Tolerance, Accuracy and Error

No component is perfect. A resistor marked 100 Ω may actually be 102 Ω. A capacitor rated at 100 nF might measure anywhere from 80 nF to 120 nF. Your multimeter shows 5.00 V, but the real voltage might be 4.97 V. Understanding tolerance, accuracy and error is what separates a circuit that always works from one that works on your bench but fails for a customer, in the cold, or after six months. This lesson explains the concepts, the numbers, and how to work with them.

What is tolerance?

Tolerance is the allowed deviation of a component's actual value from its stated (nominal) value. A 100 Ω resistor with ±5% tolerance may have any actual resistance between 95 Ω and 105 Ω and still be within specification. The manufacturer guarantees it will be in that range; they make no promise beyond that.

Tolerance is not a defect or a sign of poor quality — it is an engineering choice that balances cost against precision. Tighter tolerances require tighter manufacturing processes and cost more. For many applications, 5% or 10% tolerance is perfectly adequate. For filter networks, oscillator tuning circuits, and measurement equipment, 1% or better may be essential.

Resistor tolerance grades

Through-hole resistors indicate their tolerance with a colour band. In a four-band resistor, the fourth band is the tolerance band:

Five-band and six-band resistors are used for 1% and tighter tolerances. They use an extra digit band for the resistance value. Six-band resistors add a sixth band for temperature coefficient.

Precision 1% resistors are available in the E96 series, which has 96 values per decade (compared to 12 for the E12 series used for 10% parts). The finer spacing of E96 values means you can hit almost any target resistance using a single component.

The tolerance band is always the last band on a four-band resistor — separated from the value bands by a slightly larger gap. Gold (±5%) is the most common; brown (±1%) identifies a precision part.

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Capacitor tolerance codes

Capacitors use letter codes for tolerance, particularly on small ceramic and film capacitors where there is no room for a colour band:

Note that electrolytic capacitors often have asymmetric tolerance (e.g. −20% / +80% or −10% / +50%). The nominal capacitance can vary significantly; for decoupling and filtering applications this is usually acceptable, but for tuned circuits you need tighter tolerance types such as ceramic NP0/C0G or film capacitors.

Accuracy vs precision

These two words are often used interchangeably in everyday language, but in metrology (the science of measurement) they have distinct meanings:

• Accuracy is how close a measured or stated value is to the true value. A highly accurate component has a value very close to its marked value.
• Precision is how consistent or repeatable a value is. A precise instrument gives the same reading every time, even if that reading is not the true value.

The classic analogy uses a target:

• Accurate and precise: all shots clustered tightly near the bullseye
• Precise but not accurate: all shots clustered tightly, but far from the bullseye (systematic error)
• Accurate but not precise: shots scattered randomly, but centred around the bullseye on average (random error)
• Neither accurate nor precise: shots scattered randomly and off-centre

In electronics, a resistor with excellent precision but poor accuracy would produce the same value in every unit, but that value might be well off the marked nominal. A high-quality 1% resistor is both accurate (close to nominal) and precise (consistent unit to unit).

The four combinations of accuracy and precision. A well-calibrated, tight-tolerance component is both — it hits near the true value consistently. Systematic error shifts the cluster off-centre; random error spreads it out.

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Resolution

Resolution is the smallest change an instrument can detect or display. A digital multimeter that reads to 0.001 V has a resolution of 1 mV. It cannot tell the difference between 4.9991 V and 4.9994 V — both will display as 4.999 V.

Resolution is not the same as accuracy. A meter might have 0.001 V resolution but only ±0.5% accuracy. Its displayed reading of 5.000 V could represent a true voltage anywhere from 4.975 V to 5.025 V. The fine resolution gives you confidence about changes, not about the absolute value.

Types of error

Measurement errors fall into two broad categories:

Systematic error (bias)

An error that consistently pushes the measurement in one direction. Examples: a miscalibrated meter that always reads 2% high; a thermometer that always reads 1°C above true temperature; a component that was stored incorrectly and aged to a shifted value. Systematic errors cannot be reduced by averaging multiple measurements — they affect every reading equally.

Random error

An error that varies unpredictably from measurement to measurement. Examples: electrical noise on a sensitive measurement; thermal fluctuations in a component's value; quantisation noise in a digital instrument. Random errors can be reduced by averaging multiple measurements — the errors partly cancel.

Calculating percentage error

Percentage error tells you how far a measured or actual value is from the expected value, expressed as a proportion of the expected value:

Temperature coefficient

Most component values change with temperature. The temperature coefficient describes how much the value changes per degree Celsius of temperature change. It is typically abbreviated TC or TCR (Temperature Coefficient of Resistance) for resistors.

A positive temperature coefficient (PTC) means the value increases with temperature. A negative temperature coefficient (NTC) means it decreases.

Parts per million (ppm)

Parts per million is a convenient way to express very small relative changes. 1 ppm means 0.0001%, or 1 in 1 000 000. It is particularly common for expressing:

• Crystal and oscillator frequency accuracy: a 10 MHz crystal rated at ±5 ppm can be anywhere from 9.99995 MHz to 10.00005 MHz — a range of just 100 Hz across 10 MHz.
• Resistor temperature coefficients: a 10 kΩ resistor with a TC of 100 ppm/°C changes by 100 kΩ/°C × 100/1 000 000 = 1 Ω/°C. Over a 50°C temperature change, it shifts by 50 Ω — still less than 1% of its nominal value.
• Reference voltage stability: a precision voltage reference rated at 5 ppm/°C drifts by just 5 μV per degree Celsius for a 1 V reference.

Worst-case analysis

Worst-case analysis is the habit of checking whether your circuit still works correctly when all component tolerances stack up in the most unfavourable direction simultaneously. In a well-designed circuit, the worst-case combination of tolerances still produces acceptable results.

The steps are:

1. Identify which parameters matter for circuit performance (e.g. output voltage, resonant frequency, gain).
2. Determine how each component tolerance affects those parameters.
3. Choose the component values that push the output in the worst direction (all low, all high, or some combination).
4. Calculate the output for this worst case and check it is still within the acceptable range.

In ham radio, worst-case thinking matters whenever you are designing frequency-selective circuits (filters, matching networks), reference oscillators, power supply regulators, or measurement bridges. It also comes up when specifying component power ratings — always size resistors and capacitors for the worst-case operating conditions, not the typical ones.