Data Representation · Lesson 4

Complex figures

Some figures pack in several lines, a scaled axis, or two panels. The data are no harder to read, but there are more ways to land on the wrong number. Before you read any value, name three things: the figure, the series (by its legend name, not its color), and the axis with its units.

Scaled axes. An axis labeled thousands of µL or ×10³ means every tick number is multiplied by 1,000: a reading of 12 means 12,000. The multiplier is in the label, not on the ticks, and a common wrong choice is the right reading with the multiplier left off.

Log axes. When an axis shows the log₁₀ of a quantity, each step of 1 means 10 times as much. A reading of 3 means 10³, or 1,000, and a reading of 6 means 10⁶, which is 1,000 times as much, not 3 more. To compare two points, subtract the readings, then use that gap as a power of 10: a gap of 1 is ×10, a gap of 2 is ×100. A true log scale, with ticks at 1, 10, 100 and 1,000 spaced evenly, works the same way: each equal step up multiplies the value.

Two vertical axes, or two panels. Some graphs add a second y-axis on the right for a second variable. Match each line to its axis through the legend and the axis labels before you read it. Two charts that share an x-axis work the same way: find the x value in one chart, then go to that same x value in the other chart, on the series the question names.

Bacteria at Three Temperatures

Students grew one strain of soil bacteria in flasks of nutrient broth kept at 15°C, 25°C and 35°C. Every 2 hours for 12 hours, they estimated the number of cells per milliliter (cells/mL) in each flask and measured the total volume of carbon dioxide (CO₂) each flask had released, in microliters (µL).

Figure 1 shows the log₁₀ of the cell count, so a reading of 6 means 10⁶ cells/mL, and a reading 1 unit higher means 10 times as many cells.

Figure 2 uses the same time axis as Figure 1. Its vertical axis is in thousands of microliters, so a reading of 5 means 5,000 µL of CO₂.

Figure 1
Figure 1Line graph of the log₁₀ of the cell count (cells/mL) against time (h) for three cultures. 15°C: 4.0, 4.2, 4.6, 5.0, 5.4, 5.8 and 6.2 at 0, 2, 4, 6, 8, 10 and 12 h, rising slowly the whole time. 25°C: 4.0, 4.6, 5.4, 6.2, 7.0, 7.6 and 7.8, rising faster and then leveling off. 35°C: 4.0, 5.0, 6.0, 7.0, 8.0, 8.2 and 8.2, rising fastest and then leveling off after 8 h.3456789024681012Time (h)Cell count, log₁₀ (cells/mL)
  • 15°C
  • 25°C
  • 35°C
Show the data as a table
Figure 1: data
Time (h)15°C (cells/mL)25°C (cells/mL)35°C (cells/mL)
0444
24.24.65
44.65.46
656.27
85.478
105.87.68.2
126.27.88.2

Line graph of the log₁₀ of the cell count (cells/mL) against time (h) for three cultures. 15°C: 4.0, 4.2, 4.6, 5.0, 5.4, 5.8 and 6.2 at 0, 2, 4, 6, 8, 10 and 12 h, rising slowly the whole time. 25°C: 4.0, 4.6, 5.4, 6.2, 7.0, 7.6 and 7.8, rising faster and then leveling off. 35°C: 4.0, 5.0, 6.0, 7.0, 8.0, 8.2 and 8.2, rising fastest and then leveling off after 8 h.

Figure 2
Figure 2Line graph of the total CO₂ released, in thousands of µL, against time (h) for the same three cultures. 15°C: 0, 1, 2, 4, 6, 8 and 10 at 0, 2, 4, 6, 8, 10 and 12 h. 25°C: 0, 2, 5, 9, 14, 19 and 23. 35°C: 0, 3, 8, 15, 24, 31 and 35. All three rise the whole time; the warmer the culture, the faster it rises.0510152025303540024681012Time (h)CO₂ released (thousands of µL)
  • 15°C
  • 25°C
  • 35°C
Show the data as a table
Figure 2: data
Time (h)15°C (thousands of µL)25°C (thousands of µL)35°C (thousands of µL)
0000
2123
4258
64915
861424
1081931
12102335

Line graph of the total CO₂ released, in thousands of µL, against time (h) for the same three cultures. 15°C: 0, 1, 2, 4, 6, 8 and 10 at 0, 2, 4, 6, 8, 10 and 12 h. 25°C: 0, 2, 5, 9, 14, 19 and 23. 35°C: 0, 3, 8, 15, 24, 31 and 35. All three rise the whole time; the warmer the culture, the faster it rises.

