Errors in Scientific Measurements

In science, every measurement is subject to errors and uncertainties. Understanding these errors is fundamental for evaluating the reliability of experimental results.

Significant Figures

Significant figures are those digits in a measurement that provide real information about the measured quantity.

Rules for identifying significant figures:

- All non-zero digits are significant. - Zeros between significant digits are significant. - Leading zeros (which only indicate decimal position) are not significant. - Trailing zeros after the decimal point are significant. - Trailing zeros in whole numbers are ambiguous unless scientific notation is used.

Examples:

- 0.00234: has 3 significant figures (2, 3, 4). - 1.050: has 4 significant figures (1, 0, 5, 0). - 7.00 × 10²: has 3 significant figures (7, 0, 0). - 3000: ambiguous without additional context.

Rounding

Rounding is necessary to express results with the appropriate number of significant figures.

Rounding rules:

- If the first discarded digit is less than 5, the last retained digit does not change. - If the first discarded digit is greater than 5, or is 5 followed by non-zero digits, the last retained digit is increased by 1. - If the first discarded digit is exactly 5 (with no significant digits after), the last retained digit is rounded to the nearest even value.

Examples:

- 2.443 rounded to 3 figures: 2.44 - 2.456 rounded to 3 figures: 2.46 - 2.450 rounded to 3 figures: 2.45 (5 is kept because it's even) - 2.350 rounded to 3 figures: 2.35 (5 is kept because it's even)

Value of a Measurement

An experimental measurement must be expressed correctly, indicating both its value and its uncertainty.

Correct way to express a measurement:

- Value ± Uncertainty (with units) - Example: (15.3 ± 0.2) cm

The number of decimal places in the uncertainty determines the number of decimals that should be shown in the measurement value.

Absolute Error

The absolute error is the difference between the measured value and the value considered true.

Formula:

$E_{abs} = | \text{Measured value} - \text{Real value}|$

The absolute error has the same units as the measured quantity. For example, if we measure length in meters, the absolute error is also expressed in meters.

Example:

If we measure 5.43 cm when the real value is 5.40 cm, the absolute error is |5.43 - 5.40| = 0.03 cm.

Relative Error

The relative error is the proportion between the absolute error and the real value, usually expressed as a percentage.

Formula:

$E_{rel} = \frac{E_{abs}}{ \text{Real value}} \times 100%$

The relative error is dimensionless (without units) if expressed as a fraction, or in percentage if multiplied by 100.

Example:

Following the previous example, the relative error would be (0.03 cm / 5.40 cm) × 100% = 0.56%.

Error Propagation

When we perform operations with quantities that have errors, these propagate to the result.

Basic propagation rules:

- In addition and subtraction: absolute errors are added. - In multiplication and division: relative errors are added.

Examples:

- If we add (10.2 ± 0.1) cm + (5.3 ± 0.2) cm = (15.5 ± 0.3) cm - If we multiply (2.5 ± 0.1) m × (3.0 ± 0.2) m = (7.5 ± 0.75) m² (relative error is the sum of 4% and 6.7%)

Accuracy vs. Precision

It is important to distinguish between these two concepts:

- Accuracy: Closeness of a measurement to the real value (related to systematic error). - Precision: Reproducibility of measurements (related to random error).

An instrument can be precise (always give the same readings) but inaccurate (far from the real value), or vice versa.

Measurement Variability and Statistical Analysis

In experimental practice, exactly the same measurements are never obtained when repeating an experiment, even under identical conditions. This variability is inherent to the measurement process and is due to multiple factors:

- Random fluctuations in the physical system - Limitations of the measuring instrument - Variations in environmental conditions - Human factors in data collection

When we perform multiple measurements of the same quantity, after setting up the instrument, the most representative value is the arithmetic mean of all obtained results:

Arithmetic mean:

$ar{x} = \frac{x_1 + x_2 + ldots + x_n}{n} = \frac{sum x_i}{n}$

To quantify the dispersion of measurements, the standard deviation is used, which represents the statistical uncertainty of our measurements:

Standard deviation:

$sigma = \sqrt{ \frac{sum(x_i - ar{x})^2}{n-1}}$

The final result should be expressed as: x̄ ± error (with units), where σ represents the uncertainty in the measurement.

