Lesson 2.1 – Accuracy and Precision

Accuracy

• "Accuracy" answers “How close is your result to the actual or true value?”

  • Indicates closeness of a single measured value (or an average) to the true / accepted value.

  • Primarily influenced by systematic errors (e.g.

  • Faulty or poorly-calibrated instruments,

  • Incorrect measurement technique,

  • Consistent procedural mistakes).

  • High accuracy ⇒ low (or corrected) systematic error.

• Mathematical reminder

  • For a measured value $x<em>{\text{meas}}$ and true value $x</em>{\text{true}}$, the absolute error is $|x<em>{\text{meas}}-x</em>{\text{true}}|$.

  • Percent error formula: $\%\,\text{error}=\left(\frac{|x<em>{\text{meas}}-x</em>{\text{true}}|}{x_{\text{true}}}\right)\times100\%.$

Accuracy — Real-world examples

• Target shooting: darts/ bullets repeatedly hit the bullseye.

• Math calculations: producing the correct numerical answer on paper or calculator.

• Medical diagnostics: a test that correctly identifies a disease in 95 % of cases (high true-positive rate).

• GPS navigation: device guides you precisely to the required address; coordinates are within a few metres of reality.

• Weather forecasting: app predicts next-day temperature & precipitation that closely match what is actually observed.

Precision

• "Precision" answers “How consistent are your repeated measurements?”

  • Describes the spread (or lack of spread) in a series of readings.

  • Primarily limited by random errors (e.g.

  • Hand-eye reading variation,

  • Fluctuating environment: temperature, vibration, lighting,

  • Electronic noise in a sensor).

  • Tight clustering ⇒ small standard deviation ⇒ high precision.

• Statistical reminder

  • For $n$ repeated readings $x<em>1,\,x</em>2,\ldots,x_n$:

  • Mean: $\bar{x}=\frac{1}{n}\sum<em>{i=1}^{n}x</em>i.$

  • Precision often quantified with standard deviation $s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}.$

Precision — Real-world examples

• 100 m sprint: three times of $10.01\,\text{s},\,10.00\,\text{s},\,10.02\,\text{s}$ ⇒ tightly grouped.

• Digital balance: $5.01\,\text{g},\,5.02\,\text{g},\,5.03\,\text{g}.$

• Bathroom scale for same person: $49.5,\,49.6,\,49.5\,\text{kg}.$

• Micrometer readings: $3.450,\,3.451,\,3.449\,\text{mm}.$

• Blood-pressure monitor: $120/80,\,120/81,\,121/82.$

Additional classroom examples (Page 7):

• Automatic pipette repeatedly delivers $10.00\,\text{mL}.$

• Height recorded as $170.0\,\text{cm}$ four times.

• Archery arrow hits same mark each shot.

• Cholesterol test: $199.3,\,199.4,\,199.5\,\text{mg·dL}^{-1}.$

• Student repeatedly scoring $85\%$ on three exams.

Accuracy vs. Precision — Four Situations

• High accuracy, low precision: data scattered around the true value, but not close to one another (bullseye ➔ points near centre yet far apart).

• Low accuracy, high precision: clustered points far from centre (consistent but wrong).

• High accuracy & high precision: tight cluster centred on the true value (ideal laboratory goal).

• Low accuracy & low precision: scattered cluster far from true value (worst case).

Classroom Activity — “Guess My Age” Tables

The instructor’s actual age = 30 years. Students gave three different data groups:

Group 1 (Page 8)

• Kenneth 30, Axel 30, Gary 29, Christian 31 ⇒ Average ≈$30.0$.

  • Precise? Yes (values within $\pm1$ year).

  • Accurate? Yes (mean & most readings ≈ true age).

Group 2 (Page 9)

• Rea 55, Cherry 55, Angeline 55, Sarah 54 ⇒ Very tight group ≈$54.8$.

  • Precise? High (all within $\pm0.5$).

  • Accurate? No (≈$+25$ years error).

Group 3 (Page 10)

• Ricci 40, CJ 45, Marco 30, Jeron 35 ⇒ Wide spread 30–45.

  • Precise? Low.

  • Accurate? Only Marco hit 30; overall mean ≈$37.5$ so poor.

Pages 11–13 re-list subsets & symbols (✓, ✗) to help students classify each scenario.

Bullseye Illustration (Page 14)

• Graphic reinforces textual description above; see Four Situations list.

Seatwork — Density of Aluminium (Page 15)

True density: $2.70\,\text{g·mL}^{-1}$.

Student data (four trials each):

• Kim: $2.924,\,2.923,\,2.925,\,2.926$ (high precision, moderate positive error ⇒ low accuracy).

• Glenn: $2.316,\,2.527,\,2.941,\,2.136$ (low precision, low accuracy; large range).

