# Confidence Intervals

An interval of 4 plus or minus 2

A Confidence Interval is a range of values we are fairly sure our true value lies in.

### Example: Average Height

We measure the heights of 40 randomly chosen men, and get a:

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

175cm ± 6.2cm

This says the true mean of ALL men (if we could measure 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.

## Calculating the Confidence Interval

Step 1: find the number of samples n, calculate the mean X , and the standard deviation s of those samples.

Using our example:

• Number of samples: n = 40
• Mean: X = 175
• Standard Deviation: s = 20

Step 2: decide what Confidence Interval we want. 90%, 95% and 99% are common choices. Then find the "Z" value for that Confidence Interval here:

 Confidence Interval Z 80% 1.282 85% 1.440 90% 1.645 95% 1.960 99% 2.576 99.5% 2.807 99.9% 3.291

For 95% the Z value is 1.960

Step 3: use that Z in this formula for the Confidence Interval

X  ±  Zs√n

Where:

• X is the mean
• Z is the chosen Z-value from the table above
• s is the standard deviation
• n is the number of samples

And we have:

175 ± 1.960 × 20√40

Which is:

175cm ± 6.20cm

In other words: from 168.8cm to 181.2cm

The value after the ± is called the margin of error

The margin of error in our example is 6.20cm

## Calculator

We have a Confidence Interval Calculator to make life easier for you.

## Another Example

### Example: Apple Orchard

Are the apples big enough?

There are hundreds of apples on the trees, so you randomly choose just 30 and get these results:

• Mean: 86
• Standard Deviation: 5

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 = 5
• n is the number of samples = 30

86 ± 1.960 × 5√30 = 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,
except our samples which are 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)

But the true mean might not be inside the confidence interval but 95% of the time it will!

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

Maybe we had this sample, with a mean of 83.5 and a Standard Deviation of 3.5:

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

That does not include the true mean. Expect that to happen 5% of the time for a 95% confidence interval.

So how do we know if the sample we took 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.

## Example in Research

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

Example: the "Male" line says there were:

• 1,226 Men (47.6% of all people)
• had a "HR" (which means Hazard Reduction*) 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)

In other words the true benefit (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 in research and means "Hazard Ratio" where lower is better, so an HR of 0.92 means the subjects were better off, and 1.03 means slightly worse off.

## Standard Normal Distribution

It is all based on the idea of the Standard Normal Distribution, where the Z value is the "Z-score"

For example the Z for 95% is 1.960, and here we see the range from -1.96 to +1.96 includes 95% of all values:

From -1.96 to +1.96 standard deviations is 95%

Applying that to our sample looks like this:

Also from -1.96 to +1.96 standard deviations, so includes 95%

## Conclusion

The Confidence Interval formula is

X  ±  Zs√n

Where:

• X is the mean
• Z is the Z-value from the table below
• s is the standard deviation
• n is the number of samples
 Confidence Interval Z 80% 1.282 85% 1.440 90% 1.645 95% 1.960 99% 2.576 99.5% 2.807 99.9% 3.291