So what is the mean BMI of our sample above?

given the following information from the course material:

Calculating the mean manually for large data sets would clearly be tedious, and professional statisticians use computer tools instead. However, just glance at the normal distribution in Figure 5 and the mean jumps out at you: it’s the BMI at which the curve reaches its highest point. Note, though, that this is true for the normal distribution only. As you will learn later in the part, there are other distributions that don’t have this property.

Column chart of full set of numbers counted in each BMI group.

It appears that the mean BMI is 27.

Stretch your mind back to those maths lessons and calculate the average of the following set of figures:

3, 5, 7, 2, 8, 4, 4, 6, 7, 1
So:
Mean = $\large{\dfrac{sum}{n}}$
Mean is the mathematical term for the average in this case, and sum =  the sum of all the numbers and n = the number of numbers you are adding to make the sum.
Moving on we have:
Mean = $\large{\dfrac{3 + 5 + 7 + 2 + 8 + 4 + 4 + 6 + 7 + 1}{10}}$
Mean = ${\boldsymbol{\underline{\underline{4.7}}}}$

Why do you think we should be cautious when estimating from a sample?

A sample is only as good as the range of people who it targets. If you wanted to know the nations opinion of the current Governments policies, you would expect the sample group to be from across the country and from several different backgrounds, not a selection from sunday shoppers in the Bull Ring in Birmingham. You would not go to an ICT University to find out what brand of mobile phones people in higher education like.

the conclusions drawn from samples is only as good and thorough as the sample from which the results are derived. A brilliant example of this is the LHC ( Large Hadron Collider ) at CERN, where they collect huge amounts of information from thousands of repeats of the same experiments to make sure the validity and authenticity of their results and the conclusions drawn from them.

Suppose we found 2422 obese individuals in our sample of 10 000. What would be your estimate of the number of obese people in the whole population?

Taken from the TU100 course material:

Not really. Reflect on the problem for a moment: how can we count the number of obese people in the country? The UK’s population is about 61 million. Knocking on everyone’s door would be astronomically costly and time-consuming (setting aside the fact that people might not wish to answer questions on something as personal as their weight). A local doctor or head teacher might be able to count the number of obese people in their care, but trying to get a national figure by simple counting is a non-starter.

The only way round this is to take a sample of the population and use it to estimate the figure we are after. For example, we could weigh and measure 10 000 UK citizens, count the obese individuals and then use this number to estimate the national figure by multiplying. Try this in the next activity.

So:

UK population $= 62,000,000$

Sample size $= 10,000$

Number of obese people in sample group $= 2,422$

So the percentage of people in the study group who are obese:

$\left(\large{\dfrac{2,422}{10,000}}\right) 100\% = \boldsymbol{\underline{\underline{24.22\%}}}$

Seeing as the sample group size is so large it would make sense to assume that this percentage is a good estimate of the percentage of the total population of the UK, however:

Sample size as a percentage of the total UK population $= \left(\large{\dfrac{10,000}{62,000,000}}\right) 100\%$

Sample size as a percentage of the total UK population = 0.016 ( 2 s.f ) or as a fraction $= \large{\dfrac{1}{62,000}}$

So now that we can appreciate just how small a part of the population we have surveyed with the study, it might be sensible to not make and bold assumptions from our derived information.

John Dough weighs 132 kg and is 1.91 metres tall. What is his BMI? Is he obese?

Working with the following, taken from the TU100 course material:

In everyday speech, ‘obese’ is just another word for ‘overweight’ or, less politely, ‘very fat’. However, health professionals use a more technical definition. If your body mass index (BMI) is over 25, you are overweight; if it is over 30 you are obese. [I’ve simplified here. Health professionals take into account waist measurement as well as BMI, and there are various categories of obesity.] BMI is calculated by dividing a person’s weight in kilograms by the square of their height in metres.

So:

Body Mass Index = BMI

Person’s weight in Kg = w = 132 Kg

Person’s height in metres = h = 1.91 m

So we can write:

$BMI = \dfrac{w}{h^2}$

$BMI = \dfrac{132}{1.91^2}$

$BMI = \boldsymbol{\underline{\underline{36.18\% ( 2 d.p. )}}}$

Working with this answer, and seeing that is os greater than 30%, then we can assume that John Dough is obese.