Turning a Messy List of Numbers Into a Picture

Raw data is information exactly as it was collected, in no particular order. Arrayed data is the same information arranged in order, usually ascending or descending. Arranging it is the first step in making sense of it, because patterns that are invisible in a jumbled list become obvious once it is sorted.

This page covers everything in the ICSE Class 6 Mathematics chapter on data handling: organising raw data into a frequency table, reading and drawing pictographs, and reading and constructing bar graphs.

A frequency table records each distinct value alongside how many times it occurs, counted with tally marks. Tallies are drawn in groups of five — four upright strokes with the fifth struck diagonally across them — because a count in fives is far easier to total than a row of single strokes.

Suppose twenty students report their number of siblings as: 1, 2, 0, 1, 3, 2, 1, 0, 2, 1, 4, 2, 1, 3, 0, 1, 2, 2, 1, 3.

Sorting and tallying gives frequencies of 3 for zero siblings, 7 for one, 6 for two, 3 for three and 1 for four.

The frequency is that count, and the total of all frequencies must equal the number of observations. Here $3+7+6+3+1=20$, which matches the twenty students.

That check is not optional. If your frequencies do not add to the number of items you started with, a value has been missed or double-counted — and every later graph will inherit the error.

A pictograph shows data using a repeated symbol, and the key (or scale) states what one symbol represents. Reading it correctly means reading the key first, every time.

If a pictograph of books read uses one book symbol for 5 books, then a row of 4 symbols means $4\times 5=20$ books, not 4.

Symbols are often shown as fractions too. Half a symbol represents half the key's value, so with a key of 5, two and a half symbols mean $2.5\times 5=12.5$ — or, where only whole items make sense, a key is chosen to avoid awkward halves.

For example, a pictograph of a village's monthly water usage might use one drop for 100 litres, so seven drops mean 700 litres.

The trap is answering from the number of symbols rather than their value. A row of 3 symbols with a key of 10 means 30, and students who ignore the key answer 3 — which is the single commonest error on pictograph questions.

Choose a symbol that suits the data, then choose a scale that keeps every row short enough to read and long enough to compare.

Use this reasoning: look at the largest frequency. If the biggest value is 40, a key of 1 symbol = 1 would need forty symbols in one row, which is unreadable. A key of 1 symbol = 10 gives four symbols — comfortable.

Then draw the rows: label each category on the left, repeat the symbol the required number of times, and state the key clearly beside the graph.

For our siblings data, with frequencies of 3, 7, 6, 3 and 1, a key of 1 symbol = 1 student works well, since the largest value is only 7.

The scale must also divide the values sensibly. A key of 1 symbol = 4 would force the frequency 7 into an awkward one and three-quarter symbols, so prefer a key that leaves most values as whole or half symbols.

A bar graph represents each value by the height of a bar, with all bars the same width and equal gaps between them.

To construct one: draw the two axes, put the categories along the horizontal axis and the frequency up the vertical axis, choose a scale, then draw each bar to its height and label both axes and the graph.

Choosing the scale is the real decision. With a largest frequency of 7, a scale of 1 cm = 1 unit gives a graph 7 cm tall. If the largest frequency were 350, a scale of 1 cm = 50 units would give 7 cm again.

To read a bar graph, look at the scale first, then read each bar's height against the vertical axis. Comparisons are easy: the tallest bar is the largest value, and a bar twice the height of another represents twice the quantity — but only because the axis starts at zero.

That last point is the boundary case. If the vertical axis does not begin at 0, the bars' relative heights no longer reflect their true ratio, and comparisons by eye become misleading.