Confidence Intervals

Example: Average Height

We measure the heights of 40 randomly chosen men, and get a mean height of 175cm,

We also know the standard deviation of men's heights is 20cm.

The 95% Confidence Interval (we show how to calculate it later) is:

The "±" means "plus or minus", so 175cm ± 6.2cm means

• 175cm − 6.2cm = 168.8cm to 
• 175cm + 6.2cm = 181.2cm

And our result says the true mean of ALL men (if we could measure all their heights) is likely to be between 168.8cm and 181.2cm

But it might not be!

The "95%" says that 95% of experiments like we just did will include the true mean, but 5% won't.

So there is a 1-in-20 chance (5%) that our Confidence Interval does NOT include the true mean.

Example: Apple Orchard

Are the apples big enough?

There are hundreds of apples on the trees, so you randomly choose just 46 apples and get:

• a Mean of 86
• a Standard Deviation of 6.2

So let's calculate:

X  ±  Zs√n

We know:

• X is the mean = 86
• Z is the Z-value = 1.960 (from the table above for 95%)
• s is the standard deviation = 6.2
• n is the number of observations = 46

86 ± 1.960 × 6.2√46 = 86 ± 1.79

So the true mean (of all the hundreds of apples) is likely to be between 84.21 and 87.79

True Mean

Now imagine we get to pick ALL the apples straight away, and get them ALL measured by the packing machine (this is a luxury not normally found in statistics!)

And the true mean turns out to be 84.9

Let's lay all the apples on the ground from smallest to largest:

Each apple is a green dot,
our observations are marked  blue

Our result was not exact ... it is random after all ... but the true mean is inside our confidence interval of 86 ± 1.79 (in other words 84.21 to 87.79)

Now the true mean might not be inside the confidence interval, but in 95% of the cases it will be!

95% of all "95% Confidence Intervals" will include the true mean.

Maybe we had this sample, with a mean of 83.5:

Each apple is a green dot,
our observations are marked purple

That does not include the true mean. That can happen about 5% of the time for a 95% confidence interval.

So how do we know if our sample is one of the "lucky" 95% or the unlucky 5%? Unless we get to measure the whole population like above we simply don't know.

This is the risk in sampling, we might have a "bad" sample.

Here is Confidence Interval used in actual research on extra exercise for older people:

What is it saying? Looking at the "Male" line we see:

• 1,226 Men (47.6% of all people)
• had a "HR" (see below) with a mean of 0.92,
• and a 95% Confidence Interval (95% CI) of 0.88 to 0.97 (which is also 0.92±0.05)

"HR" is a measure of health benefit (lower is better), so it says that the true benefit of exercise for the wider population of men has a 95% chance of being between 0.88 and 0.97

* Note for the curious: "HR" is used a lot in health research and means "Hazard Ratio" where lower is better. So an HR of 0.92 means the subjects were better off, and a 1.03 means slightly worse off.