1.1. What does a Normal Distribution look like?

The Normal Distribution is one kind of probability distribution that is commonly used in the study of Psychological phenomena. Because the Normal Distribution is a type of probability distribution, it depicts the distribution of a variable at the population level, displaying the probability associated with all possible outcomes.

It is referred to as “normal” because of it’s characteristic shape:

1. Unimodal (i.e. one central peak)
2. Symmetrical (i.e. either side of the distribution is a mirror image from the center)
3. Asymptotic (i.e. the tails/extreme ends of the graph never reach the x-axis)

Directly below is a Normal Distribution.

The Normal Distribution is particularly useful in the social sciences because many psychological variables that are studied can be approximated by this distribution. In particular, the majority of scores fall in and around the centre (hence why there is a peak), and a minority of scores fall around extreme values (at very small or very high values at the tails of the graph)

1.2. Some variables that a Normal Distribution approximates

Let’s look at some rather crude, but useful examples of psychological variables that follow normal distributions:

1. Intelligence quotients (IQ)
   • The majority of participants score around the average IQ (at the centre), approximately 100
   • A minority of participants score more than 20 points below the average IQ, less than 80. They might be considered intellectually less able.
   • Another minority of participants score more than 20 points above the average IQ, greater than 120. They might be considered more intelligent than average.

2. Working memory (Digit Span Test)
   • The majority of people can typically remember 5 to 8 numbers
   • A minority of people can only remember fewer than 5 numbers. They might be considered to be impaired in memory.
   • Another minority of participants can remember more than 8 numbers. They might be considered to have above average memory abilities.

1.3. Normal vs Other Distributions

The Normal Distribution is a just one type of probability distribution. Each probability distribution has its own associated characteristics.

1.4. Normal Distribution parameters

Normal Distributions can also be described in terms of two additional characteristics:

1. Central tendency (In this case the Mean (\(\mu\)))
2. Spread (In this case the Standard Deviation (\(\sigma\)))

These characteristics are referred to as parameters, as they capture information about the entire population.

1.4.1 Normal Distribution family

The Normal Distribution is not a single probability distribution, but rather consists of a family of distributions.

Normal Distributions can differ from one another in terms of Central Tendency and Spread

Normal distributions of daily hours of sunlight (in a year period) in three different cities is displayed below

1.4.2. Effect of changes in central tendency and variability on Normal Distribution

In section 1.4.1. you will have seen how the appearance of the Normal Distribution changes as central tendency and variability changes.

Normal distributions can be entirely determined with knowledge of the two parameter values:

1. Mean (\(\mu\))
2. Standard deviation (\(\sigma\))

Let’s have a look at how changes to the mean and standard deviation can affect the Normal Distribution, through simulation (see Optional advanced section of Guided Tutorial 2 for a brief discussion on simulation).

The function rnorm() allows one to randomly draw values that come from a Normal Distribution characterised by a specific set of parameter values (i.e. mean (\(\mu\)) and standard deviation (\(\sigma\)))

?rnorm()

set.seed(...)
...=rnorm(n=..., mean = ..., sd =...)
...(..., xlab =..., main =..., xlim=c(0, 100))

set.seed(4)
data=rnorm(n=1000, mean = 50, sd =10)
hist(data, xlab= "Commission Rates (percentage)", main= "Graph of distribution of commission rates on SST", xlim=c(0, 100))

hist(rnorm(..., ..., ...),xlim=c(0, 150), xlab="...", main="...")

hist(rnorm(n=1000, mean = 50, sd = 20),xlab= "Commission rates (percentage)", main = "Graph of distribution of commission rates on SST", xlim=c(0, 100))
#You should notice that this solution is slightly different from that presented in exercise 1.4.2.3, and useful if one doesn't want to save the simulated data into an object. Also, note what the value of 400 in the `xlim` function does to the graph.

2.1 Score to probability

Probabilities are derived from the area under the curve of some probability distribution (see section 2.6.3 of Guided Tutorial 2).

Birth weight Let’s say we know that birthweight is normally distributed, with a mean (\(\mu\)) weight of 2.8kg and a standard deviation (\(\sigma\)) of 0.7kg.