• Yuan: $2.652,\,2.731,\,2.692,\,2.742$ (good precision, small negative bias ⇒ good accuracy).

• Christian: $2.254,\,2.343,\,2.763,\,2.478$ (low precision, low accuracy).

Interpretation prompt: Yuan is both most accurate (
|error| ≈ $0.05$) and fairly precise; Kim is most precise but not accurate.

Seatwork — Density of Zinc (Page 16)

Accepted value: $7.31\,\text{g·mL}^{-1}$.

Eight readings per student:

• Magda: $7.31,7.30,7.29,7.26$ (clustered, avg≈$7.29$) ⇒ High precision & high accuracy.

• Lilian: $7.65,7.65,7.32,7.37$ (fair precision around ≈$7.50$, low accuracy – high positive bias).

• Jessy: $7.04,7.55,7.64,9.32$ (poor precision and accuracy; one extreme outlier 9.32).

• Luis: $5.34,2.23,7.66,8.65$ (very poor precision and accuracy).

Answer cue: Magda is both accurate & precise.

Seatwork — Mass of NaCl (Page 17)

True mass: $35.9\,\text{g}$. Students’ measurements table (values not quoted verbatim in transcript but task asks):

• Identify set with high precision + low accuracy, and set with both high accuracy & precision.

• Approach: compute each student’s mean & spread just as previous examples.

Seatwork — 5.00 g Object (Page 18)

True mass: $5.00\,\text{g}$.

Students & four trials:

• Ara: $3.32,3.01,8.55,4.02$ (neither precise nor accurate).

• Alicia: $4.32,6.35,7.29,3.20$ (neither precise nor accurate).

• Angela: $5.76,5.77,5.78,5.75$ (very high precision, but positive bias ≈$+0.76$ ⇒ low accuracy).

• Lea: $5.01,4.99,5.00,5.00$ (high accuracy and high precision).

Answer: Angela → “high precision, low accuracy”; Lea → “accurate & precise.”

Case Study — Measuring Coin Diameter (Page 19)

Accepted diameter: $28.054\,\text{mm}$.

Data sets (four coins each):

• Student A (plastic ruler): $26.5,24.0,28.725,28.050\,\text{mm}$.

• Student B (micrometer): $28.6,25.345,26.278,23.844\,\text{mm}$ .

• Student C (vernier scale): $20.7,23.844,28.051,28.052,28.053\,\text{mm}$ (five values; three clustered near accepted, two outliers very low).

Tasks:

1. Compute each average (eliminate obvious outliers if instructed) then compare to $28.054\,\text{mm}$.

2. Accuracy ⇒ mean closeness; Precision ⇒ standard deviation.

3. Expected discussion:

   • Student C’s three-value cluster around $28.05$ is most accurate & most precise if outliers removed; else precision worsens.

   • Plastic ruler & micrometer data show wide spread (poor precision); some values closer than others but inconsistent.

Case Study — Aluminium Cylinder Density (Pages 20–21)

Accepted density: $2.702\,\text{g·cm}^{-3}$.

Student A (plastic ruler): $2.2,2.3,2.7,2.4\,\text{g·cm}^{-3}$.

• One value (2.7) matches accepted; others low (systematic under-reading of volume); spread $0.5$ ⇒ modest precision.

Student B (micrometer): $2.703,2.701,2.705,5.811\,\text{g·cm}^{-3}$.

• First three values tightly clustered around true value (high precision & accuracy).

• Fourth value 5.811 is extreme outlier (probably reading error or unit mix-up).

Instructions: compute mean excluding outlier, then discuss who is more accurate (Student B) & who is more precise (Student B if outlier omitted; otherwise precision degrades).

Fundamental Formulae & Key Reminders

• Density: $\rho = \frac{m}{V}$ (mass divided by volume).

• Cluster width quantified by standard deviation $s$; accuracy quantified by percent error.

Practical / Ethical / Philosophical Implications

• In medicine, high accuracy is vital to avoid misdiagnosis; high precision ensures repeatability of results and patient trust.

• In engineering & manufacturing, precision instruments (micrometers, vernier callipers) reduce material waste and improve safety.

• Scientific publication requires reported uncertainty; claiming accuracy without precision (or vice-versa) is misleading.

• Ethical practice: always calibrate instruments to minimise systematic errors.

Assignment (Page 22)

Produce three original images on bond paper:

• (a) Depict accuracy without precision (e.g., darts scattered around centre).

• (b) Depict precision without accuracy (e.g., darts tightly clustered away from centre).

• (c) Depict accuracy with precision (tight cluster centred on target).

Design tips:

• Clearly label true/accepted value.

• Use colour coding to highlight clusters.

• Provide short captions explaining which property is illustrated.