Units and Measurement

Let a physical quantity X be measured in an experiment and the experiment be repeated for n times. If measured values are X 1 X 2 ......X n then find the range in which all the measured values will lie. - Wired Faculty

Question

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Let a physical quantity X be measured in an experiment and the experiment be repeated for n times. If measured values are X₁ X₂......X_n then find the range in which all the measured values will lie.

Solution

When there are 'n' measured values, we cannot take any value as the true value.
Thus, true value is the mean of all the observed values.
$straight X space equals space fraction numerator straight X subscript 1 plus straight X subscript 2 plus......... space plus space straight X subscript straight n over denominator straight n end fraction$
The absolute error in measured values is,
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Now the mean absolute error is ,
$straight capital delta space straight X equals space fraction numerator ΔX subscript 1 plus ΔX subscript 2 plus.......... plus ΔX subscript straight n space over denominator straight n end fraction$
This means absolute error is the maximum possible error in the measurement.
Thus, all measured values of measurement lie in the range,
$straight X plus-or-minus ΔX$
i.e., $straight X minus ΔX space less than space straight X subscript 1 space less than space straight X space plus ΔX$

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