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# Direct observation error analisys

##### Calculates error of direct measurements for given measured value series and confidence interval.
Anton2015-11-29 16:55:42

### 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 imprefect. Therefore, measurement error estimation plays an important role. Measurement result can be written in the form : X ± ΔX, where ΔX - absolute measurement error.

## Random and systematic error

Measurement errors can be divided in two major categories: systematic error and random error.
Systematic errors stay constant or change by a known law during measurement process. For example, a measurement instrument inaccuracy or an instrument maladjustment lead 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 time of experimenter to the start/stop events. To minimize random error influence it is required to repeat the measurement several times.
The calculator below evaluates random error of direct measurement set for a given confidence interval. Some amount of theory follows the calculator.

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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 quantile function calculator gives 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: $1-\frac{1-\alpha}{2}$

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