**measurement**: comparison of an unknown quantity to a known quantity

**limit of calibration**: the smallest division on a measuring scale

**Some metric prefixes**:

10^{12} tera T

10^{9} giga G

10^{6} mega M

10^{3} kilo k

10^{2} hecto h

10^{1} deka da

10^{-1} deci d

10^{-2} centi c

10^{-3} milli m

10^{-6} micro m

10^{-9} nano n

10^{-12} pico p

*Include units when writing numbers. You may write the units in the column heading of a data table to avoid writing the same unit over and over.*

**doubtful digit**: an estimate between the divisions on a scale

The voltmeter pictured above has two scales, one with a limit of calibration of 0.1 V and one with a limit of calibration of 0.5 V. Estimating a doubtful digit would result in measurements to the nearest 0.01 V and 0.05 V for each scale respectively.

*Your answers might differ from those below by one or two hundredths of a cm. That's why it's a "doubtful digit."*

a: 0.88 cm

b: 2.40 cm*

c: 4.75 cm

d: 6.03 cm

e: 10.00 cm*

f: 14.56 cm

*Write the hundredths place digit, even if it is zero, to indicate that you are measuring to the nearest hundredth of a cm.

**significant digits / figures**: digits that result from reading a measuring instrument, including the doubtful digit

*Determining whether a digit is significant in a written number:*

- For decimal numbers, start from the right and count to the left until you get to the last non-zero digit.
- 0.3450 has 4 significant digits
- 0.003400602 has 7 significant digits
- 21.0508 has 6 significant digits

- For whole numbers, start from the left and count until only zeros remain.
- 3409 has 4 significant digits
- 340900 has 4 significant digits

**uncertainty**: the amount by which a measured value is likely to differ from the actual value being measured.

*Example: The measurement on a certain voltmeter has an uncertainty of 0.2 V. If our measured value is 15.4 V, we would write 15.4±0.2 V.*

*No measurement is perfect.*

*Uncertainty comes from estimating the doubtful digit and other sources. Using a ruler as an example, the calibration marks on the ruler are not perfect, and the person performing the measurement will not align the ruler perfectly, especially if the corners on the item being measured are rounded.*

*Measurements taken with digital meters have uncertainty.*

**Use of the doubtful digit for common measuring instruments**:

**Instrument** | **Limit of Calibration** | **Measure to nearest** | **Approximate Uncertainty** |

spring scale (5 N) | 0.20 N | 0.05 N | 0.1 N |

meter stick | 0.001 m or 0.1 cm | 0.0005 m or 0.05 cm | 0.003 m |

ruler | 0.1 cm | 0.01 cm | 0.002 m |

**Significant digits in calculations**:

- For multiplication and division, state the answer to the fewest number of significant digits in the numbers being used.
- 0.049 m x 26.1 m = 1.3 m
^{2} - 109.7 m / 5 s = 20 m/s

- For addition and subtraction, state the answer to the same accuracy as the least accurate number, by place value.
- 17.11 kg + 92.04 kg = 109.15 kg
- 29.42 s - 29.3 s = 0.1 s
- Note that addition can increase the number of significant digits, while subtraction can greatly reduce the number of significant digits.

**accuracy**: the degree of agreement between an experimental result and the accepted value

**precision**: the degree to which an experimental result can be reproduced

**error**: variations that occur in experimental results when the measuring instruments are properly used.

*Uncertainty from estimating the doubtful digit and other sources is the most common source of error. Other factors unique to an experimental procedure can also contribute to error. Discrepancies between measured values and theoretical predictions may be due not to error, but rather to factors not considered in the model upon which the prediction is based. Friction is a common cause of such discrepancies in physics experiments involving pulleys and lab carts.*

*Do not use the phrase "human error." All error is human error, because humans designed the lab measuring tool and took the measurement. The phrase is meaningless.*

**systematic error**: error in which measured values differ from actual values by a consistent, quantifiable amount or percentage

*Systematic error is correctable using math once the source of error is identified and quantified.*

**random error**: error in which measured values differ from actual values with no discernible pattern

*Random error can be characterized statistically, but individual data points cannot be made more accurate using math.*

**mistake**: an action or oversight that unnecessarily decreases the accuracy and/or precision of measured values

*Mistakes are "goofs," not sources of error.*