--- title: 'Misleading Visuals' source: 'https://academia.sh/en/courses/visualization-reporting-and-analysis-at-scale/misleading-visuals' course: 'Visualization, Reporting and Analysis at Scale' language: en updated: '2026-10-07T04:31:14+00:00' license: 'CC BY-SA 4.0' objectives: - 'Treats axis truncation, area enlargement, the dual axis and the uneven interval as members of a single family of distortions and prints the read-back error of each separately' - 'Computes that the area channel enlarges the ratio of values by its square and compares the value read back with the one in the source' - 'Shows that each of the four distortions arises from a defensible default behavior and that misleading does not require intent' --- # Misleading Visuals The true ratio of 1.2208 between east and north gives a bar ratio of 3.0769 when the axis is started from 18 m³, and a reader who takes length as proportional to value reads north as 8.019 m³ instead of 20.210. When the same ratio is encoded into a square, both sides are multiplied, so the area ratio comes out as 1.4905 and east is read as 30.123 m³. For two series with different units, nine defensible axis pairs give seven separate crossing periods. When 12 irregularly spaced periods are laid out at equal intervals, a slope of -0.2079 m³/period is read as -0.6635. All four arise from a defensible default behavior. The previous lesson measured a distortion without tying it to any choice: the area of the regions is given on the grid and the stretching is the price of printing onto a flat surface. Even so, the number read back slipped by 1.228 m³. This lesson widens the same question and treats four separate distortions as a single **family**. The common feature of the family is this: every member arises from the definition of the channel itself, none of them is a calculation error and none of them requires intent. How axis truncation distorts the ratio of bar lengths was measured in the basic charts lesson of the Analysis with Spreadsheets course, and that number is not repeated here. There the measure was **the legitimacy of the display** — is this chart acceptable at this scale. Here truncation is only one of the four members of the family and the first of four operations applied in turn to the same data. What is measured is the number each member makes the reader read back. - **CD37. The distortion arises from the definition of the channel.** The length channel is read relative to the base, the area channel grows in two dimensions at once, the position channel depends on the axis span. None of them spoils the calculation, all three spoil the read-back. - **CD38. The reader's decoder cannot be changed.** The reader takes length as proportional to value, area as proportional to value, and neighboring points as separated by an equal period interval. If the display falsifies this assumption, the error is written not to the display but to the reader. - **CD39. The area channel enlarges the ratio by its square.** If both sides of a shape are multiplied to "show a ratio as larger", the area ratio becomes the square of the ratio of values. - **CD40. A crossing on a dual axis has no counterpart in the data.** If the two series are in different units, the crossing period is determined by the choice of the two axis ranges, and although the choices are defensible they do not give a single number. - **CD41. When an uneven interval is laid out equally, the slope is measured in the number of steps.** The reader takes a step as a period; the enlargement factor is the period span divided by the number of steps. - **CD42. The series are fictional.** The consumption and reading count series of 36 periods were derived with the same seed, and that they are fictional is written in the code. ## Length: Axis Truncation ```python # misleading.py — MODEL: the measurement network of K03-K05 is # generated with the same seed. Both series of 36 periods are also FICTIONAL and # were derived with the same seed: the consumption mean (m3) and the number of # readings per period (count). import math SEED, RAW, M32 = 20260218, 1400, 0xFFFFFFFF PERIODS = ["2026-01", "2026-02", "2026-03"] REGION = [("north", 0.28, 21), ("south", 0.22, 17), ("east", 0.18, 26), ("west", 0.14, 14), ("central", 0.18, 23)] def rng(t): x = ((t ^ (t >> 16)) * 2246822507) & M32 x = ((x ^ (x >> 13)) * 3266489909) & M32 s = [(x ^ (x >> 16)) & M32] def next_value(): s[0] = (s[0] * 1664525 + 1013904223) & M32 return s[0] / 4294967296 return next_value def pick(u, w): t = 0.0 for i, x in enumerate(w): t += x if u < t: return i return len(w) - 1 _kept, _dropped = [], [] for i in range(RAW): r = rng(SEED + i) b = REGION[pick(r(), [x[1] for