Mr. Ward's Teaching Materials

112i - Data Displays with Software

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Lesson 112 — Using Software to Create and Compare Data Displays

Strand: Statistics | Descriptor: AC9M7ST02 | Duration: 45 minutes

Room/equipment: devices with spreadsheet software (as Lessons 80 and 92).

Learning Intentions

  • To create data displays using spreadsheet software.
  • To compare displays of the same data and judge which communicates best.

Success Criteria

I can:

  1. Enter data and produce a column graph in a spreadsheet.
  2. Use built-in functions to calculate summary statistics.
  3. Compare two displays of the same data critically.
  4. Recognise when software has produced a misleading display.

Warmup

(6 minutes — the default is not the answer, projected)

Show the same class data as two software-generated charts, side by side:

  • Chart A: software default — -axis auto-scaled, starting at , no axis labels, a legend saying “Series 1”.
  • Chart B: the same data, -axis from , both axes labelled with units, a title.
  1. Which looks more dramatic? Which is more honest?
  2. List three things Chart A is missing.
  3. Where does the misleading impression come from?

Answers: 1. A looks dramatic; B is honest; 2. A title, axis labels with units, a zero-based axis (and “Series 1” tells the reader nothing); 3. The auto-scaled axis — exactly the truncation trap from Lesson 83.

The principle: software chooses defaults for its convenience, not for your reader’s understanding. Every default is a decision you must review.

Activities

Activity 1 — Guided Build: from Data to Display (16 min)

Teacher-led; students follow.

Stage 1 — enter the data. Reaction times (ms) from Lesson 110, in column A with a heading in A1.

Stage 2 — summary statistics. In adjacent cells, build:

AB
1Time (ms)
2210Mean =AVERAGE(A2:A21)
3245Median =MEDIAN(A2:A21)
4198Mode =MODE(A2:A21)
5…Min =MIN(A2:A21)
6Max =MAX(A2:A21)
7Range =MAX(A2:A21)-MIN(A2:A21)

Check against the by-hand answers from Lesson 110: mean , median , range . If the sheet disagrees, the sheet is wrong — a mistyped range is the usual culprit (Lesson 80’s rule).

The MODE surprise. =MODE(…) returns an error (#N/A) because no value repeats. This is correct behaviour — there is no mode. Software failing to return a number is sometimes the right answer, and students must read the error rather than assume they broke it.

Stage 3 — the frequency table. Group the data into ms intervals using =COUNTIF or =COUNTIFS:

IntervalFormula
–=COUNTIFS(A2:A21,">=190",A2:A21,"<240")
–=COUNTIFS(A2:A21,">=240",A2:A21,"<290")
–=COUNTIFS(A2:A21,">=290",A2:A21,"<340")
……
–=COUNTIFS(A2:A21,">=850",A2:A21,"<900")

Check: the counts must total . (Expect , , then zeros, then — the outlier sits alone in a distant interval, and all the empty intervals between are information, exactly as with empty stems in Lesson 111.)

Stage 4 — the chart. Select the frequency table, insert a column chart, then fix the defaults:

  1. Add a title that states what is shown.
  2. Label both axes, with units.
  3. Set the vertical axis to start at zero.
  4. Delete the meaningless legend.

Activity 2 — Compare and Critique (14 min)

Pairs. The same data, displayed several ways.

Using the reaction-time data, produce three displays:

  1. A column graph of the grouped frequencies.
  2. A column graph with the outlier removed.
  3. A column graph with ms intervals instead of ms.

Then answer:

  1. Which display best shows a typical reaction time? Justify.
  2. What does display 1 show that display 2 hides?
  3. What did narrowing the intervals reveal or obscure?
  4. If you could publish only one, which — and what would you note alongside it?

Socratic scaffolding:

PromptPurpose
In display 1, how much of the chart is empty?Most of it — one outlier stretches the axis across a vast empty range.
So what does display 2 gain?Detail in the region where the data actually lies.
And what does it lose?The fact that an unusual reading occurred.
Q6: narrower intervals?More detail on shape, but a bumpier picture — random variation starts to look like structure.
Q7: the honest publication?Display 2 with a stated note that one reading of ms was excluded — never a silent deletion.
Looking backChoosing intervals and ranges is a judgement. Naming your choices is what keeps it honest.

Activity 3 — Inquiry: Make it Mislead, then Fix it (7 min)

Pairs — an exercise in recognising manipulation by committing it.

Take any class data set. Produce two charts of the same data:

  1. One designed to make a small difference look enormous.
  2. One that presents the data fairly.

Then write two sentences naming exactly what you did in chart 1 to create the impression.

The techniques students will discover: truncating the vertical axis; stretching the chart’s height; using unequal interval widths; omitting the zero; choosing a narrow data window.

