Online calculator: Direct measurement error analysis

Direct measurement refers to measuring exactly the thing that you're looking to measure. Examples of direct measurements: linear dimensions measurement by measurement instruments like ruler, calipers or micrometer, time intervals measurement by the stopwatch, voltage or amperage measurement by the special electrical measuring instruments.

Measurement (observation) error

Any measurement can be performed with a certain accuracy. Wherein measured value differs from true value because measurement instruments, human senses, and methodologies are imperfect. Therefore, measurement error estimation plays an important role. The measurement result can be written in the form: X ± ΔX, where ΔX - absolute measurement error.

Random and systematic error

Measurement errors can be divided into two major categories: systematic error and random error.
Systematic errors stay constant or change by a known law during the measurement process. For example, a measurement instrument inaccuracy or instrument maladjustment leads to systematic error. Usually, if the root cause of a systematic error is known, then it can be eliminated.
Random factors, which affect measurement accuracy, affect the random error. For example, measuring time intervals by a stopwatch, random error arises due to different (random) reaction times of the experimenter to the start/stop events. To minimize random error influence, it is required to repeat the measurement several times.
The calculator below evaluates the random error of the direct measurement set for a given confidence interval. Some amount of theory follows the calculator.

Random error estimation

Confidence probability

Observations

Calculation precision

Digits after the decimal point: 3

Mean value

Absolute error

Relative error

Student t (two sided)

In most cases measurement result distribution is subject to normal distribution law. Therefore true value equals to the limit:
$x_0=\lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n} x_i$
In case of limited number of measurements, mean value is the nearest to true:
$\bar{x}=\frac{1}{n}\sum_{i=1}^{n} x_i$

According to the Gauss error theory, standard deviation characterizes random error of particular measurement:
$S_n=\left. \sqrt{\frac{\sum_{i=1}^{n}{(x_i-\bar{x})^2}}{n-1}} \right$ , standard deviation square is called the variance. When the variance increases, results scatter raises as well, i.e. random error increases.

To estimate whole result set error, instead of particular measurement error we need to find standard deviation of mean, which characterises $\bar{x}$ deviation from the true value $x_0$ .
According to error addition law, mean error is less than error of particular measurement. Standard deviation of mean equals to:
$S_{\bar{x}}=\frac{S_n}{\sqrt{n}} = \left. \sqrt{\frac{\sum_{i=1}^{n}{(x_i-\bar{x})^2}}{{n}({n-1})}} \right$
Absolute random error Δх equals to:
$\Delta_{x}=t_{\alpha,k}S_{\bar{x}},$ , where $t_{\alpha,k}$ - Student t-value for the given confidence probability $\alpha$ and degrees of freedom k = n-1.
Student t-value can be obtained by a table or by using our Student t-distribution quantile function calculator. You should be aware that the quantile function calculator gives a one-sided Student t-value. Two-sided t-value for a given confidence probability $\alpha$ equals to one-sided t-value for the same degrees of freedom and confidence probability equals to: $1-\frac{1-\alpha}{2}$