x in REGION])] pick(r(), [0.06, 0.24, 0.30, 0.24, 0.11, 0.05]) pick(r(), [0.08, 0.14, 0.27, 0.34, 0.17]) (_dropped if r() < 0.046 else _kept).append( {"subscriber_id": 10001 + i, "region": b[0], "base": b[2]}) VALID = {a["subscriber_id"] for a in _kept} READINGS = [] for a in _kept + _dropped: r = rng(SEED + 7000 + a["subscriber_id"]) for d in range(pick(r(), [0.05, 0.12, 0.21, 0.62])): t = 0.0 if r() < 0.038 else math.floor( a["base"] * math.exp((r() + r() + r() - 1.5) * 0.62) * (1 - d * 0.05) * 100 + 0.5) / 100 if a["subscriber_id"] not in VALID: r() continue READINGS.append({"subscriber_id": a["subscriber_id"], "region": a["region"], "period": PERIODS[d], "consumption_m3": t}) r() NAMES = [b[0] for b in REGION] VALUE = {a: sum(o["consumption_m3"] for o in READINGS if o["region"] == a) / sum(1 for o in READINGS if o["region"] == a) for a in NAMES} N = 36 _g = rng(SEED + 505) CONSUMPTION = [round(VALUE["north"] - 0.140 * t + 2.20 * math.sin(2 * math.pi * t / 12) + (_g() + _g() + _g() - 1.5) * 0.60, 3) for t in range(N)] _h = rng(SEED + 917) READ_COUNT = [round(1066 + 6.4 * t + 40 * math.sin(2 * math.pi * t / 12) + (_h() + _h() + _h() - 1.5) * 26) for t in range(N)] WIDTH = 40 def length(v, base, top): """Encoder: value -> length. The bar is rounded to a whole character.""" return max(0, round(WIDTH * (v - base) / (top - base))) def decode_length(u, base, top): """Decoder: length -> value. The reading resolution is one character.""" return base + u / WIDTH * (top - base) a, b = VALUE["east"], VALUE["north"] TRUE_RATIO = a / b print(f"east {a:.3f} m3, north {b:.3f} m3, true ratio {TRUE_RATIO:.4f}") for base in (0.0, 18.0): ua, ub = length(a, base, a), length(b, base, a) print(f"axis from {base:4.0f} m3: east |{'#' * ua}") print(f" north |{'#' * ub}") print(f" bar ratio {ua / ub:.4f}; a reader who takes length as proportional " f"to value reads north as {decode_length(ub, 0.0, a):.3f} m3") TRUNCATED = length(a, 18.0, a) / length(b, 18.0, a) print(f"truncation ratio {TRUE_RATIO:.4f} -> {TRUNCATED:.4f}, enlargement factor " f"{TRUNCATED / TRUE_RATIO:.3f}; read-back error at north " f"{decode_length(length(b, 18.0, a), 0.0, a) - b:+.3f} m3") ``` ``` east 24.674 m3, north 20.210 m3, true ratio 1.2208 axis from 0 m3: east |######################################## north |################################# bar ratio 1.2121; a reader who takes length as proportional to value reads north as 20.356 m3 axis from 18 m3: east |######################################## north |############# bar ratio 3.0769; a reader who takes length as proportional to value reads north as 8.019 m3 truncation ratio 1.2208 -> 3.0769, enlargement factor 2.520; read-back error at north -12.191 m3 ``` The two sets of bars carry the same two numbers. On the axis that starts from zero the ratio comes out as 1.2121; the true ratio is 1.2208, and the difference comes from the bars being rounded to whole characters, that is, from the reading resolution. On the axis that starts from 18 m³ the ratio of the same two numbers becomes **3.0769**. The real number is on the last line. A reader who takes length as proportional to value reads north from the truncated display as **8.019 m³**; the true value is 20.210 and the read-back error is **-12.191 m³**. Neither did the data change nor was a calculation done wrongly. Where this decision comes from matters. The means of the two regions are squeezed between 20.210 and 24.674, and on an axis that starts from zero the two bars look almost the same. "Make the difference visible" is a design requirement, and the previous course had treated it as a legitimate decision — since what is read on a line chart is the slope, the axis may be truncated there. When the same decision is applied to the length channel, it falsifies the reader's decoder. The decision is the same, the channel is different, and the result is an error of 12 m³. ## Area: The Square of the Ratio ```python # area.py — the decision to "show it twice as large" enlarges the shape in two # dimensions at once; the reader reads the area and the ratio returns squared. SIDE = 12 def encode_area(ratio): """Encoder: ratio -> area. Both sides are multiplied by the ratio.""" k = SIDE * ratio return k * k def decode_area(ar): """Decoder: area -> value. The reader takes the area as proportional to the value.""" return b * ar / (SIDE * SIDE) area_a, area_b = encode_area(TRUE_RATIO), encode_area(1.0) print(f"north square {SIDE}x{SIDE} = {area_b:.0f} area units") print(f"east square {SIDE * TRUE_RATIO:.2f}x{SIDE * TRUE_RATIO:.2f} = {area_a:.1f} area units") print(f"area ratio {area_a / area_b:.4f} = square of the true ratio {TRUE_RATIO ** 2:.4f}") print(f"east