Data Representation and Analysis

Measures of centre

Three different answers to "what is a typical value?", and questions expect you to know which is which.

• Mean — the total divided by how many there are: $\bar{x} = \dfrac{\sum x}{n}$. Uses every value, so a single extreme value drags it.
• Median — the middle value once the data is in order. Unaffected by extremes.
• Mode — the most frequent value. The only one that works for non-numerical data.

Frequency tables

When each value $x$ occurs $f$ times, you do not list them all out:

$\sum f$ is the total number of items, not the number of rows in the table.

For grouped data, use the midpoint of each class as $x$. The answer is then an estimate, because you no longer know where inside each class the values sat — and CXC expects the word "estimate" in your answer.

Measures of spread

A centre alone says nothing about how tightly the data clusters. Two sets can share a mean and look nothing alike.

• Range — largest minus smallest. Easy, but ruined by one outlier.
• Interquartile range — $Q_3 - Q_1$, the width of the middle half. Ignores the extremes.
• Semi-interquartile range — half of that, $\dfrac{Q_3 - Q_1}{2}$. This is the one CSEC Add Maths usually asks for.
• Standard deviation — the typical distance of a value from the mean.

Variance and standard deviation

For a plain list, $\sum f$ is just $n$ and the formula becomes $\dfrac{\sum x^2}{n} - \bar{x}^2$.

Work in this order: find $\bar{x}$, find $\sum fx^2$, subtract, then take the root. Set the working out in columns — $x$, $f$, $fx$, $fx^2$ — and total each one. Almost every lost mark here comes from disorganised arithmetic rather than from not knowing the formula.

Quartiles and the cumulative frequency curve

A cumulative frequency curve plots the running total against the upper boundary of each class. With $n$ values in total, read across from:

So for 200 values you read at cumulative frequencies of 50, 100 and 150.

What happens when the data changes

Worth understanding rather than memorising, because it is quick marks:

• Add a constant to every value — the mean rises by that constant; the standard deviation is unchanged, because every value moved together and the spread is identical.
• Multiply every value by a constant — the mean and the standard deviation are both multiplied by it.
• Add a new value equal to the mean — the mean does not move.