Objectives

The objectives of this tutorial are:

1. Introduce the concept of a distribution
2. Distinguish between empirical and theoretical distributions
3. Introduce the normal distribution
4. Summarise a distribution using central tendency, spread and skew
5. Introduce the central limit theorem

Introduction

This tutorial introduces the idea of a distribution. We will look at the distribution of the human height observations for woman from the HEIGHT.CSV data file and the distribution of cortisol levels in wolves from the WOLF.CSV data file.

We will also look at an important theoretical distribution, the normal distribution, and introduce the central limit theorem to explain why the normal distribution (a theoretical distribution) is so often used to analyse real data.

To start with we will import the data from the HEIGHT.CSV file and extract just the data for females.

# ************************************
# Import  data ------------------------

human = read.table('HEIGHT.CSV', header=T, sep=',') # Import human height data
humanF = subset(human, SEX=='Female')               # Extract data for females

And the data from WOLF.CSV

wolf = read.table('WOLF.CSV', header=T, sep=',')    # Import wolf data set

What is a distribution?

What is a distribution?

Review this online lesson about distributions before continuing.

Online Lesson:Distributions

This lesson is athttps://www.ucd.ie/ecomodel/OnlineLessons/lesson3_distributions_Website.html

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

# Display a histogram of female heights from the HEIGHT.CSV dataset
ggplot(data=humanF,               # Define data to be plotted
       aes(x=HEIGHT)) +
  geom_histogram(bins=30) +       # Draw a histogram using 30 bins
  labs(x='Height',y='Count') +    # Set axis titles
  theme_bw() +                    # Set background to white
  theme(axis.title = element_text(size=20),  # Set fontsize of axis title
        axis.text =  element_text(size=16))  # Set fontsize of axis labels

Visualising a distribution

# Calculate the median height, ignoring missing data
median(humanF$HEIGHT, na.rm=TRUE)       

[1] 1.628

# Calculate the median absolute deviation of 
# height, ignoring missing data
mad(humanF$HEIGHT, na.rm=T)        

[1] 0.0622692

# Calculate the range of the data, ignoring missing data
range(humanF$HEIGHT, na.rm=TRUE)

[1] 1.409 1.829

# Calculate 1% and 99% quantiles, ignoring missing data
quantile(humanF$HEIGHT, prob=c(0.01, 0.99), na.rm=TRUE)

     1%     99% 
1.47704 1.77896 

The Normal distribution

The normal distribution (sometimes called a Gaussian distribution) is an extremely important theoretical distribution in data analysis. The distribution is completely described by two parameters:

• the mean (commonly represented by µ)
• the standard deviation (commonly represented by σ)

These two parameters describe the bell shape of the normal distribution. The normal distribution is symmetric, meaning that the distribution can be reflected about the mean and look the same.

Generating random numbers

R can generate random numbers from a specified theoretical distribution. Here are three examples.

Normal distribution

Use thernorm()function to generate random data drawn from a normal distribution (you can specify the two parameters of the normal distribution: the mean and standard deviation).

Below is the code to generate 10,000 values with mean=2 and standard deviation=0.5.

# .................................................
# Generate 10000 data point drawn from a normal distribution
# .................................................


normal_data = rnorm(n=10000, mean=2, sd=0.5)

# Produce a histogram of the generated data
ggplot(data=as.data.frame(normal_data),     # Define data to be plotted
       aes(x=normal_data)) + 
  geom_histogram(bins=30) +                 # Draw a histogram using 30 bins
  labs(x='x',                               # Set axis titles
       y='Count') + 
  theme_bw() +                              # Set background to white
  theme(axis.title = element_text(size=20), # Set fontsize of axis title
        axis.text =  element_text(size=16)) # Set fontsize of axis labels

Binomial distribution

Use therbinom()function to generate the number of success given the number of attempts and the probability of a success.

The binomial distribution can be used to simulate the chance of seeing a species at a location given the probability that the species exists in the habitat.

Below is the code to generate the number of times a species is seen in a habitat out of 20 replicate habitats and a probability of occurence of 0.69 (69%). This is repeated 10,000 times and the distribution from these 10,000 draws from a binomial distribution plotted.

# ................................................. # Generate 10000 data points drawn from a binomial distribution # ................................................. binomial_data = rbinom(n=10000, size=20, prob=0.69) # Produce a histogram of the generated data ggplot(data=as.data.frame(binomial_data), # Define data to be plotted aes(x=binomial_data)) + geom_bar() + # Draw a bar plot labs(x='Times species seen in 20 locations',# Set axis titles y='Count') + theme_bw() + # Set background to white theme(axis.title = element_text(size=20), # Set fontsize of axis title axis.text = element_text(size=16)) # Set fontsize of axis labels

Categorical distribution

Use thesample()to randomly sample from a set of categorical outcomes.

# .................................................
# Generate categorical data 
# .................................................

# Sample from a set of possible outcomes using sample()

# The outcomes are all the days of the week
week = c('Monday','Tuesday','Wednesday',
         'Thursday','Friday','Saturday','Sunday')

# Sample 1 day from the week
sample(week, size=1)

[1] "Thursday"

We can repeat this sampling many times (70 times, say) using a for-loop

# Create a blank data frame
categorical_data = data.frame(day=rep(NA, times=70))   

for (i in 1:70) {
  categorical_data$day[i] = sample(week, size=1)
}

# Produce a histogram of the generated data
ggplot(data=categorical_data,     # Define data to be plotted
       aes(x=day)) + 
  geom_bar() +                              # Draw a barplot
  labs(x='Day of week',                     # Set axis titles
       y='Count') + 
  theme_bw() +                              # Set background to white
  theme(axis.title = element_text(size=20), # Set fontsize of axis title
        axis.text =  element_text(size=16), # Set fontsize of axis labels
        axis.text.x = element_text(angle=90,# Rotate x axis labels
                                   hjust=1))# Justify text

Using a theoretical distribution

A theoretical distribution is used in data analysis to:

• provide a mathematical description of a random variable
• provide a distribution which canmimic observations
• provides a link between statistical theory and observed data

A Normal distribution is a theoretical distribution. A Normal distribution with mean = 1.63 m and standard deviation = 0.064 produces datasimilarto our observations of female heights (humanF$HEIGHT)

Below we have graphed histograms of the 1552 real human height observations (left) and a random sample of 1552 values drawn from a Normal distribution (right). On top of these two histograms the Normal distribution is represented as the red line (with mean = 1.63 m and standard deviation = 0.064 m).

The data produced by this Normal distribution is very similar to the real height observations. This will allow us to use a Normal distribution (with mean = 1.63 m and standard deviation = 0.064 m) as a mathematical basis for an analysis of these height data.

Summary of topics

• A distribution describes all possible outcomes and their relative frequencies
• A measure of central tendency (e.g. median, mean) and a measure of spread (e.g. median absolute deviation, standard deviation) are descriptive statistics that start to describe a distribution
• A histogram and quantile-quantile plot can be used to visualise a distribution
• The normal distribution is the single most important theoretical distribution in data analysis
• The normal distribution is completely described by two parameters: mean and standard deviation
• A common departure from normally distributed data is right-skew.
• Normal data can be generate in R using thernorm()function
• Categorical data can be generated using thesample()function

Further Reading

All these books can be found in UCD’s library

• Mark Gardner, 2012Statistics for Ecologists Using R and Excel(Pelagic, Exeter) [Chapter 4]
• Tenko Raykov and George A Marcoulides, 2013Basic statistics: an introduction with R(Rowman and Littlefield, Plymouth)
• John Verzani, 2005Using R for introductory statistics(Chapman and Hall, London)