Standard deviation example In addition to standard deviation, when working with samples and we want to estimate the precision of the mean as an estimator of the true value, we use the standard error of the mean:

Standard error:

$sigma_{ar{x}} = \frac{sigma}{sqrt{n}}$

The standard error decreases as the number of measurements increases, indicating that the mean is more precise when based on more data. It is especially useful when we want to compare our results with a theoretical value or with results from other experiments.

Solved Examples

Analysis of significant figures in calculations
Problem: Calculate the area of a rectangle whose sides measure 4.52 cm and 3.1 cm. Express the result with the correct number of significant figures.

Step 1: Identify significant figures

- 4.52 cm has 3 significant figures - 3.1 cm has 2 significant figures

Step 2: Perform the calculation

$ \text{Area} = 4.52 \text{ cm} \times 3.1 \text{ cm} = 14.012 \text{ cm}^2$

Step 3: Round to the correct number of significant figures

In multiplications, the result should have the same number of significant figures as the factor with fewer figures (in this case, 2).

Result: Area = 14 cm²

Calculation of absolute and relative error
Problem: The mass of an object is measured with a balance, obtaining 245.3 g. If the real mass of the object is 243.8 g, calculate the absolute and relative error.

Step 1: Calculate the absolute error

$E_{abs} = | \text{Measured value} - \text{Real value}|$ $E_{abs} = |245.3 \text{ g} - 243.8 \text{ g}| = 1.5 \text{ g}$

Step 2: Calculate the relative error

$E_{rel} = \frac{E_{abs}}{ \text{Real value}} \times 100%$ $E_{rel} = \frac{1.5 \text{ g}}{243.8 \text{ g}} \times 100% = 0.615%$

Result:

- Absolute error: 1.5 g - Relative error: 0.62% (rounded to 2 significant figures)

Statistical analysis of repeated measurements
Problem: The oscillation period of a pendulum is measured 8 times, obtaining the following values (in seconds): 2.14; 2.16; 2.13; 2.17; 2.15; 2.12; 2.16; 2.14. Calculate the mean, standard deviation and standard error.

Step 1: Calculate the arithmetic mean

$ar{x} = \frac{2.14 + 2.16 + 2.13 + 2.17 + 2.15 + 2.12 + 2.16 + 2.14}{8} = \frac{17.17}{8} = 2.146 \text{ s}$

Step 2: Calculate deviations of each measurement

xᵢ (s)xᵢ - x̄ (s)(xᵢ - x̄)² (s²)
2.14-0.0060.000036
2.160.0140.000196
2.13-0.0160.000256
2.170.0240.000576
2.150.0040.000016
2.12-0.0260.000676
2.160.0140.000196
2.14-0.0060.000036

Step 3: Calculate the standard deviation

$sigma = \sqrt{ \frac{sum(x_i - ar{x})^2}{n-1}} = \sqrt{ \frac{0.001988}{7}} = \sqrt{0.0002840} = 0.0168 \text{ s}$

Step 4: Calculate the standard error

$sigma_{ar{x}} = \frac{sigma}{sqrt{n}} = \frac{0.0168}{sqrt{8}} = \frac{0.0168}{2.828} = 0.0059 \text{ s}$

Result: T = (2.146 ± 0.017) s

If we want to express the uncertainty of the mean: T = (2.146 ± 0.006) s

Error propagation in addition and multiplication
Problem: A student measures the sides of a rectangle: L₁ = (15.3 ± 0.2) cm and L₂ = (8.7 ± 0.1) cm. Calculate the perimeter and area with their respective uncertainties.

Perimeter calculation (addition):

$P = 2(L_1 + L_2) = 2 \times (15.3 + 8.7) = 2 \times 24.0 = 48.0 \text{ cm}$

In additions, absolute errors are added:

$Delta P = 2 \times (Delta L_1 + Delta L_2) = 2 \times (0.2 + 0.1) = 0.6 \text{ cm}$

Perimeter: P = (48.0 ± 0.6) cm

Area calculation (multiplication):

$A = L_1 \times L_2 = 15.3 \times 8.7 = 133.11 \text{ cm}^2$

In multiplications, relative errors are added:

$ \frac{\Delta A}{A} = \frac{\Delta L_1}{L_1} + \frac{\Delta L_2}{L_2} = \frac{0.2}{15.3} + \frac{0.1}{8.7} = 0.0131 + 0.0115 = 0.0246$ $Delta A = A \times 0.0246 = 133.11 \times 0.0246 = 3.27 \text{ cm}^2$

Area: A = (133 ± 3) cm² (appropriately rounded)

Comparison of accuracy and precision
Problem: Two students measure the length of a table (real value: 120.0 cm) using different instruments. Analyze the accuracy and precision of each method.

