Sorting Numbers Before You Count Them Saves Every Mistake

Because almost every question becomes easy once the numbers are sorted. The largest and smallest values sit at the two ends, so the range is one subtraction, and equal values fall together, so counting them cannot go wrong.

Sorting is the cheapest step in the whole chapter and prevents the most errors. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: raw and arrayed data, the range, and frequency distribution tables.

Raw data is information exactly as collected, in no particular order. Arrayed data is the same information arranged in ascending or descending order.

Take the marks of seven students, collected as they were handed in:

$$23,45,12,45,30,12,38$$

That is raw data. Arranged in ascending order it becomes

$$12,12,23,30,38,45,45$$

and in descending order

$$45,45,38,30,23,12,12$$

Each separate value is called an observation, and the number of observations here is 7. This arranged form is also called an array.

A teacher's attendance register is raw data; the same names printed alphabetically are arrayed.

Two points matter when arranging. Repeated values are written as many times as they occur — both 12s and both 45s appear, so the count stays 7. And the number of observations never changes with sorting: if your arrayed list has 6 entries where the raw list had 7, a value has been dropped.

The range is the difference between the highest and the lowest observation:

$$\text{Range}=\text{highest observation}-\text{lowest observation}$$

Worked example. For the marks $12,12,23,30,38,45,45$:

$$\text{Range}=45-12=33$$

Another. Daily temperatures of $28°C$, $31°C$, $27°C$, $33°C$ and $30°C$ give

$$33-27=6°C$$

The range tells you how spread out the data is. A small range means the values are bunched closely together; a large range means they are scattered widely.

So two classes could both average 30 marks while one has a range of 6 and the other a range of 40 — the same average describing very different classes.

The range carries the same unit as the data, so a temperature range is in $°C$ and a height range in cm.

The limitation worth knowing is that the range uses only two values and ignores everything between them. One unusually high or low observation stretches it completely, which is why the range describes the extremes rather than the typical spread.

List each distinct value, put a tally mark against it for every occurrence, then total the tallies. That total is the frequency.

Tally marks are grouped in fives, with the fifth drawn across the previous four, which makes counting in fives possible at a glance.

Worked example. Twenty students report their shoe sizes:

$$5,6,7,5,6,6,7,8,5,6,7,6,5,7,6,8,5,6,7,6$$

Working through the list once, marking a tally against each value as you meet it, gives:

- Size 5 — frequency 5
- Size 6 — frequency 8
- Size 7 — frequency 5
- Size 8 — frequency 2

The frequencies must add to the number of observations:

$$5+8+5+2=20$$

That check catches a missed or double-counted entry immediately, and is worth doing every time.

The table now answers questions the raw list hid. The most common size is 6, only two students take size 8, and the range is $8-5=3$.

The habit to build is tallying in a single pass through the data. Scanning the whole list again for each separate value is slow and is exactly where entries get counted twice — and the frequency total is what tells you it happened.