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calculations.html
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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>BMI Calculations</title>
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<body>
<a href="https://github.com/vaishnavsudarshan/bmicalc" class="github-corner" aria-label="View source on GitHub"><svg width="80" height="80" viewBox="0 0 250 250" style="fill:#64CEAA; color:#fff; position: absolute; top: 0; border: 0; right: 0;" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"></path><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"></path><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"></path></svg></a><style>.github-corner:hover .octo-arm{animation:octocat-wave 560ms ease-in-out}@keyframes octocat-wave{0%,100%{transform:rotate(0)}20%,60%{transform:rotate(-25deg)}40%,80%{transform:rotate(10deg)}}@media (max-width:500px){.github-corner:hover .octo-arm{animation:none}.github-corner .octo-arm{animation:octocat-wave 560ms ease-in-out}}</style>
<div class="container w-50">
<a class="btn btn-outline-primary" href="index.html" role="button">\(\leftarrow\) Back to the main page</a>
<h1>How BMI is calculated</h1>
<p align="justify">
For all ages, the formula for BMI is simple.
<div class="card text-dark bg-light mb-3" style="max-width: 18rem;">
<div class="card-body">
<p class="card-text">
$$
\Large{\text{BMI} = \frac{w_{kg}}{h_{m}^2}}
$$
</p>
</div>
</div>
This is for when the mass in kilograms, and the height is in meters.
</p>
<p align="justify">
When the height is in feet and weight is in pounds, the formula for BMI becomes:
<div class="card text-dark bg-light mb-3" style="max-width: 18rem;">
<div class="card-body">
<p class="card-text">
$$
\Large{\text{BMI} = 703 \times \frac{w_{lb}}{h_{in}^2}}
$$
</p>
</div>
</div>
</p>
<h1>How BMI Percentile is calculated</h1>
<p align="justify">
The way to find the BMI <i>percentile</i> for children and teens is a bit harder than the BMI itself because
it depends on the distribution of BMI across the population of kids with a certain age, weight, height and
gender.
</p>
<p align="justify">
First, we obtained the <a href="https://www.cdc.gov/growthcharts/html_charts/bmiagerev.htm">growth
charts</a>
from the Center for Disease Control and Prevention (CDC).
These growth charts provide the BMI percentile based on the age and BMI, and there is a different chart for
each gender.
</p>
<p align="justify">
Next, we used a technique called <i>linear interpolation</i> to obtain the BMI percentile given the gender,
age and BMI.
</p>
<!-- https://math.preview.excalidraw.com/#json=1WlBhWooN2hZm0VZ0b2vx,56yOO6tKQLNdQY-T2PfCrg -->
<p align="justify">
If the BMI for a kid lies between two values \(\text{BMI}_{\text{low}}\) and \(\text{BMI}_{\text{high}}\)
and we know the percentiles for these as \(p_{\text{low}}\) and \(p_{\text{high}}\), the percentile for a
given BMI can be calculated as:
<div class="card text-dark bg-light mb-3" style="max-width: 30rem;">
<div class="card-body">
<p class="card-text">
$$
p = p_{\text{low}} + \left(\frac{p_{\text{high}}-p_{\text{low}}}{\text{BMI}_{\text{high}}-\text{BMI}_{\text{low}}}\right) \times
\left(\text{BMI} - \text{BMI}_{\text{low}}\right)
$$
</p>
</div>
</div>
<img src="linear-interpolation.svg"/>
</p>
<h1>How weight is calculated given target BMI percentile</h1>
<p align="justify">
Given a target BMI percentile, \(p_{\text{target}}\), we can also use linear interpolation to calculate the BMI value. Suppose \(p_{\text{target}}\) lies between two percentile values in the CDC grow charts, \(p_{\text{low}}\) and \(p_{\text{high}}\). If the BMI value for \(p_{\text{low}}\) in \(\text{BMI}_{\text{low}}\) and the BMI value for \(p_{\text{high}}\) is \(\text{BMI}_{\text{high}}\), then the BMI value for \(p_{\text{target}}\) can be calculated as:
<div class="card text-dark bg-light mb-3" style="max-width: 32rem;">
<div class="card-body">
<p class="card-text">
$$
\text{BMI} = \text{BMI}_{\text{low}} + \left(\frac{\text{BMI}_{\text{high}}-\text{BMI}_{\text{low}}}{p_{\text{high}}-p_{\text{low}}}\right) \times
\left(p_{\text{target}} - p_{\text{low}}\right)
$$
</p>
</div>
</div>
<!-- https://math.preview.excalidraw.com/#json=5B3cylLP7YDsCyQK9fyeJ,D-HbVPZ4BPKUVoyDhvujaw -->
<img src="linear-interpolation-reverse.svg"/>
</p>
</div>
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</body>
</html>