# How to Lose Weight, Mathematically

## The Harris-Benedict Equation

The amount of energy expended by animals at rest, including humans, is known as the basal metabolic rate, or BMR. Think of it as the amount of fuel burned every 24 hours to keep your body at 37°C degrees as you sleep. We can estimate the rate for each individual by using the so-called Harris–Benedict equation (Mifflin and St. Jeor, 1990):

Determining the basal metabolic rate (BMR) is useful because, combined with an estimation for your physical activity level (PAL), it can be used to determine your total energy expenditure (TEE) also known as the amount of calories you need to consume daily in order to maintain your body weight over time. Subtract a certain percentage from that daily, and your body weight over time begins behaving like a mortgage — slowly but surely it gets paid off as both the principle (your body weight) and the payments (your necessary daily calorie deficit) decrease.

The following essay helps you to estimate your daily energy expenditure (TEE) and in turn, help you mathematically determine how many calories you should consume daily in order to lose weight over time.

# Calculate your Daily Energy Expenditure

The below process proceeds in four (simple!) steps. As with anything, your mileage may vary, but indeed even being consciously aware of your calorie consumption has been shown to correlate positively with weight decrease, so what do you have to lose? Except fat

## Step 1: Calculate your Basal Metabolic Rate (BMR)

The first step is simple, we want to estimate your BMR based on four values:

- Your gender
- Your height in centimeters
- Your weight in kilograms
- Your age in years

For this we’ll use the Harris-Benedict equations, specifically the latest versions proposed by Mifflin and St. Jeor (1990). As they are the result of regressions, they are slightly different for men and women:

Example: Determining my Basal Metabolic Rate (BMR)

I'm a 30 year old male, 176cm tall and weigh 79kg. Plugging these four values into the Harris-Benedict equation above:(10 x 79) + (6.25 x 176) - (5 x 30) + 5 = 1745My BMR is 1745 calories per day

## Step 2: Estimate your Physical Activity Level (PAL)

The second step requires you to evaluate how sedentary/active you are in your daily life, your physical activity level (PAL). For someone who works in an office and does not exercise, that would be ‘sedentary’, and a value of PAL = 1.53. For someone who works construction or another active job, or runs an hour per day, the value would be higher. Finally, if the person works in a very strenuous profession, is an athlete, both works physically and exercises, the value would be even higher than that. The following table approximates an individual’s TEE based on these example lifestyles (WHO, 2001):

**Sedentary Active Very Active**

1.53 1.76 2.25

I exercise, but I don’t do it every day. I work in an office, but I also walk to and from work for close to an hour every day, so I’d put myself in the middle category, active. For our example, my PAL is hence approximately 1.76.

## Step 3: Determine your Total Energy Expenditure (TEE)

The third step is to determine how much energy you expel every 24 hours. This because, if you want to maintain your current body weight, that’s how much energy you will need to consume in order to maintain a net calorie equilibrium. We calculate the the TEE by multiplying the numbers we found in step 1 and 2 together:

Example: Determining my Total Energy Expenditure (TEE)

My total energy expenditure (the amount of fuel my body consumes every 24 hours) is approximately equal to:TEE = BMR x PAL = 1745 x 1.76 = 3071 calories

That is, at my gender, height, weight, age, and activity level I need to consume 3071 calories per day in order to maintain my weight over time. Any less, and over time, I lose weight. Any more, and over time, I gain weight.

## Step 4: Determine Necessary Daily Calorie Deficit

How much weight to you want to lose? Is it 1 kilogram, or more like 20 kilograms? Unfortunately, metabolisms are such that everyone loses and gains weight differently, even if they eat the same things, and so prescribing a certain daily calorie deficit in order to achieve a certain weight is impossible without more information.

What is possible, is figuring out approximately how many calories one consumes on average each day, and making a conscious decision to consume less than that. We can figure out approximately how much less that is than our current consumption by calculating what our BMR would be if we were at our goal weight:

Example: Determining my Necessary Daily Calorie Deficit

Say I had reached my goal weight. What would my Basal Metabolic Rate be? Plug this number into the same equations as did before. Let's say I wanted to reach a goal weight of 70 kilograms. Everything else stays the same:BMR: (10 x 70) + (6.25 x 176) - (5 x 30) + 5 = 1650 calories

PAL: 1.76

TEE: 1650 calories x 1.76 = 2904 calories2904 / 3071 calories = 0.946

1 - 0.946 = 0.054 =5.4% fewer calories at 70kg

Everyone’s metabolism is different, but at the very least, I know that if I want my body weight to be 70 kilograms instead of 79 kilograms, without increasing my activity level, I have to learn to consume at least** 5.4%** fewer calories every day over time. At my current daily energy expenditure, that equates to:

`3071 calories x 0.054 = `**166 fewer calories per day**

# Some notes

I’m expecting an avalanche of opposition to this essay, so before that allow me to first make some notes/disclaimers:

