1.1. Descriptive statistics

In tutorial 1, we calculated descriptive statistics for samples (including central tendency and variability estimates such as the mean and standard deviation).

A sample refers to the group of observations drawn from the total population of interest (e.g. a sample of 72 female anorexic patients were studied out of the nation-wide hospital registry of female anorexic patients)

Descriptive statistics allow us to summarise the characteristics of a sample

1.2. Inferential statistics

Unlike descriptive statistics, inferential statistics are calculated with respect to the population.

A population includes all possible elements (e.g. the entire female anorexia patient hospital registry)

Inferential statistics allow us to characterise a population.

One does not always have access to an entire population, but probability theory can be used to infer characteristics about a population

Exercise 2.0.1

Exercise 2.0.2

Exercise 2.0.3

Exercise 2.0.4

Exercise 2.1.1

Exercise 2.1.2

2.2.1. Mutually exclusive events

Two or more events are mutually exclusive when they cannot occur at the same time.

1. Some examples:
   • Turning left OR right in a car are mutually exclusive because one cannot turn both left and right at the same time (i.e. simultaneously).
   • You cannot have both a King and an ace on one playing card
   • You cannot get both heads and tails on a coin toss

2.2.2. Exhaustive events

Events are exhaustive if they encompass the entire range of possible outcomes. The probabilities of all possible events sum to 1

1. Some examples:
   • values 1,2,3,4,5,6 on a die are exhaustive of all possible values that can be obtained on dice (i.e. no other value is possible on a die)
   • 7 continents on earth (not, say, 5 or 6)

2.2.3. “NOT” rule

The probability of an event NOT occurring is equal to 1-probability of event occurring

“NOT” rule probability formula:

\[ \begin{aligned} P(NOT Event) ={1-P(Event)} \end{aligned} \]

1- 6/41

2.2.4. “OR” rule

The probability of either one OR other events occurring is equal to the sum of the individual probabilities probability of event 1 + probability of event 2. Otherwise referred to as the ADDITIVE rule

“OR” rule probability formula:

\[ \begin{aligned} {P(EventA~OR~EventB)} ={P(EventA)+P(EventB)} \end{aligned} \]

6/41 + 2/41

6/41 + 2/41 + 5/41

2.3.1. Independent events

We call events independent when they have no effect on each other. In other words, events are independent when the probability of one event occurring does not influence the probability of another event occurring.

1. Some examples:
   • When throwing a die, if you obtained a 5 on the previous turn, this will not affect what number you roll on the next turn/roll
   • Today’s weather in Cape Town vs weather in Cape Town in exactly a year’s time (we assume that today’s weather will have no bearing on the weather experienced in another year’s time)

Continuing with the smartie example…

2.3.2 Sampling with replacement

The probability of events occurring is independent when there is sampling with replacement

Sampling with replacement refers to the fact that after a sampling unit is drawn (e.g. a smartie from a smartie box), the event space from which it was drawn is returned to its original state (e.g. the total number of smarties in the smartie box does not change as a result of removing a smartie).

Let’s illustrate with the smartie example

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2.3.3. “AND” rule

The probability of BOTH events occurring is equal to the multiplication of the individual probabilities: probability of event 1 X probability of event 2. Otherwise referred to as the MULTIPLICATIVE rule

“AND” rule probability formula: \[ \begin{aligned} {P(EventA~and~EventB)} ={P(EventA)*P(EventB)} \end{aligned} \] This formula can also be extended to include more than 2 events (i.e. calculating the probability of more than two events occurring)

6/41*2/41

6/41*2/41*4/41

6/41*5/40*4/39

2.4.1. Cumulative probabilities

We can add up probabilities of mutually exclusive, independent events, as we have done previously with the additive rule (see OR Rule, exercise 2.3.2.1)

Child IQ scores on WISC
Let’s take a look at the wiscsem.csv dataset that was first introduced in tutorial 1, where 174 children were assessed on various measures of IQ using the Weschler Intelligence scale (WISC).

Below is a frequency distribution (i.e. empirical) of comprehension scores on the WISC (which can serve as an approximation of a theoretical probability distribution when the sample size is large), with its associated frequency table directly below

Frequency table of comprehension scores

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69/174

2.4.2. Theoretical distributions

Theoretical probability distributions refer to what we expect to see, as opposed to empirical distributions, which refer to what we actually see in data we have collected. We have only dealt with empirical distributions up until this point.

Unlike empirical distributions, theoretical probability distributions aim to approximate some real-world phenomenon in a population, rather than just in a specific sample.

A theoretical distribution may be approximated by an empirical distribution when the sample size is large, and the sample is representative.

Heights of South African males
Simulated height data of South African males are stored in the objects named heights.smalln and heights.largen. The data in each vector object has approximatey the same mean (approx 168) and standard deviation (approx 35), except there are 100 observations stored in heights.smalln, and 10000 observations stored in heights.largen. Height is measured in centimeters

hist(heights.smalln)

hist(heights.largen)

2.4.3. Area under the curve

Probabilities can be obtained from theoretical distributions by finding an appropriate area under the curve

Integral calculus methods are used to calculate probabilities from under the curve, but R has built in functions to do this for us, so we don’t have to use calculus.

See the various normal distributions below that each display a distribution of Z scores (i.e., with a mean of 0, and an SD of 1), each with a shaded area under the curve.

3.1. Simulation

Briefly put, simulation is a method used to artificially generate data. Simulated data differs from real data in that the former is fictitious, whilst the latter was actually collected in the real world

Simulation has several advantages:

1. Simulation is not limited by sample size (we could simulate extremely large numbers of data points with relative ease).
2. Simulation allows us to better understand the true nature of theoretical distributions underlying variables (Remember that larger samples get us closer to the theoretical distribution)
3. Simulation allows for pure random sampling, which is typically not possible in real life (there is typically selection bias present in real-world samples).

set.seed(4)

set.seed(10)
sample(x=heights.smalln, size = 10, replace = TRUE)

set.seed(20)
sample(x=..., size = ..., replace = ...)

set.seed(20)
sample(x=heights.smalln, size = 20, replace = FALSE)

set.seed(...)
sim.dat=sample(x=heights.smalln, size = 20, replace = FALSE)
...(...)

set.seed(32) #Seed can be any value except 20
sim.dat =sample(x=heights.smalln, size = 20, replace = FALSE)
hist(sim.dat) 

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