value read back {decode_area(area_a):.3f} m3, true {a:.3f} m3, " f"error {decode_area(area_a) - a:+.3f} m3") ``` ``` north square 12x12 = 144 area units east square 14.65x14.65 = 214.6 area units area ratio 1.4905 = square of the true ratio 1.4905 east value read back 30.123 m3, true 24.674 m3, error +5.449 m3 ``` The second member of the family does not touch the axis at all. To show that east is 1.2208 times larger than north, the square of east is scaled by 1.2208; the scaling is applied to both sides at once because the square must remain a square. The result is that the area ratio is **1.4905**, and this number is the square of the ratio of values. The reader sees the area, not the side. A reader who takes area as proportional to value reads east as **30.123 m³**; the true value is 24.674 and the read-back error is **+5.449 m³**. The enlargement factor is 1.2208, that is, as large as the ratio itself — the larger the ratio, the larger the distortion. This member differs from axis truncation in two respects. First, it cannot be corrected: making the ratio of the sides the square root of the value corrects the area, but then the ratio of the sides no longer belongs to the data, and someone who reads the side is mistaken. Second, the decision itself is invisible; the axis base is a number and can be written down, but "which dimension was scaled" stands nowhere in the display. ## Position: Dual Axis and Uneven Interval ```python # dual.py — two series in two separate units. When each axis is stretched to # its own range, the crossing period is determined by the choice of axis. ROWS = 12 def position(v, low, high): return max(0, min(ROWS - 1, round((v - low) / (high - low) * (ROWS - 1)))) def options(v): lo, hi = min(v), max(v) return {"from zero": (0.0, hi), "data range": (math.floor(lo), math.ceil(hi)), "wide": (math.floor(lo) - (hi - lo) * 0.5, math.ceil(hi))} oc, on = options(CONSUMPTION), options(READ_COUNT) print(f"{'consumption axis':18s}{'count axis':15s}{'crossing period':>15s}") crossing = set() for na, (a_lo, a_hi) in oc.items(): for nb, (b_lo, b_hi) in on.items(): k = next((t for t in range(N) if position(CONSUMPTION[t], a_lo, a_hi) <= position(READ_COUNT[t], b_lo, b_hi)), None) crossing.add(k) print(f"{na:16s} {nb:14s} {str(k):>15s}") print(f"{len(oc) * len(on)} defensible axis pairs give " f"{len(crossing)} different crossing periods") print(f"the units of the two series differ (m3 and count); the crossing period is " f"a number with no counterpart in the data") ``` ``` consumption axis count axis crossing period from zero from zero 5 from zero data range 25 from zero wide 19 data range from zero 0 data range data range 18 data range wide 8 wide from zero 0 wide data range 19 wide wide 17 9 defensible axis pairs give 7 different crossing periods the units of the two series differ (m3 and count); the crossing period is a number with no counterpart in the data ``` The dual axis is the hardest member of the family to defend, because its reason for existing is legitimate: the consumption mean is in m³, the number of readings per period is in count, and putting the two on a single axis flattens one series into a straight line. Stretching each series to its own range is the natural solution. The price is in the table. Nine defensible axis pairs give **seven separate crossing periods**, and among them are completely different answers such as period 0 and period 25. The sentence "in this period one overtook the other" that the reader draws from this display corresponds to nothing in the data; the impression that two different units are comparable is entirely a product of the axis choice. In the other three members the number read back had an error; here the number read back **has no source**. ```python # interval.py — if the periods on hand are laid out at equal intervals, the # slope is spoiled. HELD = [0, 1, 2, 3, 4, 5, 6, 12, 18, 24, 30, 35] def slope(xs, ys): n = len(xs) mx, my = sum(xs) / n, sum(ys) / n return (sum((xs[i] - mx) * (ys[i] - my) for i in range(n)) / sum((x - mx) ** 2 for x in xs)) y = [CONSUMPTION[t] for t in HELD] gaps = [HELD[i + 1] - HELD[i] for i in range(len(HELD) - 1)] correct = slope(HELD, y) drawn = slope(list(range(len(HELD))), y) print(f"{len(HELD)} periods on hand, spread from period {HELD[0]} to period " f"{HELD[-1]}; gaps {min(gaps)}-{max(gaps)} periods") print(f"true slope {correct:+.4f} m3/period ({HELD[-1] - HELD[0]} periods of spread)") print(f"slope read from the display drawn at equal intervals {drawn:+.4f} " f"m3/period ({len(HELD) - 1} steps), enlargement {drawn / correct:.2f} times") print(f"{'step':>4s}{'true gap':>15s}{'true slope':>13s}{'drawn slope':>14s}") for