Statistics and Probability

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This Bicen Maths video clip shows everything you need to memorise on Measures of Spread for A Level Maths.

If you use a TI-Nspire GDC there are instructions for calculating statistics from a list which is useful for this topic.

Official Guidance, clarification and syllabus links:

Calculation of mean using formula and technology.

Students should use mid-interval values to estimate the mean of grouped data.

Modal class for equal class intervals only.

Calculation of standard deviation and variance of the sample using only technology, however hand calculations may enhance understanding.

Variance is the square of the standard deviation.

Examples:

If three is subtracted from the data items, then the mean is decreased by three, but the standard deviation is unchanged.

If all the data items are doubled, the mean is doubled and the standard deviation is also doubled.

Using technology. Awareness that different methods for finding quartiles exist and therefore the values obtained using technology and by hand may differ.

Formula Booklet:

Mean, \(\bar{x}\), of a set of data

\( \bar{x} = \frac{\sum_{i=1}^{k}f_i x_i}{n} \), where \(n = \sum_{i=1}^{k}f_i\)

Some of the key formulae related to central tendency and dispersion:

$$ \text{Population Mean} (\mu) = \frac{\sum_{i=1}^{N} x_i}{N} \\ \text{Sample Mean} (\bar{x}) = \frac{\sum_{i=1}^{n} x_i}{n} \\ \text{Population Variance} (\sigma^2) = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} \\ \text{Sample Variance} (s^2) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} $$

There are actually four different methods for calculating the quartiles of a data set. These are explained on the Wikipedia page on quartiles. For discrete data, it turns out that there is no universal agreement on how to determine quartile values.

For example, consider the ordered data set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49.

Here are the lower quartiles calculated using four different methods:

Method	Lower Quartile
Method 1	15
Method 2	25.5
Method 3	20.25
Method 4	15

This variation in results highlights the nuances in statistical definitions that we often overlook. It's fascinating (and humbling) to realise that even concepts we thought were straightforward can be approached in multiple ways, depending on the methodology used.

This video on Mean, Standard Deviation and Variance is from Revision Village and is aimed at students taking the IB Maths AA SL/HL level courses.