ B) = P(A) + P(B)\)
5.1.3 Bayes’ theorem
\(P(A|B)= \frac{P(B|A) * P(A)}{P(B)}\), where \(P(B)= P(A) * P(B|A) + P(A’) * P(B|A’)\).
5.2 Univariate statistics
5.2.1 Mean
mean()
The mean is calculated as \(\bar{y} = \frac{\sum{y}}{n}\).
For measures of central tendency and dispersion, we use Greek letters to refer to population values and Latin letters to refer to samples:
| Population | Sample | |
|---|---|---|
| Mean | \(\mu\) | \(\bar{y}\) |
| Variance | \(\sigma^2\) | \(s^2\) |
| Standard deviation | \(\sigma\) | \(s\) |
5.2.2 Median, quartiles and adjacent values
median(), quantile(), IQR()
With data ordered by rank, the median value is the middle value or the \((\frac{n+1}{2})^{th}\) value, the lower quartile, \(Q1\), is the \((\frac{n+1}{4})^{th}\) value, the upper quartile, \(Q3\), is the \((\frac{3*(n+1)}{2})^{th}\) value. The inter-quartile range is \(IQR = Q3 – Q1\). The upper adjacent value is the upper value that is less than \(Q3 + (1.5 * IQR)\). The lower adjacent value is the lower value that is more than the \(Q1 – (1.5 * IQR)\). Outliers are defined as values that lie outside the range \(Q1 - 1.5*IQR\) or \(Q3 + 1.5*IQR\).
5.2.3 The sum of squares
The sum of squares is given by \(SS = \sum{y^2} - \frac{(\sum{y})^2}{n}\) where \(n\) is the number of observations.
5.2.4 Measures of dispersion
var(), sd(), length()
When dealing with populations the following formulas are used:
Variance: \(\sigma^2 = \frac{SS}{n}\)
Standard deviation: \(\sigma = \sqrt{\frac{SS}{n}} = \sqrt{\sigma^2}\)
When dealing with samples the following formulas are used:
Variance: \(s^2 = \frac{SS}{n-1}\)
Standard deviation: \(s = \sqrt{\frac{SS}{n-1}}\)
5.3 The Binomial distribution
dbinom(), pbinom(), qbinom(), rbinom()
The binomial distribution describes the expected distribution for two mutually exclusive outcomes. The formula is given by:
\(P(x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}\)
where \(P(x)\) is the probability of \(x\) ‘successes’ occurring, \(n\) is the number of trials, \(p\) is the probability of \(x\) in a single trial, and \(q\) is the probability of not \(x\) in a single trial with \(p+q=1\).
A binomially distributed variable has mean = \(n*p\) and variance = \(n*p*q\).
5.4 The Poisson distribution
dpois(), ppois(), qpois(), rpois()
The Poisson distribution for a sample is defined as follows:
\(P(x) = \frac{\lambda^x e^{-\lambda}}{x!}\)
where \(\lambda\) is the mean and \(x\) is the value (count) of interest. Note that in a sample, we use the sample mean \(\bar{y}\) as our estimate for \(\lambda\)
5.5 Z scores
scale()
The standard normal distribution is \(Norm(\mu=0, \sigma=1)\). Z scores are used to convert any normal distribution to the standard normal distribution:
\(z = \frac{y-\mu}{\sigma}\)
where \(y\) is the value, \(\mu\) is the mean of the population and \(\sigma\) is the standard deviation of the population.
5.6 Samples taken from a population
sd(), sqrt(), length(), mean(), sqrt()
5.6.1 Standard error of the mean
The standard error is the standard deviation of sample means:
\(SEM = \sigma_{\bar{y}} = \frac{\sigma}{\sqrt{n}}\)
where \(\sigma\) is the population standard deviation and \(n\) is the sample size. If we are testing a sample taken from a population of known population mean and population standard deviation we use the formula:
\(z = \frac{\bar{y}-\mu}{\sigma_{\bar{y}}}\)
where \(\bar{y}\) is a particular sample mean, \(\mu\) is the population mean, and \(\sigma_{\bar{y}}\) is the standard error of the mean.
5.7 The t distribution
dt(), pt(), qt(), rt()
The t-distribution describes the distribution of \(T\) statistics expected under the null hypothesis.
5.7.1 The one sample t-test
t.test()
The T statistic is calculated as:
\(T = \frac{\bar{y}-\mu}{s_{\bar{y}}}\)
Where \(\bar{y}\) is the sample mean, \(\mu\) is the hypothesized population mean, and \(s_{\bar{y}}\) is the standard error such that \(s_{\bar{y}} = \frac{s}{\sqrt{n}}\) with sample standard deviation \(s\).
Use t.test(), or find the p-value associated with your T with pt(T, df-1).
5.7.2 Confidence intervals for sample means
t.test(), qt(), mean(), sd(), length(), sqrt()
The 95% CI for a mean is calculated as:
\(\bar{y} \pm t_{\alpha/2, df=n-1} * s_{\bar{y}}\)
5.8 ANOVA
anova(lm()), aov()
An ANOVA table contains the following:
| Source of variation | Sum of Squares | Degrees of Freedom | Mean Square | F |
|---|---|---|---|---|
| Between groups | \({SS}_{groups}\) | \({df}_{groups}\) | \(\frac{{SS}_{groups}}{{df}_{groups}}\) | \(\frac{{MS}_{groups}}{{MS}_{error}}\) |
| Error | \({SS}_{error}\) | \({df}_{error}\) | \(\frac{{SS}_{error}}{{df}_{error}}\) | |
| Total | \({SS}_{total}\) | \({df}_{total}\) |
with:
\({SS}_{groups} = \sum{n_j (\bar{y_j} - \bar{\bar{y}})^2}\)
\({SS}_{error} = \sum{(y_{ij} - \bar{y_j})^2}\)
\({SS}_{total} = {SS}_{groups} + {SS}_{error} = \sum{(y_{ij} - \bar{\bar{y}})^2}\)
where \(y_{ij}\) is the \(i^{th}\) observation in group \(j\), \(\bar{y_j}\) is the mean for group \(j\), and \(\bar{\bar{y}}\) is the grand mean. With \(N\) total observations, \(n_j\) observations per group, and \(k\) groups, the degrees of freedom are:
\({df}_{groups} = k - 1\)
\({df}_{error} = N - k = k (n_j-1)\)
\({df}_{total} = {df}_{groups} + {df}_{error} = N-1\)
The critical value (CV) for samples of equal size is given as:
\(CV = q \sqrt{\frac{{MS}_{error}}{n}}\)
where \(q\) either comes from a table online (the studentized range distribution), or from qtukey(1-alpha, k, df_error).
5.9 Regression & correlation
lm(), cor(), confint(), predict()
We do not bother calculating regression coefficients by hand.