Worked example Easy

Show the whole passage

Bacteria at Three Temperatures

Students grew one strain of soil bacteria in flasks of nutrient broth kept at 15°C, 25°C and 35°C. Every 2 hours for 12 hours, they estimated the number of cells per milliliter (cells/mL) in each flask and measured the total volume of carbon dioxide (CO₂) each flask had released, in microliters (µL).

Figure 1 shows the log₁₀ of the cell count, so a reading of 6 means 10⁶ cells/mL, and a reading 1 unit higher means 10 times as many cells.

Figure 2 uses the same time axis as Figure 1. Its vertical axis is in thousands of microliters, so a reading of 5 means 5,000 µL of CO₂.

Figure 1
Figure 1Line graph of the log₁₀ of the cell count (cells/mL) against time (h) for three cultures. 15°C: 4.0, 4.2, 4.6, 5.0, 5.4, 5.8 and 6.2 at 0, 2, 4, 6, 8, 10 and 12 h, rising slowly the whole time. 25°C: 4.0, 4.6, 5.4, 6.2, 7.0, 7.6 and 7.8, rising faster and then leveling off. 35°C: 4.0, 5.0, 6.0, 7.0, 8.0, 8.2 and 8.2, rising fastest and then leveling off after 8 h.3456789024681012Time (h)Cell count, log₁₀ (cells/mL)
  • 15°C
  • 25°C
  • 35°C
Show the data as a table
Figure 1: data
Time (h)15°C (cells/mL)25°C (cells/mL)35°C (cells/mL)
0444
24.24.65
44.65.46
656.27
85.478
105.87.68.2
126.27.88.2

Line graph of the log₁₀ of the cell count (cells/mL) against time (h) for three cultures. 15°C: 4.0, 4.2, 4.6, 5.0, 5.4, 5.8 and 6.2 at 0, 2, 4, 6, 8, 10 and 12 h, rising slowly the whole time. 25°C: 4.0, 4.6, 5.4, 6.2, 7.0, 7.6 and 7.8, rising faster and then leveling off. 35°C: 4.0, 5.0, 6.0, 7.0, 8.0, 8.2 and 8.2, rising fastest and then leveling off after 8 h.

Figure 2
Figure 2Line graph of the total CO₂ released, in thousands of µL, against time (h) for the same three cultures. 15°C: 0, 1, 2, 4, 6, 8 and 10 at 0, 2, 4, 6, 8, 10 and 12 h. 25°C: 0, 2, 5, 9, 14, 19 and 23. 35°C: 0, 3, 8, 15, 24, 31 and 35. All three rise the whole time; the warmer the culture, the faster it rises.0510152025303540024681012Time (h)CO₂ released (thousands of µL)
  • 15°C
  • 25°C
  • 35°C
Show the data as a table
Figure 2: data
Time (h)15°C (thousands of µL)25°C (thousands of µL)35°C (thousands of µL)
0000
2123
4258
64915
861424
1081931
12102335

Line graph of the total CO₂ released, in thousands of µL, against time (h) for the same three cultures. 15°C: 0, 1, 2, 4, 6, 8 and 10 at 0, 2, 4, 6, 8, 10 and 12 h. 25°C: 0, 2, 5, 9, 14, 19 and 23. 35°C: 0, 3, 8, 15, 24, 31 and 35. All three rise the whole time; the warmer the culture, the faster it rises.

According to Figure 1, at 8 h, the log₁₀ of the cell count of the 15°C culture was:

  1. 5.4Answer
  2. 5.8
  3. 7.0
  4. 8.0

How to solve it

  1. Name the coordinates: Figure 1, series 15°C, time 8 h.
  2. Go up from 8 h on the time axis to the 15°C line, the lowest of the three at that time.
  3. Read across to the vertical axis: 5.4. The 7.0 and 8.0 at the same time belong to the 25°C and 35°C lines.

Why each choice is right or wrong

  • A. Correct. In Figure 1, the 15°C line passes through 5.4 at 8 h.
  • B. Incorrect. 5.8 is the 15°C reading at 10 h, one gridline too late on the time axis.
  • C. Incorrect. 7.0 is the 25°C line at 8 h: the right time, but the wrong series.
  • D. Incorrect. 8.0 is the 35°C line at 8 h: the right time, but the wrong series.

Worked example Medium

Show the whole passage

Bacteria at Three Temperatures

Students grew one strain of soil bacteria in flasks of nutrient broth kept at 15°C, 25°C and 35°C. Every 2 hours for 12 hours, they estimated the number of cells per milliliter (cells/mL) in each flask and measured the total volume of carbon dioxide (CO₂) each flask had released, in microliters (µL).