i in (0, 6, 7, len(gaps) - 1): print(f"{i:4d} {gaps[i]:14d} {(y[i + 1] - y[i]) / gaps[i]:+12.4f} " f"{y[i + 1] - y[i]:+13.4f}") TABLE = [("axis truncation", "length", f"{TRUE_RATIO:.4f}", f"{TRUNCATED:.4f}", f"{TRUNCATED / TRUE_RATIO:.2f}x", "axis base"), ("area enlargement", "area", f"{TRUE_RATIO:.4f}", f"{area_a / area_b:.4f}", f"{(area_a / area_b) / TRUE_RATIO:.2f}x", "whether side or area was scaled"), ("dual axis", "position", "none", f"{len(crossing)} separate periods", "-", "range of the two axes"), ("uneven interval", "position", f"{correct:.4f}", f"{drawn:.4f}", f"{drawn / correct:.2f}x", "period gaps")] print(f"{'distortion':18s} {'channel':8s} {'source':>8s} {'read back':>18s} {'growth':>9s} unwritten decision") for name, channel, src, back, grow, row in TABLE: print(f"{name:18s} {channel:8s} {src:>8s} {back:>18s} {grow:>9s} {row}") ``` ``` 12 periods on hand, spread from period 0 to period 35; gaps 1-6 periods true slope -0.2079 m3/period (35 periods of spread) slope read from the display drawn at equal intervals -0.6635 m3/period (11 steps), enlargement 3.19 times step true gap true slope drawn slope 0 1 +1.3370 +1.3370 6 6 -0.1895 -1.1370 7 6 -0.0655 -0.3930 10 5 -0.3822 -1.9110 distortion channel source read back growth unwritten decision axis truncation length 1.2208 3.0769 2.52x axis base area enlargement area 1.2208 1.4905 1.22x whether side or area was scaled dual axis position none 7 separate periods - range of the two axes uneven interval position -0.2079 -0.6635 3.19x period gaps ``` The fourth member is the one that appears most often and most unnoticed. There are 12 periods on hand and they are spread from period 0 to period 35; the gaps vary between 1 and 6 periods. The display lays them out in order and at equal intervals, because laying out what is on hand in order is the default behavior. The slope is read as **-0.6635** instead of -0.2079 m³/period, **3.19 times** larger. The factor is close to the period span divided by the number of steps: 35 divided by 11, that is, 3.18. The step table shows how this comes about step by step — at step 0 the true gap is one period and the two numbers coincide, at step 6 the gap rises to six periods and the drop of -0.1895 per period is read in the display as -1.1370. The last table puts the four members side by side. Its last column is what the four have in common: **in each of them, the information that would correct the read-back is left outside the display**. The axis base is a number and can be written down; so can the period gaps. "Which dimension was scaled" and "what is the range of the two axes" are known only where the display is made and never reach the reader. All four arose from a defensible default: make the difference visible, enlarge the shape by the ratio, stretch the two series to their own ranges, lay out what is on hand in order. Misleading required no intent in any of them. ## Summary - The four distortions are a single family: they arise from the definition of the channel itself, do not spoil the calculation, and spoil the read-back. - When the axis is started from 18 m³, the ratio of 1.2208 is read as 3.0769, and a reader who takes length as proportional to value reads north as 8.019 m³ instead of 20.210, an error of -12.191 m³. - When the same ratio is encoded into a square, both sides are multiplied at once, so the area ratio is 1.4905, that is, the square of the ratio; east is read as 30.123 m³ and the error is +5.449 m³. - In two series with different units, nine defensible axis pairs give seven separate crossing periods; the number read back has no counterpart in the source. - When 12 irregularly spaced periods are laid out equally, a slope of -0.2079 m³/period is read as -0.6635; the factor is close to the period span divided by the number of steps, 35 divided by 11, that is, 3.18. - In all four, the information that would correct the read-back is left outside the display, and all four arise from a defensible default behavior. ## Next Step This lesson showed that distortion does not require intent; in every member the display had been built correctly and not a single number had been calculated wrongly. One assumption still stands untouched: that the reader is able to read the channel. The next lesson measures this. How many series remain indistinguishable from one another when color is the only encoding channel is counted, and how many this number falls to with **redundant encoding**, made up of shape, direct labeling and ordering, is measured. The contrast ratio calculation and the color system are the business of the interface courses and are not repeated; what is counted here is only what is specific to the data.