Discrete data lists exact values (e.g. "number of pets") with a frequency for each — because every individual value is known, any average calculated from it is exact. Grouped data uses class intervals (e.g. $10\le t<20$) because the data is continuous or too spread out to list individually — since the actual values inside each class are unknown, any mean calculated from it is only an estimate.

1. Add a column multiplying each value $x$ by its frequency $f$, to get $fx$.
2. Sum the frequency column to get $\sum f$ (the total number of observations), and sum the $fx$ column to get $\sum fx$.
3. Divide: $\text{Mean}=\sum fx \div \sum f$.

Example: if $\sum fx=35$ and $\sum f=20$, the mean is $35\div20=1.75$ — note the mean doesn't need to be a value that actually appears in the data.

Mode: the data value $x$ with the highest frequency — not the frequency number itself, the value it corresponds to.

Median: build a cumulative frequency (CF) column by adding frequencies progressively down the table. Find the median position, $(\sum f+1)\div2$, then find which row's CF first reaches that position — the $x$-value in that row is the median. If the position isn't a whole number (e.g. the $10.5$th value), the two middle data values average together.

Since exact values are unknown within each class, we assume every observation sits exactly at the midpoint of its class: $x=\dfrac{\text{lower limit}+\text{upper limit}}{2}$. This midpoint stands in for the true (unknown) value in the $fx$ calculation. Because real data is rarely spread perfectly evenly around the midpoint, the result is always labelled an estimate, never an exact mean.

Modal class: the interval with the highest frequency.

Median class: build the cumulative frequency column, find $N/2$ (where $N=\sum f$), and identify the first class whose CF exceeds that value — at Foundation level you only need to name this interval, not calculate an exact value inside it.

The true smallest and largest values are hidden somewhere inside the first and last class intervals — you only know the interval they fall in, not the precise number. This is why grouped-data questions rarely, if ever, ask for the range.

The table shows the time, $t$ minutes, taken by 30 students to complete a puzzle.

Find an estimate for the mean time, and state the modal class.

1. Midpoints: $5, 15, 25$.
2. $fx$ values: $6\times5=30$; $14\times15=210$; $10\times25=250$.
3. $\sum f = 6+14+10=30$. $\sum fx = 30+210+250=490$.
4. Estimated mean $=490\div30=16.3$ minutes (3 s.f.).
5. Modal class: highest frequency is 14, so the modal class is $10\le t<20$.