If one were interested in knowing the likelihood (i.e. probability) of any new born child having a weight equal to or under 1.6 kg, the area under the curve from the value 1.6kg and lower would represent the associated probability (see birth weight distribution below). The pnorm() command can be used to extract the probability associated with a value that comes from a normal distribution.

pnorm(q=1.6, mean =..., sd=... )

pnorm(q=1.6, mean = 2.8, sd = 0.7)

1-pnorm(q=..., mean =..., sd=... )

1-pnorm(q=5, mean = 2.8, sd = 0.7)

pnorm(q=..., mean =..., sd=..., lower.tail = ... )

pnorm(q=5, mean =2.8, sd=0.7, lower.tail = FALSE )

pnorm(q=5, mean =..., sd=... ) - pnorm(q=1.6, mean =..., sd = ...)

pnorm(q=5, mean =2.8, sd=0.7) - pnorm(q=1.6, mean =2.8, sd=0.7)

1-pnorm(q=2.8, mean =2.8, sd=0.7)

2.2 Probability to score

Just like one can obtain the associated probability of some value (e.g. birth weight), by extracting the area under the curve. One can also extract the score from a probability, using the qnorm() command

Using the same example for birth weight distribution with a mean (\(\mu\)) of 2.8kg and standard deviation (\(\sigma\)) of 0.7kg, one may wish to find the birth weight associated with the smallest 5% of infants

qnorm(p=0.05, mean =..., sd=...)

qnorm(p=0.05, mean =2.8, sd=0.7)

qnorm(p=0.95, mean =..., sd=...)

qnorm(p=0.95, mean =2.8, sd=0.7)

qnorm(p=..., mean =..., sd=...)
qnorm(p=..., mean =..., sd=...)

qnorm(p=0.25, mean =2.8, sd=0.7)
qnorm(p=0.75, mean =2.8, sd=0.7)

3.1. Generating a Standard Normal Distribution

A Standard Normal Distribution can be simulated like any other Normal Distribution using the rnorm() command, where the mean (\(\mu\)) and standard deviation (\(\sigma\)) are now specified as 0 and 1 respectively (or simply omitted, since they are default arguments the function assumes)

set.seed(...)
...=rnorm(n=..., mean =..., sd=...)
hist(...,xlab="...", main"...")

set.seed(4)
dat.SN=rnorm(n=1000, mean =0, sd=1)
hist(dat.SN,xlab="Standard Normal Distribution", main="Histogram of Standard Normal Distribution")

3.2. Transform Normal to Standard Normal Distribution (converting to Z-scores)

Any Normal Distribution can be converted to a Standard Normal Distribution by converting all possible raw scores to z-scores

Z-score formula:

\[ \begin{aligned} Zscore =\frac{X - \mu}{\sigma} \end{aligned} \]

\(X\) = any raw score in the range of possible values of some Normal Distribution
\(\mu\) = population mean of Normal Distribution
\(\sigma\) = population standard deviation of Normal Distribution

Let’s look at simulated birth weight scores, in the object vector named “birth.weight”, that comes from a Normal Distribution of mean (\(\mu\)) 2.8kg and standard deviation (\(\sigma\)) 0.7kg.

...(...,...)

hist(birth.weight, xlab ="Birth Weights in kgs", main="Plot of Distribution of Birth Weights")

(...-2.8)/0.7

(1-2.8)/0.7

...=(birth.weight-...)/...
...(...,xlab ="...", main="...")

zscore =(birth.weight-2.8)/0.7
hist(zscore, xlab ="Z-scores", main="Graph of Distribution of Z-scores")

...=scale(...)
...(..., xlab="...", main="...")

zscore2 =scale(birth.weight)
hist(zscore2, xlab="Z-scores", main="Histogram of distribution of Z-scores")

3.3. Z-scores to probability

The pnorm() command can be used to extract the probability associated with a z-score that comes from a Standard Normal Distribution

pnorm(q =..., mean=..., sd=...)

pnorm(q = 2, mean=0, sd=1)

3.4. Probability to z-scores

The qnorm() command can be used to calculate the z-score associated with a particular probability

qnorm(p=..., mean=..., sd=...)

qnorm(p=0.90, mean =0, sd=1)

Exercise 4.1.1

Exercise 4.1.2

Exercise 4.1.3

Move on to and complete Statistics Tutorial Assignment 3 on Amathuba (Activities | Assignments)