Measurement is the foundation of all mathematical concepts and this is not possible to imagine the world without measurements. The perfect measurements will increase the level of accuracy if they are based on international standards. Still, always measurement is suspected to small errors in mathematics or a level of uncertainty too. In simple words, the 100 percent accurate measurements are not possible in the practical world.
Error Formulas
Errors are simply defined as the difference between the measured value and the actual value. For example, when two operators use the same device for the measurement then this is not necessary that results would be the same. The difference that occurs between the actual value and the measured value is named as the ERROR.
\[\ Percentage\;Error=\frac{Approximate\;Value-Exact\;Value}{Exact\;Value}\times 100\]
\[\ Standard\;Error =SE_{\overline{x}}=\frac{S}{\sqrt{n}}\]
Where,
s is the standard deviation
n is the number of observation
\[\ Sampling\;Error=\pm \sqrt{\frac{2500}{Sample\;Size}}\times 1.96\]
\[\ E=Z\left(\frac{\alpha}{2}\right)\left(\frac{\alpha}{\sqrt{n}}\right)\]
Here,
$z$ $(\frac{\alpha }{2})$ = represents the critical value.
$z$ $(\frac{\sigma }{\sqrt{n}})$ = represents the standard deviation.
To learn the mathematics concepts deeply, you should know the different terms that could define the errors like sampling error, standard error, marginal error or percent error etc. Let us discuss on each of terms one by one with respective formulas. If you would understand these definitions and formulas deeply then there are chances that you could calculate the values as accurate as possible.
Sampling Error Formula
The error that arises due to sampling is named as the sampling error. This is the error usually related to the statistical analysis because of the wrong samples of the observations are taken. For example, the weight of 2000 citizens of a country are noted down and you need to calculate the average of weights now then it could be the same as the average weight of two million people.
To determine the weight of the whole population, the sampling technique is used. The difference between sample values and the population is termed as the sampling error. This is not possible to calculate the exact value of the population of you don’t know the value of sampling error and that could be found with sample modeling only.
So, the sampling error Formula in mathematics could be written as below –
\[\ Sampling\;Error=\pm \sqrt{\frac{2500}{Sample\;Size}}\times 1.96\]
Percent Error Formula
There could be a manufacturing error in measuring instruments too. This is not possible to assure them the exact. To know what type of error could be available here, we should know about the percentage error formula too. This is the absolute difference between measured value and the actual value and you should multiply the values by hundred too.
\[\ Percentage\;Error=\frac{Approximate\;Value-Exact\;Value}{Exact\;Value}\times 100\]
The Margin of Error Formula
The margin of errors is generally found in random sampling or the result of a survey. It is assumed that result of a sample is highly closers to the one would get from the population has been queried. In easy words, the margin of error is the product of critical value with the standard deviation. This is given by E and it could be written as –
\[\ E=Z\left(\frac{\alpha}{2}\right)\left(\frac{\alpha}{\sqrt{n}}\right)\]
Here,
\[\ Z\left(\frac{\alpha}{2}\right)= represents\;the\;critical\;value.\]
\[\ Z\left(\frac{\alpha}{\sqrt{n}}\right) = represents\;the\;standard\;deviation. \]
Standard Error Formula
Standard Error is the important statistical measure that is related to the standard deviation. The accuracy of a sample that could be presented by the population is given through the standard error formula and it could be written as below –
\[\ Standard\;Error =SE_{\overline{x}}=\frac{S}{\sqrt{n}}\]
Where,
s is the standard deviation
n is the number of observation
Where,S is the standard deviation, and n is the number of observations.