Student A (ruler graduated in mm):

Measurements: 119.8; 119.9; 119.7; 120.1; 119.8 cm

Mean: 119.9 cm; Standard deviation: 0.15 cm

Student B (measuring tape):

Measurements: 121.2; 121.1; 121.3; 121.0; 121.2 cm

Mean: 121.2 cm; Standard deviation: 0.11 cm

Accuracy analysis:

- Error of A: |119.9 - 120.0| = 0.1 cm → More accurate - Error of B: |121.2 - 120.0| = 1.2 cm → Less accurate

Precision analysis:

- Deviation of A: 0.15 cm → Less precise - Deviation of B: 0.11 cm → More precise

Conclusion: Student A is more accurate but less precise. Student B is more precise but less accurate (possible systematic error in the instrument).

1. How many significant figures does the number 0.00340 have? (1 points)

2. How many significant figures does the number 2300 have? (1 points)

3. What is the value of 3.1416 rounded to 3 significant figures? (1 points)

4. If a measurement has a value of 15.3 m and the real value is 15.0 m, what is the absolute error? (1 points)

5. If a measurement has a value of 98.5 g and an absolute error of 0.2 g, what is the relative error expressed as a percentage? (1 points)

6. When adding two measurements with errors, how is the absolute error of the result calculated? (1 points)

7. How many significant figures does the number 4.070 × 10³ have? (1 points)

8. Which of the following instruments probably has the highest precision? (1 points)

9. Determine the number of significant figures: (8 points)

a ) 0.00045

b ) 3.1400

c ) 7000

d ) 7.000 × 10³

e ) 0.0703

f ) 230.00

g ) 1.20 × 10⁻⁴

h ) 800

10. Calculate the absolute and relative error (%): (10 points)

a ) Measured value: 24.3 cm, Real value: 24.0 cm. Absolute error = ? cm

b ) Measured value: 24.3 cm, Real value: 24.0 cm. Relative error = ? %

c ) Measured value: 0.578 g, Real value: 0.600 g. Absolute error = ? g

d ) Measured value: 0.578 g, Real value: 0.600 g. Relative error = ? %

e ) Measured value: 1.05 m, Real value: 1.00 m. Absolute error = ? m

f ) Measured value: 1.05 m, Real value: 1.00 m. Relative error = ? %

g ) Measured value: 150 kg, Real value: 152 kg. Absolute error = ? kg

h ) Measured value: 150 kg, Real value: 152 kg. Relative error = ? %

i ) Measured value: 2.34 × 10³ N, Real value: 2.30 × 10³ N. Absolute error = ? N

j ) Measured value: 2.34 × 10³ N, Real value: 2.30 × 10³ N. Relative error = ? %

11. A student measures the mass of an object five times, obtaining the following values (in grams): 24.3; 24.5; 24.2; 24.4; 24.1. What is the best estimate of the object's mass? (1 points)

12. A series of voltage measurements is performed, obtaining (in volts): 12.1; 12.0; 12.2; 12.0; 11.9; 12.1. What is the approximate standard deviation of these measurements? (1 points)

13. Calculate the mean and standard deviation: (6 points)

a ) Length measurements (cm): 5.2; 5.1; 5.3; 5.2; 5.0. Mean = ? cm

b ) Length measurements (cm): 5.2; 5.1; 5.3; 5.2; 5.0. Standard deviation = ? cm

c ) Temperature measurements (°C): 25.3; 25.5; 25.4; 25.2; 25.3; 25.4. Mean = ? °C

d ) Temperature measurements (°C): 25.3; 25.5; 25.4; 25.2; 25.3; 25.4. Standard deviation = ? °C

e ) Time measurements (s): 3.45; 3.51; 3.49; 3.47; 3.52. Mean = ? s

f ) Time measurements (s): 3.45; 3.51; 3.49; 3.47; 3.52. Standard deviation = ? s