# What I Learned After I Stepped On My Bathroom Scales: BMI, Correlation and Models

#### Written by Robert Coyle

- Is this a healthy weight for somebody my age and height?
- How does my weight compare to those around me?

An accurate measure of a healthy weight is given by a person’s body mass index (BMI). This can be calculated by dividing a person’s weight (in kg) by their height (in m) squared. For example, given that my height is 1.80m, I have a BMI of 76/(1.80^{2}) = 23.5. A person is considered underweight if their BMI is under 18.5 and overweight if their BMI is over 25. A BMI between 18.5 and 25 indicates an optimal weight.

Next, I wanted to compare my weight/BMI to people around me. To do this I thought of ten male family and friends with the intention of obtaining their weight and height in order to calculate their BMI. I was able to obtain the necessary details from nine out of the ten people. My uncle was not contactable and his details were not recorded. The information is summarised along with the age and BMI of each person in the table below.

*Table 1: BMI Values*

Looking at the list of BMIs I observed mine as being located somewhere in the middle of those recorded.

Thinking again about my uncle, whose BMI we were unable to calculate, I wondered if it would be possible to estimate his BMI using the information from the other recorded BMI values. Would his BMI be similar to somebody close to his age such as one of my neighbours since their ages were closest to his from the people I had asked? To base my uncle’s BMI on somebody close to his age it would be necessary to see if a relationship/correlation exists between age and BMI.

To see if a correlation existed between age and BMI, I decided to re-arrange the entries in the table starting with the smallest age and finishing with the oldest. The re-arranged data is presented in the table below:

*Table 2: Age Versus BMI *

It is possible to see above that as age increases then (in the majority of cases) so also does BMI. We can see this more clearly if we graph the points.

*FIgure 1: BMI Values*

Using the data recorded in Table 1, a first attempt to create a formula for estimating BMI given a person’s age is:

BMI = 0.17*Age + 18

The table below compares the actual BMI values with the estimates when using the formula.

*Table 3: Estimates for BMI*

The estimates are comparable with the actual BMI values for the people with ages between 4 and 38. However we see for the two people aged 60 and 75, the BMI estimates are not close to the actual values. This can be seen visually in Figure 2 where the line BMI = 0.17*Age + 18 is plotted with the points from Figure 1.

*Figure 2: Graph of actual BMI values with estimate line*

Considering that the line BMI = 0.17*Age + 18 provides inaccurate estimates for BMI for my two neighbours aged 60 and 75, it would be incorrect to use this line to estimate my uncle’s BMI.

For example, we used the line BMI = 0.17*Age + 18 to model a person’s BMI.

What I have described in the blog is a first step in selecting a model to match the given data. Note that only a small sample is considered in order to demonstrate the concept, and there may be other confounding factors involved.

The next step is to consider improved models which estimate BMI more accurately. This is a task that faces our Expert Modelling team in Creme Global every day. That is, given a set of data, we seek to find a (statistical) model which best describes the data. Once a model is established, it can be used to answer questions, predict outcomes and ultimately to help people make better decisions.