Put the independent-variable values in one column and give each repeated response its own column. State quantities and units in headings. Record the actual precision of the instrument and add a notes column for relevant qualitative observations, such as a collision or an unusual path. Distinguish a missing reading from zero.
A zero-second descent would mean a measured zero, not “we forgot to time it”. Use a clear missing-data code with a reason. Keep the original order or trial IDs so you can check whether equipment drift or practice affected results. A processed summary should be traceable to these raw records.
Book notes: Design a table before starting
Put the independent-variable values in one column and give each repeated response its own column. State quantities and units in headings. Record the actual precision of the instrument and add a notes column for relevant qualitative observations, such as a collision or an unusual path. Distinguish a missing reading from zero.
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Times 1.2, 1.4 and 1.3 s have a mean of:
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Calculate a centre and a spread
Calculate a centre and a spread
For descent times of 1.2, 1.4 and 1.3 s, the mean is (1.2 + 1.4 + 1.3)/3 = 1.3 s. The range is 1.4 − 1.2 = 0.2 s. The mean summarises centre; the range gives a simple description of variability in these trials. A mean alone conceals the difference between tightly grouped and widely scattered results.
Use the number of actual valid readings in the denominator. A missing reading must not silently become zero. Do not automatically discard an unusual reading. First inspect the raw notes and method; an exclusion needs an identified reason and a transparent record.
Book notes: Calculate a centre and a spread
For descent times of 1.2, 1.4 and 1.3 s, the mean is (1.2 + 1.4 + 1.3)/3 = 1.3 s. The range is 1.4 − 1.2 = 0.2 s. The mean summarises centre; the range gives a simple description of variability in these trials. A mean alone conceals the difference between tightly grouped and widely scattered results.
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The range of 1.6, 1.7 and 1.8 s is:
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Keep representations linked to evidence
Keep representations linked to evidence
A photograph, sketch or video can preserve spatial or behavioural features not captured by a number. Include a scale, date/time and clear label where relevant. A hand-drawn diagram can simplify a view, but the simplification should be acknowledged; a digital image also depends on exposure, framing and processing.
Separate observation notes from inferred explanations. “Spinner contacted desk” records an event; “wing length caused instability” proposes a mechanism. Store processed values alongside, not instead of, the original records, and explain transformations so another person can check them.
Book notes: Keep representations linked to evidence
A photograph, sketch or video can preserve spatial or behavioural features not captured by a number. Include a scale, date/time and clear label where relevant. A hand-drawn diagram can simplify a view, but the simplification should be acknowledged; a digital image also depends on exposure, framing and processing.
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How should a missing timing reading be recorded?
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Worked example
Worked example: Summarise the 5 cm wing-length row.
Add the three readings: 1.6 + 1.7 + 1.8 = 5.1 s.
Mean = 5.1/3 = 1.7 s.
Range = 1.8 − 1.6 = 0.2 s.
Keep all three raw readings and the stated fixed-height conditions.
Mean descent time 1.7 s, range 0.2 s across three synthetic trials; this is teaching data, not a class measurement.
Synthetic teaching dataset: paper-spinner descent times over the same fixed height.
Wing length (cm)
Trial 1 (s)
Trial 2 (s)
Trial 3 (s)
3
1.2
1.4
1.3
5
1.6
1.7
1.8
7
2
2.2
2.1
Compare pH readings without confusing the scales
Synthetic pH data for three anonymous supplied aqueous samples at 25 °C. These are calculation records only: do not taste, mix or handle unknown liquids.Text description
Sample A has repeated pH readings 4.1, 4.0 and 3.9. Sample B has readings 6.1, 6.0 and 5.9. Sample C has readings 8.1, 8.0 and 7.9. A and B are acidic; C is alkaline under these conditions. The unitless pH scale is logarithmic.
Sample A has repeated pH readings 4.1, 4.0 and 3.9. Sample B has readings 6.1, 6.0 and 5.9. Sample C has readings 8.1, 8.0 and 7.9. A and B are acidic; C is alkaline under these conditions. The unitless pH scale is logarithmic.
Synthetic repeated readings from a pH meter
Sample
Reading 1 (pH)
Reading 2 (pH)
Reading 3 (pH)
A
4.1
4
3.9
B
6.1
6
5.9
C
8.1
8
7.9
Calculate the mean recorded pH and range for each sample. Compare acidity and identify what the repetitions cannot establish.
Mean of recorded pH values for A = (4.1 + 4.0 + 3.9) ÷ 3 = 4.0; range = 4.1 − 3.9 = 0.2.
For B, mean recorded pH = 6.0 and range = 0.2; for C, mean = 8.0 and range = 0.2. pH has no unit; these are ranges of readings, not uncertainty bounds for the true value.
At 25 °C, A and B are below neutral pH 7 (acidic), while C is above it (alkaline). A has lower pH than B. A two-pH difference represents a 100-fold hydrogen-ion activity ratio, not “two times as acidic”. For dilute solutions, concentration is often used as an approximation to activity.
A mean of logged pH readings is a descriptive summary; it is not the pH of a mixture or a mean hydrogen-ion concentration. Similar repeated readings do not establish calibration accuracy.
A: mean recorded pH 4.0, range 0.2; B: mean recorded pH 6.0, range 0.2; C: mean recorded pH 8.0, range 0.2. A is more acidic than B; C is alkaline. Check calibration with suitable reference buffers; these data do not establish sample identity, acid strength, safety or a universal property.
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Activities
Activities
Activity 1
Calculate means and ranges for the 3 cm and 7 cm rows.
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Activity 2
Create a table heading and missing-data rule for a new spinner measurement.
Five questions from this lesson’s eight-question bank
Answer the five-question Quick Check. Use the feedback to repair an idea before retrying.
Short answer
Explain and evaluate
Q1. Calculate mean and range for 2.4, 2.6 and 2.5 s.
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Explain and evaluate
Q2. Explain why “mean = 1.7” is an incomplete report.
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Explain and evaluate
Q3. A group replaces a missing value with zero. Explain the consequence.
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Answers and model responses
Activities
Activity 1
3 cm: mean 1.3 s, range 0.2 s. 7 cm: mean 2.1 s, range 0.2 s.
Activity 2
Use wing length (cm), trial time (s) with repeat columns and notes. Use a stated missing-data code, preserve its reason and omit it from the mean denominator; never treat it as zero.
Short Answer Model Answers
Q1: Mean = 7.5/3 = 2.5 s; range = 2.6 − 2.4 = 0.2 s. Include units and retain the raw values.
Q2: It lacks a stated quantity/unit and context. It also omits repeats/spread and conditions, so readers cannot assess what was measured or how variable it was.
Q3: Zero is a measured response, not missing data. Including it depresses the mean and misrepresents the observations. Mark missing data and use actual valid readings with a documented denominator.
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Revisit and reflect
Revisit without looking
Build tables with quantities, units and raw repeats.
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Check the key idea
Mean descent time 1.7 s, range 0.2 s across three synthetic trials; this is teaching data, not a class measurement.