Functional Skills Qualifications Functional Skills Maths — Handling Data and Statistics Syllabus

When reading a value from a chart with a scale, how do you work out the value of each small division?

Find the difference between two labelled gridlines and divide by the number of small divisions between them; that gives the value of each small square.

Which chart is most suitable for showing how each category compares as a proportion of a whole?

A pie chart, because each sector shows the fraction/percentage of the total that each category makes up.

Which chart is most suitable for showing a trend or change over time?

A line graph, because joined points clearly show how a quantity rises or falls over time.

Which chart is most suitable for comparing the frequencies of separate categories?

A bar chart, because the heights of the bars can be compared directly.

Which chart/graph is most suitable for showing the relationship (correlation) between two variables?

A scatter graph, because it plots paired data and reveals whether the variables are linked.

Why is a pie chart usually a poor choice when you need to read off exact values?

Because it shows proportions, not actual amounts; reading precise frequencies from sector angles is difficult, and many similar-sized slices are hard to compare.

Define the mean of a data set and give its formula.

The mean is the arithmetic average. $$\text{mean} = \frac{\text{sum of all values}}{\text{number of values}}$$

Define the median of a data set.

The median is the middle value when the data are arranged in order of size. If there are two middle values, the median is their mean (halfway between them).

Define the mode of a data set.

The mode is the value (or category) that occurs most often. A data set can have no mode, one mode, or more than one mode.

For an ordered list of $n$ values, how do you find the position of the median?

The median is at position $\frac{n+1}{2}$. For example, with $n=9$ values it is the $\frac{9+1}{2}=5\text{th}$ value.

Find the median of the data set $3, 7, 8, 12, 15, 20$.

There are $6$ values, so the median is the mean of the $3\text{rd}$ and $4\text{th}$: $\frac{8+12}{2}=10$.

Find the mean of $4, 6, 9, 5$.

$$\frac{4+6+9+5}{4}=\frac{24}{4}=6$$

Find the mode of $2, 3, 3, 5, 7, 3, 8$.

$3$, because it appears most often (three times).

Which average can be used for categorical (non-numerical) data, such as favourite colour?

Only the mode, because you cannot add or order categories to find a mean or median.

Why can a single very large or very small value (an outlier) make the mean misleading?

Because the mean uses every value in its calculation, an extreme value pulls it towards itself, so it may no longer represent a typical value. The median is less affected by outliers.

Define the range of a data set and give its formula.

The range measures spread. $$\text{range} = \text{highest value} - \text{lowest value}$$

Find the range of $12, 5, 9, 20, 3$.

$20 - 3 = 17$.

What does the range tell you about a data set, and is it an average?

The range tells you how spread out (consistent or variable) the data are — a small range means values are close together. It is a measure of spread, not an average.

When comparing two data sets, which two types of measure should you use together, and why?

An average (mean, median or mode) to compare typical values, and the range to compare consistency/spread. Using both gives a fairer comparison than either alone.

Team A has a higher mean score but a larger range than Team B. What does this tell you?

Team A scores higher on average, but Team B is more consistent/reliable because its smaller range means its scores vary less.

How do you find the mean from a (ungrouped) frequency table?

Multiply each value by its frequency to get $value \times frequency$ for each row, add these up, then divide by the total frequency: $$\text{mean}=\frac{\sum fx}{\sum f}$$

What does it mean to interpret data rather than just read it?

Interpreting means drawing conclusions, spotting trends or patterns, making comparisons, and explaining what the figures mean in context — not just stating individual values.

What is a trend in a data set?

A trend is a general direction or pattern over time or across categories, such as values generally increasing, decreasing, or staying steady.

Why should conclusions drawn from a small sample be treated with caution?

A small sample may not be representative of the whole population, so conclusions based on it may be unreliable and could be due to chance.

On the probability scale, what range of values can a probability take?

Any value from $0$ to $1$ inclusive, $$0 \leq P \leq 1$$ which may be written as a fraction, decimal or percentage.

On the probability scale, what do $0$, $\frac{1}{2}$ and $1$ represent in words?

$0$ = impossible, $\frac{1}{2}$ = even chance (equally likely as not), and $1$ = certain. Values between describe unlikely (near $0$) or likely (near $1$) events.

What is the formula for the probability of an event when all outcomes are equally likely?

$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$$

A fair six-sided dice is rolled. What is the probability of rolling an even number?

Even numbers are $2,4,6$, so $$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$

What is the formula linking the probability of an event happening to it not happening?

$$P(\text{not } A) = 1 - P(A)$$ because all probabilities of an outcome and its complement add up to $1$.

If the probability it rains tomorrow is $0.3$, what is the probability it does not rain?

$1 - 0.3 = 0.7$.

What is the expected number of times an event occurs over $n$ trials?

$$\text{expected frequency} = P(\text{event}) \times n$$ For example, rolling a $6$ on a fair dice $60$ times: $\frac{1}{6}\times 60 = 10$ times.

When listing the possible outcomes of two events (e.g. two coins), what tool helps ensure you list them all systematically?

A sample space diagram (a two-way grid) or a systematic list, which sets out every combination so none is missed. For two coins: HH, HT, TH, TT.

Two fair coins are tossed. What is the probability of getting two heads?

The outcomes are HH, HT, TH, TT (four equally likely), so $$P(\text{two heads}) = \frac{1}{4}$$