## On Expectations

For a long time it was believed that a weekly calorie deficit of 3500kcal would result in a 0.5 kilogram of weight loss. More recent studies have shown that this indeed varies widely from person to person. However, if we consider this number to be an average, I can for myself use this as a metric to calculate both A) How much of a deficit I should be in per day in order to reach a goal weight within a certain amount of time and 2) How long it will take me to reach my goal weight given a certain daily calorie deficit. Ideally, of course, one would estimate what the number is for ourselves individually and use this as guidance instead. However, lacking a better approximation:

Example: Necessary calorie deficit to lose 9 kg in six months

Let's say I want to lose 9 kilograms within six months, 180 days. For this first calculate the sum total calorie deficit for the period, then divide it by the number of days until my goal date:9kg x (2 x 3500 kcal) = 63,000 calories / 180 days

63,000 calories / 180 days = 350 calories / dayMy daily calorie intake in order to lose 9 kilograms in 180 days then hence becomes:

3071 calories - 350 calories =2,721 calories per day

Of course, as we lose weight, our BMR changes and so we require less and less energy in order to maintain our body weight. In effect, we should be discounting into the future using a geometric series so that that over time, my calorie deficit decreases relative to my body weight. This, however, will take us too far away from the purpose of this article. It is worthwhile mentioning, though, that we can see what happens if we use the Harris-Benedict equation to calculate my basal metabolic rate (BMR) if I maintained a total energy expenditure (TEE) of 2,721 calories per day for 180 days (no discounting):

`BMR = TEE / PAL = 2721 / 1.76 = 1546 calories`

(10 x weight in kg) + (6.25 x 176) - (5 x 30) + 5 = 1546 calories

(1546 - 5 + 150 - 1100)/10 = weight in kg = **59.1 kilograms**

I’ve shot more than 10kg past my goal. This, because, my TEE x 0.054 at 70kg is different than it is at 79kgs.

## On Accuracy

As many experts have pointed out, looking at the numbers alone, the successful, long-term maintenance of a stable body weight is a bit of a mathematical mystery, as even a tiny percentage of a calorie surplus over time would predict a massive weight gain down the road. Perhaps the answer lies in our metabolism, appetite regulation or hormones. Those in favor of ketogenic and/or paleo diets will argue that carbohydrates promote insulin spikes, which may promote fat retention.

As with the Body-Mass-Index (BMI), the BMR equations to not account for body composition, and so will calculate identical results for muscular and fat people of the same dimensions. As muscle and fat require differing Total Energy Expenditure to maintain, the BMR is certainly a bad measure for comparative studies.

## On Calories

A calorie is a measure of energy, typically defined formally as:

`A `**calorie** is the amount of energy required to raise the temperature of 1 liter of room temperature water by one degree Celsius.

The claim that a calorie deficit (or surplus) is a sufficient metric for one to be able to determine their future weight fluctuations is indeed hotly contested. The tautology that “a calorie is a calorie” is often disputed by proponents of the counterfactual argument that the body does not consume every macronutrient in the same way, regardless of their energy contents. Although one dietary calorie is certainly known to contain 4,184 joules of energy (as it was defined to do so), we do in fact not know if our bodies are agnostic to this fact. That is, as to whether dietary calories coming form fat, carbs or protein are absorbed in the same way in terms of fat retention. We do know that:

**Protein** = 4 calories per gram

**Carbohydrates** = 4 calories per gram

**Fats** = 9 calories per gram

i.e. that fats are more energy dense than are proteins or carbohydrates. That should come as no surprise, as the ultimate high-fat food, cooking oil, can indeed also be used as fuel to run a car. Sugar (the ultimate carbohydrate), not so much. Anecdotally, a common composition of macronutrients often promoted is the following:

`30% of calories from protein`

50% of calories from carbohydrates

20% of calories from fats

# History

J. Arthur Harris and Francis G. Benedict published the first version of the Harris-Benedict equation in their 1918 paper *A Biometric Study of Human Basal Metabolism* published in

where *h* is ‘total heat production per 24 hours’, *w* is ‘weight in kilograms’, *s* is ‘height in centimeters’ and *a* is ‘age in years’. The equation is valid for w = {25.0, 124.9}, w = {151, 200}, a = {21, 70}.

The equation was derived from regressions done on the measurements of 136 men, 103 women and 94 newborns. They revised their regression in the 1919 paper *A Biometric Study of Basal Metabolism in Man**.* Since, their regression has been re-run on other data sets multiple times, and so have also been revised multiple times. Roza & Shizgal (1984) in their paper *The Harris Benedict equation reevaluated: resting energy requirements and the body cell mass* found that better approximations are given by:

Finally, Mifflin & St. Jeor (1990) revised the equations again in their paper *A New Predictive Equation for Resting Energy Expenditure in Healthy Individuals*, to obtain the versions used in this essay:

# Disclaimer

The most concise argument for the case that energy expenditure is **not** the best approximation of how humans maintain their bodyweight over time is in my opinion made by author Gary Taubes. I especially recommend his book “Good Calories, Bad Calories” and his excellent 2017 appearance on Sam Harris’ Making Sense Podcast #74 - “What Should We Eat”.

None of the contents of this or any essay I write should be considered medical advice. For the best results, do your own reading or consulting with a nutritionist or medical professional.