Figure 1 shows the log₁₀ of the cell count, so a reading of 6 means 10⁶ cells/mL, and a reading 1 unit higher means 10 times as many cells.

Figure 2 uses the same time axis as Figure 1. Its vertical axis is in thousands of microliters, so a reading of 5 means 5,000 µL of CO₂.

Figure 1
Figure 1Line graph of the log₁₀ of the cell count (cells/mL) against time (h) for three cultures. 15°C: 4.0, 4.2, 4.6, 5.0, 5.4, 5.8 and 6.2 at 0, 2, 4, 6, 8, 10 and 12 h, rising slowly the whole time. 25°C: 4.0, 4.6, 5.4, 6.2, 7.0, 7.6 and 7.8, rising faster and then leveling off. 35°C: 4.0, 5.0, 6.0, 7.0, 8.0, 8.2 and 8.2, rising fastest and then leveling off after 8 h.3456789024681012Time (h)Cell count, log₁₀ (cells/mL)
  • 15°C
  • 25°C
  • 35°C
Show the data as a table
Figure 1: data
Time (h)15°C (cells/mL)25°C (cells/mL)35°C (cells/mL)
0444
24.24.65
44.65.46
656.27
85.478
105.87.68.2
126.27.88.2

Line graph of the log₁₀ of the cell count (cells/mL) against time (h) for three cultures. 15°C: 4.0, 4.2, 4.6, 5.0, 5.4, 5.8 and 6.2 at 0, 2, 4, 6, 8, 10 and 12 h, rising slowly the whole time. 25°C: 4.0, 4.6, 5.4, 6.2, 7.0, 7.6 and 7.8, rising faster and then leveling off. 35°C: 4.0, 5.0, 6.0, 7.0, 8.0, 8.2 and 8.2, rising fastest and then leveling off after 8 h.

Figure 2
Figure 2Line graph of the total CO₂ released, in thousands of µL, against time (h) for the same three cultures. 15°C: 0, 1, 2, 4, 6, 8 and 10 at 0, 2, 4, 6, 8, 10 and 12 h. 25°C: 0, 2, 5, 9, 14, 19 and 23. 35°C: 0, 3, 8, 15, 24, 31 and 35. All three rise the whole time; the warmer the culture, the faster it rises.0510152025303540024681012Time (h)CO₂ released (thousands of µL)
  • 15°C
  • 25°C
  • 35°C
Show the data as a table
Figure 2: data
Time (h)15°C (thousands of µL)25°C (thousands of µL)35°C (thousands of µL)
0000
2123
4258
64915
861424
1081931
12102335

Line graph of the total CO₂ released, in thousands of µL, against time (h) for the same three cultures. 15°C: 0, 1, 2, 4, 6, 8 and 10 at 0, 2, 4, 6, 8, 10 and 12 h. 25°C: 0, 2, 5, 9, 14, 19 and 23. 35°C: 0, 3, 8, 15, 24, 31 and 35. All three rise the whole time; the warmer the culture, the faster it rises.

Based on Figure 1, at 6 h, the 35°C culture had about how many times as many cells per mL as the 15°C culture?

  1. 1.4 times
  2. 2 times
  3. 100 timesAnswer
  4. 1,000 times

How to solve it

  1. Read both lines at 6 h in Figure 1: 35°C is at 7.0 and 15°C is at 5.0.
  2. Subtract the log readings: 7.0 − 5.0 = 2.
  3. A gap of 2 on a log₁₀ axis is a factor of 10², so the 35°C culture had 100 times as many cells per mL.

Why each choice is right or wrong

  • A. Incorrect. 1.4 times comes from dividing the readings, 7.0 ÷ 5.0. Those are log values, not cell counts, so they can’t be divided directly.
  • B. Incorrect. 2 times uses the gap between the readings, 7.0 − 5.0 = 2, as the factor. On a log₁₀ axis, a gap of 2 means 10², not 2.
  • C. Correct. At 6 h, the 35°C line is at 7.0 (10⁷ cells/mL) and the 15°C line is at 5.0 (10⁵ cells/mL). A gap of 2 units on a log₁₀ axis means 10 × 10 = 100 times as many.
  • D. Incorrect. 1,000 times would need a gap of 3 units (10³). The readings at 6 h, 7.0 and 5.0, are only 2 units apart.

Trap Right point, wrong line or wrong scale

With three lines on one graph, a wrong choice is often a real value from a neighboring line at the right time. Find the series name from the question in the legend and trace only that line. Then check the axis label for a multiplier such as thousands or ×10³: a choice that is the right reading with the multiplier left off, such as 12 µL for 12 thousand µL, is a trap. On a log axis, remember that equal steps multiply; they don’t add.