Class debrief question: every one of these charts is accurate. Where does the dishonesty live? (In the framing, not the numbers — a reader who checks the axis is protected, which is why checking axes is the first move on any unfamiliar chart. Third visit to this idea: Lessons 83, 86, and now 112.)

Checks for Understanding

(5 minutes — exit ticket, on paper)

  1. Write the spreadsheet formula for the mean of values in B2 to B30.
  2. Write the formula for the range of the same values.
  3. =MODE(…) returns an error. What does that tell you about the data?
  4. Name three things you must check or fix on a software-generated chart.
  5. Reasoning. Two charts of identical data give opposite impressions. What should a reader check first?

Answers: 1. =AVERAGE(B2:B30); 2. =MAX(B2:B30)-MIN(B2:B30); 3. No value repeats — the data has no mode; 4. Any three of: title, axis labels with units, axis starting at zero, meaningless legend removed, sensible intervals; 5. Where each vertical axis starts and what one step represents — truncation is the commonest distortion.

Common Misconceptions

MisconceptionHow to pre-empt it
Accepting software defaults as correct.The warmup’s two charts; the four-point fix list.
Treating an error message as a broken spreadsheet.The MODE error is the right answer for this data.
Deleting an outlier silently.Activity 2 Q7 requires the exclusion to be stated.
Believing more intervals is always better.Q6 — narrow intervals can turn noise into apparent structure.
Trusting a chart because a computer made it.The make-it-mislead inquiry.
Skipping the by-hand check.Stage 2 compares against Lesson 110’s verified figures.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). =COUNTIF(A2:A21,">200") returns on a set of values. How many values are or less?

Answer

Three.

E2 (AMC Junior style). A chart’s vertical axis runs from to and the bars rise from to . A headline claims a ” surge”. Assess it.

Answer

The rise is on a base of — about . The bars look three times taller only because the axis hides the first units. The claim confuses bar height with value.

E3 (Challenge). A spreadsheet’s =AVERAGE gives and =MEDIAN gives for the same values. What does the difference suggest, and which display would show it?

Answer

The mean exceeds the median, suggesting a right-skew with high outliers. A column graph of grouped frequencies (or a stem-and-leaf plot) would show a long tail to the right.

E4 (Challenge). Why might =MODE on continuous measurement data return a value that is statistically meaningless?

Answer

Any repeat is usually an artefact of rounding, not a genuine peak in the data. With enough decimal places, no value repeats at all — which is why the mode is rarely used for continuous variables (Lesson 110’s closing point).

Homework

(Paper-based; no devices assumed.)

  1. Write the formula for: (a) the mean of C2:C25 (b) the largest value in C2:C25 (c) the range of C2:C25 (d) the median of C2:C25.
  2. =COUNTIFS(A2:A41,">=60",A2:A41,"<70") returns . Explain in words what has been counted.
  3. A software chart is produced with: no title, a legend reading “Series 1”, a vertical axis from to , and no axis labels. List every fix needed and say why each matters.
  4. A data set’s mean is and its median is . What does this suggest about its shape?
  5. A data set’s mean is and its median is . What does this suggest, and what should be investigated?
  6. =MODE(B2:B50) returns an error. Give two possible reasons and say which is more likely for height data measured to the nearest millimetre.
  7. Sketch by hand two column graphs of the same five values (): one with the axis from , one from . Describe the different impressions.
  8. Reasoning. Explain why a by-hand check of a spreadsheet’s output is still worth doing.
  9. Reasoning. Explain why removing an outlier from a chart can be legitimate, and what must accompany it.
  10. Challenge. A class of has test marks with mean , median , minimum and maximum . Sketch the shape you would expect a grouped column graph to show, and explain your reasoning.

Answers: Q1 — (a) =AVERAGE(C2:C25) (b) =MAX(C2:C25) (c) =MAX(C2:C25)-MIN(C2:C25) (d) =MEDIAN(C2:C25). Q2 — the number of values from up to but not including . Q3 — add a title (says what is shown), label both axes with units (says what is measured), start the axis at zero (prevents exaggeration), remove the legend (it carries no information). Q4 — roughly symmetric, no strong outliers. Q5 — a strong right-skew or high outliers; investigate the largest values and whether they are genuine. Q6 — no value repeats, or the data is text; for millimetre-precision heights, no repeats is far more likely. Q8 — the sheet computes whatever formula was typed, including a wrong range; only an independent check catches a mis-specified model. Q9 — legitimate when the outlier obscures the pattern and the exclusion is stated openly with the value given; silent deletion is not. Q10 — the mean below the median suggests a left-skew: a bulk of marks in the s–s with a tail of low marks pulling the mean down, consistent with the minimum of .

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