12.1 Reading Tables & Graphs

Every ICAS Mathematics paper contains several Chance & Data questions, and at every year level many of them come down to one thing: reading a display carefully. These questions rarely need hard calculation. They test whether you can extract the right number from a table, picture graph, column graph, dot plot or line graph — and whether you notice when a graph has been drawn to trick you. Because personal calculators are not allowed, all the arithmetic here is mental, which is exactly how ICAS intends it.

Frequency Tables and Tallies

A frequency table organises raw data by counting how often each value or category occurs. The count for each category is called its frequency. When data is collected by hand, counts are usually recorded as tallies: four vertical strokes, with the fifth drawn diagonally across them to make a bundle of five.

To find the total number of students surveyed, add the frequencies: 7 + 8 + 5 + 2 = 22. A classic ICAS trap is to count the tally rows (4) instead of summing the frequencies (22).

Picture Graphs

A picture graph (pictograph) uses symbols to represent quantities. The single most important habit is to read the key first. If one smiley face stands for 4 students, then three and a half faces stand for 3.5 × 4 = 14 students. ICAS regularly includes half-symbols to test whether you can multiply by the scale rather than just counting pictures.

Bar and Column Graphs

A column graph draws vertical columns whose heights show frequency (a bar graph is the same idea with horizontal bars). To compare columns, read each height against the scale on the vertical axis.

Worked example. A column graph shows books read in a month: Year 3 reaches 10, Year 4 reaches 14, Year 5 reaches 8. How many more books did Year 4 read than Year 5? Subtract: 14 − 8 = 6. Notice the question asks for the difference — ICAS often asks 'how many more' or 'how many fewer', not just 'how many'.

Comparing columns across categories

Questions at Papers A–D level frequently ask which two categories together equal a third, or which category is closest to the average height. Work through the options mentally rather than guessing from the picture.

Dot Plots

A dot plot places one dot for each data value above a number line. A dot plot of goals scored by players might show:

• 0 goals: 2 dots
• 1 goal: 5 dots
• 2 goals: 4 dots
• 3 goals: 1 dot

From this you can read the total number of players (2 + 5 + 4 + 1 = 12), the most common value (1 goal — the tallest stack) and the spread (0 to 3 goals). Paper D lists dot plots explicitly, so practise translating between a dot plot and a frequency table: each stack height is a frequency.

Line Graphs

A line graph shows how a quantity changes over time — temperature through a day, height over a year, rainfall per month. Read it in three steps: find the time on the horizontal axis, go up to the line, then across to the vertical axis for the value. The steepest segment shows the greatest change, not the highest point. If ICAS asks 'between which two months did sales increase the most?', compare slopes, not heights.

Sector (Pie) Graphs

A sector graph — usually called a pie chart — shows how a whole is divided into parts, and Paper E names it explicitly. The complete circle represents the whole data set, and each slice (sector) shows one category's share.

Because a full circle is 360°, two conversions cover almost every question:

• Fraction of the data → angle: multiply by 360. A quarter of the data is a 90° sector.
• Angle → share of the data: divide by 360. A 72° sector is 72/360 = 1/5 = 20% of the total.

Worked example. Sixty students name their favourite sport, and the football sector measures 120°. How many chose football? The sector is 120/360 = 1/3 of the circle, so 1/3 of 60 = 20 students.

Worked example (reverse). If 9 students out of 45 walk to school, what angle represents walking? The fraction is 9/45 = 1/5, so the angle is 1/5 × 360 = 72°.

The habit that protects marks here: a sector graph shows proportions, not counts. Two pie charts can have identical slices while describing groups of wildly different sizes, so a question like 'did more girls or more boys choose football?' cannot be answered from the slices alone unless you are told both totals. That comparison trap appears regularly in ICAS.

Two-Way Tables

A two-way table sorts data by two categories at once. To answer a question, find the cell where the relevant row and column meet.

Worked example. How many students both walk to school and bring a helmet? Read the intersection: 6. How many students were surveyed altogether? Use the corner total: 40. A common error is adding the row and column totals together, which double-counts every student.

Venn Diagrams

A Venn diagram sorts items using overlapping circles, and Paper G lists it alongside two-way tables. The questions test one skill above all: handling the overlap correctly.

Worked example. In a class of 30 students, 18 play soccer, 14 play cricket and 7 play both.

• The overlap holds the 7 who play both.
• Soccer only = 18 − 7 = 11.
• Cricket only = 14 − 7 = 7.
• Neither sport = 30 − (11 + 7 + 7) = 5 students.

The classic error is writing 18 straight into the soccer-only region. A figure quoted for a whole circle always includes the overlap, so subtract the overlap before filling in the outer part. Fill a Venn diagram from the middle outwards — overlap first, then the single-category regions, then 'neither' — and the arithmetic looks after itself.

Stem-and-Leaf Plots

A stem-and-leaf plot lists data compactly while keeping every original value visible, and it is named in Papers F through J. The stem is the leading digit (usually tens) and each leaf is a single trailing digit (units).

Key: 3 | 4 means 34.

Reading this plot, there are 2 + 4 + 5 + 2 = 13 values, running from 23 to 53. Because the leaves are already in order, the statistics fall out almost for free:

• Median = the 7th of the 13 values. Counting through 23, 27, 31, 34, 34, 38, 40 gives a median of 40.
• Mode = 34, the only value that repeats.
• Range = 53 − 23 = 30.

That is exactly why ICAS favours this display: it looks like a picture but behaves like an ordered list, so median, mode and range questions are fast — provided you read the key first. The most common mistake is treating a leaf as a whole number, reading '4 | 0' as 4 instead of 40.

Misleading Graphs

ICAS frequently tests whether you can spot a graph designed to deceive. The two most common tricks:

1. Truncated axis — the vertical axis starts at, say, 50 instead of 0, so a column of 55 looks twice the height of a column of 52 when the real difference is tiny.
2. Uneven scale — the axis jumps 0, 5, 10, 25, 30, so equal vertical steps no longer mean equal amounts.

Other tricks include uneven bar widths and pictures that grow in two dimensions (a symbol twice as tall is also twice as wide, so it looks four times as big). When an ICAS question asks why a graph is misleading, the answer almost always refers to the scale or the axis, not the data itself.

Strategy for Read-and-Interpret Questions

• Read the title, axis labels, scale and key before looking at the data.
• Check what the question actually asks: a single value, a total, a difference, or a comparison.
• Keep mental arithmetic simple — all numbers are chosen to be manageable without a calculator.
• Re-read the display after choosing an answer; misreading the scale is the most common way to lose these marks.