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Random and systematic error in your scales.... And breathe!

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Random and systematic error in your scales.... And breathe!

Postby Butterfly_Dawn » Tue Nov 09, 2010 5:27 pm

I know some people like to weigh... a lot... and place a lot of faith in changes in their weight of a few grams here or there. As a PhD student, I was forced to learn one or two things about measurement and measurement error that is kind of applicable here and would like to share it with you. I promise I'll make it as painless as possible... Here goes... Stats 101...

In every measurement there is error, error is part of life. I'm not talking "whoops I added two numbers wrong" kind of error, I'm talking unreliability. Every scale has unreliability of some magnitude. This error is one of two forms:

Systematic error is also known as bias. This is what you get if your scales are badly calibrated and they weigh consistently either heavy or light. Systematic error is responsible for the "but the doctor's scales said I was 5kg lighter!" effect: since scales aren't calibrated to each other, or to a cental standard very often, they will weigh differently to each other.

This is not a problem so long as they are precise and you only weigh on one set of scales and you're only interested in the trend of your weight change either up or down, really the number is just a number. If you're trying to get to a specific weight on a specific set of scales (e.g. the police force or army recruitment scales) then you better check how they weigh compared to yours.

Random error is about precision. These errors are caused by unknown and uncontrollable factors and cause variations in reading despite weighing the same target weight. This is what is responsible for the "but I just got on the scales 5 min ago and I SWEAR it said I was 200g lighter and I haven't eaten anything in between!" effect. Each set of scales can have different amounts of random error, my scales might vary wildly over a 5kg range, yours might be within a 100g range.

This is a problem if you measure a lot and take a lot of stock by small changes in your weight.

Let's look at an example of both types with something other than body weight so we can take the "but it could be sweat/drink/bowel weight" arguments out of the equation.

Bob takes a 20kg bag of flour (measured by those spunky guys at the atomic weights place so he knows it's 20kg) and puts it on his set back room scales. They read 25kg, so he checks a few times, still 25kg. "Bonus!" thinks Bob "Standard error has meant I can sell this 20kg of flour to my unsuspecting customers as being 25kg!"

Bob then puts it on his shop scales to check it again and it reads 18kg, dismayed, Bob checks it again, and again, and again, and gets completely different readings each time. "Oh dear" thinks Bob "random error means I can't really accurately measure this bag, sometimes I'll be selling my customers short, and sometimes they'll get more than they should!"

Clearly, Bob needs a new set of scales for his shop front and to calibrate his back of shop ones (and a new work ethic)

How to work around error
I'm focusing on random error because IMHO it's the most important for weight changes. Since we know all measurements have random error, we can work out (with just a teensy weensy bit of maths) how much error we have in a measurement. One term for this is "standard deviation". Once we know that we can put error bars on all our measurements. That is, we can say "my weight is 88kg, plus or minus 600g". Then, if our next weight is different than our past one by MORE than 1200g, we can be sure that it's meaningfully different because the standard deviation doesn't overlap. It may not be statistically different, but that's a story for another day...

In this example, we can see that at 6 months and 12 months, the low carb diet weight error bars overlap, so we can be sure that the difference is not significant because the TRUE weight might have been the same, but they just read differently because of random error.


If you're feeling keen, you can calculate the standard deviation for your scales using the following:

1) Get something you know is a certain weight (or just use yourself, I promise the minute variations due to sweat in 5 min is not going to make a jot of difference). Weigh it/yourself a number of times in a row and record the numbers.
2) Now take those numbers and add them together, then divide the answer by the number of numbers you had to start with (read: add all the numbers up and divide by how many there are). This is the mean.
3) Now, subtract the mean (the answer from step 2) from every number in the original list to get the list of deviations. It's OK to get negative numbers here.
4)Next, square the resulting list of numbers (read: multiply them with themselves).
5) Add up all of the resulting squares to get their total sum.
6) Divide your result by one less than the number of items in the list.
7) To get the standard deviation, take the square root of the resulting number

Here's an example (in small numbers):
your list of numbers: 1, 3, 4, 6, 9, 19
mean: (1+3+4+6+9+19) / 6 = 42 / 6 = 7
list of deviations: -6, -4, -3, -1, 2, 12
squares of deviations: 36, 16, 9, 1, 4, 144
sum of deviations: 36+16+9+1+4+144 = 210
divided by one less than the number of items in the list: 210 / 5 = 42
square root of this number: square root (42) = about 6.48

Now you have the standard deviation you can know that your weight is 88kg plus or minus (insert standard deviation here). You might have figured out that the more wildly your scale varies, the bigger the standard deviation to the more you need to apparently "lose" to make it vaguely interesting. If you find that disheartening, see below (or get a better set of scales that are more precise).

Also note that you might have combinations of random and systematic error, so you might have:
- a very accurate but biased set of scales (i.e. it fairly consistently weighs about 5kg heavy/light),
- an inaccurate and biased set of scales (it varies all over the place but on average weighs heavy/light),
- an inaccurate, unbiased set of scales (varies wildly, but averages around about where it should) or
- an accurate, unbiased set of scales (the holy grail, not much variation and tells it like it is, prepare to fork out plenty for this little sucker and have it calibrated regularly :wink: )

The other way to work around error is to keep a record of your weights and plot them on a graph (excell is your friend here and it can also put in error bars for you once you know the standard deviation). What this does is allows you to see your current weight in the context of all your other weigh ins so you can see if there's a general trend of increase or decrease (which is what it's all about). When you see a 200g increase in the context of repeated 1ky per week losses, it loses it's worry. By comparison, if you've been kidding yourself that each week's 500g is just a small gain, when you look at it over the past month and see 2kg gained you'll have some info to make you reassess. IMHO trends over time is what you should be looking at whether you use error bars or not.

If you really really must weigh in every day (fine, whatever) one thing I have seen people do is average their weight over the week. That is, they record their weight each day of the week, add them all up, then divide by 7. This gives you a bit more of an accurate picture, but I still reckon it's best plotted on a graph.
35kg lost. (November 2005 - October 2006)
15kg gained again (as at October 2010).
Back to the drawing board - Let's do this thing!

"You can't change the winds, but you can change the sails"

"Reach out and take control of what lands in your lap"

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