# Direct observation error analysis

Calculates error of direct measurements for given measured value series and confidence interval.

#### 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.

In most cases measurement result distribution is subject to normal distribution law. Therefore true value equals to the limit:

In case of limited number of measurements, mean value is the nearest to true:

According to the Gauss error theory, **standard deviation** characterizes random error of *particular measurement*:

, 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 deviation from the true value .

According to error addition law, mean error is less than error of particular measurement. Standard deviation of mean equals to:

Absolute random error Δх equals to:

, where - Student t-value for the given confidence probability 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 equals to one-sided t-value for the same degrees of freedom and confidence probability equals to:

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