Statistics and Probability

Furthermore

Official Guidance, clarification and syllabus links:

Class intervals will be given as inequalities, without gaps.

Frequency histograms with equal class intervals.

Not required: Frequency density histograms.

Use of box and whisker diagrams to compare two distributions, using symmetry, median, interquartile range or range. Outliers should be indicated with a cross.

Determining whether the data may be normally distributed by consideration of the symmetry of the box and whiskers.

Formula Booklet:

Interquartile range

\( IQR = Q_3 - Q_1\)

Cumulative frequency graphs are graphical representations that display the cumulative frequencies of a dataset. They are particularly useful for determining various statistical measures such as the median, quartiles, percentiles, range, and interquartile range (IQR). The median is the value that separates the higher half from the lower half of a dataset. Quartiles divide a rank-ordered dataset into four equal parts. The IQR is the range between the first quartile (Q1) and the third quartile (Q3) and represents the middle 50% of the data. Percentiles, on the other hand, divide the dataset into 100 equal parts.

Box and whisker diagrams, also known as box plots, are another tool used for data analysis. They provide a visual representation of the distribution of a dataset. These diagrams are especially useful for comparing two distributions in terms of symmetry, median, interquartile range, and overall range. The symmetry of the box and whiskers can give insights into whether the data may be normally distributed. A symmetric box plot suggests that the data is likely to be normally distributed, whereas an asymmetric one indicates otherwise.

Key formulae:

$$ \text{Median} = \frac{n + 1}{2} \text{th term} \\ \text{Q1} = \frac{n + 1}{4} \text{th term} \\ \text{Q3} = \frac{3(n + 1)}{4} \text{th term} \\ \text{IQR} = Q3 - Q1 \\ \text{Range} = \text{Maximum value} - \text{Minimum value} $$

Example:

Consider a dataset: \( \{3, 7, 8, 5, 12, 14, 21, 13, 18\} \).
Arranging in ascending order: \( \{3, 5, 7, 8, 12, 13, 14, 18, 21\} \).
Using the formulae:

$$ \text{Median} = \text{5th term} = 12 \\ \text{Q1} = \text{2.5th term} = 6 \\ \text{Q3} = \text{7.5th term} = 16 \\ \text{IQR} = 16 - 6 = 10 \\ \text{Range} = 21 - 3 = 18 $$

Thus, for this dataset, the median is 12, Q1 is 6, Q3 is 16, IQR is 10, and the range is 18.

If you use a TI-Nspire GDC there are instructions useful for drawing box plots.

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 Bicen Maths video clip shows everything you need to memorise on Representation of Data for A Level Maths.