## Confindence Intervals

Confidence intervals are a range of values within which you can say an unknown value is expected to lie with a specified degree of certainty or probability. For example, from a sample of 10 journeys you might say that the you are 95% certain that the average time it takes me to get to school (from all journeys I have made to school not just the sample of 10) lies within the range 10-12.5 minutes.

When taking a random sample it is much better to use a confidence interval for your results rather than just giving the mean, because it gives an idea of how reliable your mean is. This is because you can calculate an average value from a set of completely random results but this doesn’t mean that you can have any certainty that the next result will be similar to the mean (since we have stated that the results are completely random they are no more likely to be close to the mean than any other value).

**Calculating a confidence interval (for a normal distribution)**

When calculating a confidence interval you must first decide on the percentage certainty that you are going to use for the interval (a common value to use is a 95% confidence interval)

You then use the reverse tables for the normal distribution to work the value of the standardised normally distributed variable to use.

**Note**: You must use the value half way between the certainty level and 100%, ie if you want a 95% confidence interval use 97.5% since you only want 2.5% on either side of the distribution.

You can now calculate the interval. To do this you need the standard deviation and mean of the sample. However to correct the standard deviation for the entire sample of possible tests divide by the square root of the number of items in your sample

ie) if you sample of n items is Y and the entire sample of possible results is X

sd(X) = sd(Y)/√n

Now the confidence interval is

X – z sd(X), X + z sd(X)

Where z is the value obtained from the inverse tables.

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