Analysing Data

When you have a list of numbers, four summaries do most of the heavy lifting.

• The mean adds everything up and shares it out equally: $\text{mean} = \dfrac{\text{sum of values}}{\text{number of values}}$.
• The median is the middle value once the list is in order. If there are two middle values, take the mean of those two.
• The mode is the value that appears most often. A set can have no mode, one mode or more than one mode.
• The range is a measure of spread: $\text{range} = \text{largest} - \text{smallest}$.

For the list $2, 4, 4, 7, 8$:

• Mean $= \dfrac{2 + 4 + 4 + 7 + 8}{5} = \dfrac{25}{5} = 5$.
• Median (middle of five values, sorted) $= 4$.
• Mode (most common) $= 4$.
• Range $= 8 - 2 = 6$.

Each summary tells a slightly different story. Together they sketch the data quickly and accurately — which is exactly what you want when comparing two groups.

When data repeats, a frequency table is tidier than a list. The frequency $f$ is just how many times that value $x$ appears.

For the table below (shoe sizes of $20$ students):

Shoe size $x$	4	5	6	7	8
Frequency $f$	2	5	7	4	2

The total frequency is $\sum f = 2 + 5 + 7 + 4 + 2 = 20$.

• Mode: the value with the highest frequency, so the modal shoe size is $6$.
• Median: with $20$ values, the median is between the $10^{\text{th}}$ and $11^{\text{th}}$ values once they are in order. Adding the frequencies as you go — $2, 7, 14, \ldots$ — both the $10^{\text{th}}$ and $11^{\text{th}}$ values fall in the size-$6$ row, so the median is $6$.
• Mean: multiply each value by its frequency, add the products, then divide by $\sum f$:

$\bar{x} = \frac{\sum fx}{\sum f} = \frac{4(2) + 5(5) + 6(7) + 7(4) + 8(2)}{20} = \frac{8 + 25 + 42 + 28 + 16}{20} = \frac{119}{20} = 5.95.$

• Range: $8 - 4 = 4$.

The trick is to remember that the frequency replaces writing the same value out multiple times. A working column for $fx$ alongside the table keeps your numbers tidy and saves silly arithmetic slips.

If the data covers a wide range, it is often grouped into intervals. You can no longer read off exact values, so you find the modal class and an estimated mean.

Mass $m$ (kg)	$40 \le m < 50$	$50 \le m < 60$	$60 \le m < 70$	$70 \le m < 80$
Frequency $f$	6	12	8	4

• Modal class: the class with the highest frequency — $50 \le m < 60$ (frequency $12$).
• Median class: with $\sum f = 30$, the median is the $15^{\text{th}}$ -ish value. Cumulative frequencies $6, 18, 26, 30$ tell you the $15^{\text{th}}$ value is in $50 \le m < 60$ — the median class.
• Estimated mean: pick the midpoint of each class as $x$, multiply by the frequency, and divide:

$\bar{x} \approx \frac{\sum fx}{\sum f} = \frac{45(6) + 55(12) + 65(8) + 75(4)}{30} = \frac{270 + 660 + 520 + 300}{30} = \frac{1750}{30} \approx 58.3 \text{ kg}.$

The result is an estimate because every student in the $50 \le m < 60$ class is treated as if they were exactly $55$ kg. That is the trade-off when data is grouped — convenience now, a small loss of accuracy later.

When you write the final answer, say it is an estimate — that single word reminds the reader that midpoints stand in for the real values inside each class.

When you compare two groups, the rule is simple: use one average (mean or median) and one spread (the range), and write a sentence that names the group and the units.

Suppose two classes sit the same $10$-mark quiz:

	Class A	Class B
Mean	$6.4$	$7.1$
Range	$5$	$9$

A good comparison reads:

"Class B has a higher mean ($7.1$ marks vs $6.4$ marks), so on average they scored higher than Class A. However, Class A has a smaller range ($5$ vs $9$), so their scores were more consistent."

Two ideas pop out. The average answers "who did better on the whole?" — a higher mean (or higher median) means higher scores on average. The range answers "who was more consistent?" — a smaller range means the scores cluster more tightly.

Two reminders to keep your comparison full marks. Always use the same pair of statistics for both groups (do not compare A's mean with B's median). And always write a sentence in context — name the groups, give the numbers and the units, and say what the difference means in plain English.

Different averages work best in different situations.

• Mean uses every value. It is the most informative when the data is fairly evenly spread, but a single huge or tiny value (an outlier) can pull the mean a long way.
• Median is the middle value, so a single outlier barely moves it. The median is a safer choice for skewed data — like income or house prices.
• Mode is the only average that works for categorical data (favourite colour, shoe brand, eye colour), because you cannot add up words.

For the salaries (in thousands) $20, 22, 24, 25, 110$:

• Mean $= 40.2$ — dragged upwards by the outlier $110$.
• Median $= 24$ — the middle value, unaffected by the outlier.

The median is the better summary of "typical" salary here. The mean tells you about the total payroll, but the median tells you about the middle person. Picking the right average is part of analysing data — and showing you understand the difference is exactly what comparison questions reward.

Mean, median, mode and range are the language of data — and they keep coming back:

• Charts and graphs in the next subtopic let you see averages and ranges at a glance.
• Probability uses relative frequency, which is built on the same idea of "how often does this value happen?".
• Cumulative frequency and scatter graphs in later years extend the same comparison skills.
• In real life, summaries are how journalists, scientists and economists describe huge data sets in a sentence.

Practise the four summaries on lists, tables and grouped tables until they feel automatic. The comparison sentence — average, spread, context — is the part that often makes the difference between a good and a top answer.