** This page continues on from "Understanding Averages" page. **Remember back to the beginning of all this. We took a sample of 11 beard lengths and calculated the arithmetic mean. This was not the true mean (or population mean) because we couldn’t measure the beard-length of every single member of the RSPB.

So, as The Grand Wizard Beard Master of the RSPB might reasonably ask: How confident are we that our sample mean is a reasonable approximation of the true mean? (Bearded Empires have risen and fallen on the answers to such questions).

Let’s say you decide to take another set of eleven measurements and calculate the mean of them. You would probably get an answer slightly different to the answer you got from the original measurements.

If you did this many times, you would end up with a large number of means, all a bit different from each other.

If you took all of these separate means and calculated an overall mean for the whole lot, you would end up with a value that was the same as the population mean (the mean you’d get if you could measure every one of them).

(Statisticians call this The Central Limit Theorem, the RSPB call it The Best Ever Assessment of Raw Data Protocol, for reasons only they can fathom).

Just like in the example above, we could work-out the standard deviation of this set of sample means. It would however take ages and we haven’t got ages. Happily, we don’t need to collect multiple samples because we can calculate it using just one sample like this:

This standard deviation of lots of means is now called the STANDARD ERROR (SE) of the mean

(The other standard error of the mean is that they think that their attitude will make them happy).

Now, because statisticians say so, (you can look this up in any statistics book if you really want to), it can be shown that we can be 95% confident that the population mean will fall within 1.96 X SE of a sample mean.

This lets us answer the question posed at the top of this page:

Is our estimate any good?

As you can see, the Druids’ error bars are very small, we can be 95% confident that our population mean lies within those bars. In other words we think our Druid estimate is pretty dashed spiffing. Our Eco-Warriors’ estimate has big error bars, so we wouldn’t be so certain about their mean being close to the population mean. The Ecologists’ error bars are also quite big but not as big as those unkempt Eco-Warriors.

Keep on reading for a bit more on how you might use this.........................................

Look at the graph below:

We can be 95% confident that the Greenies mean IQ is significantly different from either the Reddies or the Yellowies. Things are not nearly so clear-cut between the Reddies and the Yellowies. Notice how the error bars of the Reddies and Yellowies overlap, but there’s clearly no overlap between both of them and the Greenies. (The Greenies may not be very bright, but they can lift heavy things).

Please do not take offence from the colour references above, they do not refer to skin or hair.

Also have a look at ** Box and Whisker Plots **

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