Math Vale

📊 Statistics Steppe · Statistics

Reading Data Displays

Pull exact answers out of tables, bar charts, pictographs, two-way tables and line graphs without being misled by the picture.

In short

  • Read the labels, the scale and the key before reading any number; most errors here are made before the arithmetic starts.
  • A frequency table totals its frequency column, not its rows, and a grouped table hides the individual values it summarises.
  • In a two-way table a cell satisfies both descriptions, a total satisfies one, and the row and column totals must both reach the same grand total.
  • On a line graph the overall change uses only the two end readings, and the steepest rise is not the highest point.

Read the labels before the numbers

A display is an argument in picture form, and the picture is only as honest as its labels. Before you take a single number off it, find four things:

  • What is being counted? Bundles, people, minutes, gold.
  • What are the categories or the time steps?
  • What does one unit on the scale mean? A gridline may be worth 1, or 5, or 100.
  • Is there a key? A pictograph symbol usually stands for more than one thing.

Skipping this is where most marks are lost, and none of it is arithmetic. A bar reaching halfway between the 40 and 50 gridlines is 45, not 4.5 and not "about the middle".

Frequency tables, grouped and ungrouped

A frequency table lists each value or category with how often it occurred. The total is the sum of the frequency column — not the number of rows. Six crops listed does not mean six bundles.

A grouped frequency table sorts the data into bands: 10-14, 15-19, 20-24. Grouping is a trade. You gain a readable summary; you lose the individual values, so from a grouped table you can no longer say what the largest measurement actually was, only which band it fell in.

To answer "how many were at least 20 cm?", decide which bands qualify and add only those frequencies. Take care at the boundary: 20-24 starts *at* 20, so "at least 20" includes it and "under 20" does not.

Two-way tables

A two-way table sorts the same group of people twice at once — by how they travelled down the side, by whether they carried a lantern along the top.

  • A cell is where one row meets one column, so the people in it satisfy both descriptions.
  • A row total counts everyone with that row's property, whatever the column says.
  • The grand total, in the corner, is everybody.

Row totals add down to the grand total, and column totals add across to the same grand total. That gives a free check on every reading you take, and it also lets you fill in a missing cell: if a row total is 72 and one of its two cells is 40, the other must be 32.

Graphs over time, and pictographs

A line graph shows a quantity changing over time, and the shape carries the meaning: rising, falling, level, or steepening.

  • The overall change between two days only needs the two end readings: later minus earlier, negative if it fell.
  • The steepest rise is not the highest point. A line can reach its peak by climbing gently after a sharp jump earlier.

A pictograph replaces the scale with a key, and the key is the whole point. If one symbol stands for 25 sacks, four symbols mean 100 sacks, not 4. Half a symbol means half the key's value.

Whatever the display, finish by asking whether the answer is the right size. A total must be at least as big as its biggest part, and a difference must be smaller than the larger of the two things being compared.

Worked examples

Example 1

A grouped table records reed heights: 5-9 cm: 12, 10-14 cm: 20, 15-19 cm: 31, 20-24 cm: 17, 25-29 cm: 8. How many reeds were at least 15 cm tall?

  1. Decide which bands qualify. "At least 15 cm" means 15 and above, so the 15-19, 20-24 and 25-29 bands all count.
  2. The 10-14 band does not, because every reed in it is under 15 cm.
  3. Add only the qualifying frequencies: 31 + 17 + 8.
  4. 31 + 17 = 48, and 48 + 8 = 56.
  5. So 56 reeds were at least 15 cm tall. Check: the whole table holds 12 + 20 + 31 + 17 + 8 = 88 reeds, and 56 is a sensible majority of them.

Example 2

A two-way table shows: by boat with a lantern 48, by boat without 24; on foot with a lantern 15, on foot without 33. How many travellers carried no lantern?

  1. "No lantern" is a column of the table, so it takes in both rows.
  2. The by-boat cell in that column is 24; the on-foot cell is 33.
  3. Column total = 24 + 33 = 57 travellers.
  4. Check with the grand total: 48 + 24 + 15 + 33 = 120 travellers in all, and 48 + 15 = 63 carried a lantern.
  5. 63 + 57 = 120, which agrees, so the reading is right.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

The table records the bundles gathered at Dice Landing this week. How many bundles were gathered altogether?

Answer: 21 bundles

  1. The column reads 9, 1, 5, 6.
  2. Total = 9 + 1 + 5 + 6 = 21
  3. So the answer is 21 bundles.

Problem 2

Difficulty 3 of 5

Surveyors at Survey Tower measured the height, in centimetres, of every reed in a plot and grouped the results. How many reeds were under 20 cm tall?

Answer: 67 reeds

  1. Bands that qualify: 0-9, 10-19.
  2. Their frequencies are 43, 24.
  3. Total = 43 + 24 = 67

Problem 3

Difficulty 4 of 5

Every traveller crossing at Coin Ferry was recorded twice over: by how they travelled, and by whether they carried a lantern. How many travellers walked and carried no lantern?

Answer: 16 travellers

  1. The "On foot" row meets the "No lantern" column at 16.
  2. Check against the grand total: 56 + 72 + 48 + 16 = 192.
  3. So the answer is 16.

Common mistakes

  • Counting the rows or categories instead of adding the frequencies they hold.
  • Ignoring a pictograph key and reading four symbols as four rather than as four times the key value.
  • Taking the wrong side of a boundary, so "at least 20" quietly excludes the 20-24 band.
  • Reading a row total when a column total was asked for, or a single cell when a total was wanted.
  • Naming the highest point on a line graph as the biggest rise, when the biggest rise is the steepest climb between two points.

What you should be able to do

  • Read a value, a total and a difference from a frequency table or bar chart.
  • Use the key of a pictograph to turn symbols back into counts.
  • Read a cell, a row total and a column total from a two-way table.
  • Describe how a quantity changes over time from a line graph.

Where this fits in the curriculum

Common Core

  • 6.SP.B.4

    Grade 6 — Display numerical data in plots on a number line, including dot plots, histograms and box plots.

  • 3.MD.B.3

    Grade 3 — Draw a scaled picture graph and a scaled bar graph, and solve one- and two-step problems using the information in them.

Ontario

  • G6.D1

    Grade 6 — Manage, analyse and use data to make convincing arguments and informed decisions, in various contexts drawn from real life.

    Ontario numbers its Data expectations per grade document and the specific numbering could not be verified line by line here, so this tag names the strand's OVERALL expectation rather than guessing at a specific one.

  • G7.D1

    Grade 7 — Manage, analyse and use data to make convincing arguments and informed decisions, in various contexts drawn from real life.

    Ontario numbers its Data expectations per grade document and the specific numbering could not be verified line by line here, so this tag names the strand's OVERALL expectation rather than guessing at a specific one.

SAT

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