What is the physical significance of the height of the the bell shaped curve in normal distribution?

Question

Hello enthusiasts and explorers!!

Going through the concepts of normal distribution, one can understand that the height and the width of the bell shaped curve in a normal distribution graph tells about the spread of the data, but even going deeper into the concept, I am failing to understand the physical significance or how to verbally term the height of the given parameters in hand. In many of the articles for example the one below:

http://www.stat.yale.edu/Courses/1997-98/101/normal.htm

There is no mention of formal meaning of the height of the graph.

Can anyone provide some insights on the same ?

thanks in advance..

Answer 1

I agree with the OP that the interpretation of the y-axis of a PDF is quite cryptic. Here's my informal, intuitive take on it, for the real thing refer to proper sources.

It's useful to keep in mind that the y-axis is a density; density, in general, is an amount of something per unit of something else. In the case of a PDF, density could be taken as the expected number of events occurring in a unit of the x-axis (whatever that x happens to be). Since the unit of measurement matters, the y-axis can have values above 1 which is often cause of puzzlement.

The example below could represent the PDF of human height in one sex. It doesn't matter if you measure height in centimetres or metres but the density changes because relatively few people fall in 1-cm interval whereas almost everybody is in a 1-m interval.

(Similarly, I guess, you could measure population density as people per km^2 and find, typically, values greater than 0 or people per m^2 and find values less than 0)

We could then ask the percentage of people falling in the interval 170-171 cm, the answer doesn't change by changing the unit of measurement (R code):

integrate(dnorm, 170, 171, mean= 178, sd= 7)
0.0321063 with absolute error < 3.6e-16

integrate(dnorm, 1.70, 1.71, mean= 1.78, sd= 0.07)
0.0321063 with absolute error < 3.6e-16

~0.03 is the value of the y-axis at ~170 cm on the plot on the left.

Asking for the percentage of people falling in a given 1-metre interval would return ~100% for most sensible intervals.

This is a simulated example from measuring 1 million people and asking the same question, the answer is practically the same as above:

set.seed(1234)
cm <- rnorm(n= 1000000, mean= 178, sd= 7)

# Number of people with height between 170 and 171 cm divided by sample size
length(cm[cm >= 170 & cm < 171])/length(cm)
0.032245

# Using metres
m <- rnorm(n= 1000000, mean= 1.78, sd= 0.07)
length(m[m >= 1.70 & m < 1.71])/length(m)
0.03228

---

Code for plot:

par(mfrow= c(1,2))
plot(150:210, dnorm(x= 150:210, mean= 178, sd= 7), type= 'l', xlab= 'Height (cm)', ylab= 'Density')
plot(150:210/100, dnorm(x= 150:210/100, mean= 178/100, sd= 7/100), type= 'l', xlab= 'Height (m)', ylab= 'Density')