Normal (Gaussian) Distribution
Introduction to Global Health Data Science
Normal Distribution
In this slide deck, we explore the normal distribution, one of the primary inferential tools in statistical science!
The normal distribution is commonly called the Gaussian distribution after Carl Friedrich Gauss, who wrote down the equations governing it in the early 1800’s. It is also sometimes referred to as a bell curve.
Birth Weight
We consider birth weight data for US (full term) births in 2024 from the National Center for Health Statistics. For simplicity, we consider weight rounded to the nearest pound.
If we want to estimate the probablity that a baby delivered at term weighs less than 6 pounds, we just add up the probabilities associated with the bars for birth weights under 6 pounds (this will be an error-prone estimate given that weights of 5.5 were rounded up to 6 for simplicity, but just imagine each baby weighed exactly an integer number of pounds for now). For births of 8-9 pounds, we just add up the probabilities associated with those two bars, and so forth.
Continuous Distribution
Birth weight is a continuous quantity, and in hospitals it is measured in g for much higher resolution. Let’s consider the distribution of birth weight in g.
The probability density curve can be used to get the probability of any range of birth weights we would like to investigate.
For example, we can calculate probabilities under the normal distribution, such as the probability a baby weighs between 3000 and 3500 grams.
Continuous Probability Distributions
- Density curves, like histograms, can have a wide variety of shapes. The area under a density curve is always 1.
- How do you find the area of interest in a plot?
- Calculus! \[Pr(a \leq X \leq b)=\textrm{area between a and b below the curve}=\int_a^b f(x)dx\] where \(f(x)\) represents the density curve
- While you will need to know calculus for courses like STA 240, for STA 198/ GLHLTH 298, we can rely on R to provide the probabilities we need.
- We will see density curves for several important distributions – normal, \(t\), \(\chi^2\), and \(F\) random variables (coming this fall!)
Normal Distribution
For the normal distribution, \[f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left\{-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right\},\] where the mean is given by \(\mu\), the variance by \(\sigma^2\), and the standard deviation by \(\sigma\). The notation \(N(\mu,\sigma^2)\) is often used.
Also called the Gaussian distribution or bell curve, this distribution is symmetric.
Which of these normal curves has the biggest mean? Standard deviation?
Standard Deviation Rule for the Normal Distribution
A useful rule for normal distributions is that roughly 68% of the area under the curve is within one standard deviation \(\sigma\) of the mean, 95% is within \(2\sigma\), and 99.7% is within \(3\sigma\).
The symmetry of the normal distribution also allows us to calculate the probability of values falling in the tails. (This birth weight distribution has mean 3250g and sd 600g.)
5% of the data are further than two standard deviations, 2 \(\sigma\), from the mean, 2.5% in each tail.
For our distribution of term birth weight, normal with mean 3250g and sd 600g, often denoted \(N(\mu=3250,\sigma^2=600^2)\), what is the probability of a birth weight between 2050g and 4450g?
What is the probability that a randomly chosen term baby will have a birth weight greater than 3850g?
What is the probability that a randomly chosen term baby will have a birth weight less than 2500g?
Z-scores
How many standard deviations below the mean is a 2500g term baby?
Recall \(\mu=3250g\) and \(\sigma=600g\)
2500g is 750g below the mean
750g is \(\frac{750}{600}=1.25\) standard deviations
\(\frac{2500-3250}{600}=-1.25\) is known as a z-score
Z-scores are used in many health settings, including
Evaluating health effects related to body mass index (BMI)
Measuring child growth (height-for-age, weight-for-age, weight-for-height, BMI-for-age) and identifying malnourished children
Testing safety of food, water, and environmental samples
Reporting bone density scan results
The z-score is a standardized normal variable that tells us now many standard deviations above (positive z-scores) or below (negative z-scores) the mean our original value (2500g) is. That is, \[z=\frac{x-\mu}{\sigma}=\frac{\text{value - mean}}{\text{standard deviation}}.\]
The distribution and areas are unchanged by the conversion to z-scores.
The distribution of the z-scores is known as the standard normal distribution. Here are a few important reference points for this distribution.
A standard normal table allows you to calculate values based on the standard normal distribution. It tells you how much area is under the normal curve to the left of the specified value (lower tail area). Sometimes the table shows the complement of this probability (upper or right tail area). Because the standard normal distribution is symmetric with mean zero, \(Pr(Z \leq 0)=0.5\).
With modern computing, we don’t need to rely on these tables to get the desired probabilities, but you often find them in the back of statistics textbooks.
This table shows the probability under the curve less than the indicated value.
What is the probability of a term baby less than 2500g (z-score \(\frac{2500-3250}{600}=-1.25\))?
What is the probability of a term baby less than 2500g (z-score -1.25)?
Using R to calculate normal probabilities
Just like we used the pbinom command to calculate \(P(Y \leq y \mid \pi, n)\) for binomial data, we can use pnorm to calculate \(P(Y \leq y \mid \mu=0, \sigma^2=1)\) for standard normal data. For example, to get the probability of a baby less than 2500g (z=-1.25), we just take pnorm(-1.25) yielding the value 0.1056498.
We can even just have R calculate the z-score for us using the code pnorm(2500,mean=3250,sd=600), which yields the value 0.1056498.
Probabilities between two values
To get the probability that a random term baby would weigh between 2500g and 3500g, use pnorm(3500,mean=3250,sd=600)-pnorm(2500,mean=3250,sd=600) to get the value 0.5558891.
You try it!
Coquerel’s sifaka has birth weight distribution with mean 100g and sd 7.5g.
What is the probability a new baby sifaka would weigh less than 80g?
What is the probability it would weigh more than 110g?
What is the probability it would weigh between 90 and 110g?
Homework (Practice): Birth Weights of British Shorthair kittens
A veterinarian is studying the birth weights of British Shorthair kittens. Suppose that, for this exercise, birth weights are approximately normally distributed with a mean of 100 grams and a standard deviation of 10 grams.
1. Meet Biscuit. Biscuit weighed 120 grams at birth.
- How many standard deviations from the mean is Biscuit’s birth weight?
- What is Biscuit’s z-score?
- Would you consider Biscuit’s birth weight unusual? Why or why not?
2. Meet Crumpet. Crumpet weighed just 85 grams at birth.
Use R to calculate the proportion of kittens expected to weigh less than Crumpet. Write your answer as a complete sentence describing what the probability means.
3. A whole litter! The veterinarian considers kittens weighing between 80 and 120 grams to be within a particular monitoring range. What proportion of British Shorthair kittens would we expect to fall within this range?
Hint: Think about how you could use two calls to pnorm() to find the area between two values.
4. Put it into words. A client asks, “Should I be worried that my kitten weighs less than most other kittens?”
In a few sentences, provide guidance for the client, citing data reported in a 2023 study that shows British shorthair kittens in the bottom 10% of birth weight are at elevated mortality risk and kittens in the bottom 0.5% of birth weight are at very high risk of mortality, and translating these values into the relevant birth weights.
No worries here, Odin was a little guy who is now happily overseeing STA198 prep!
