- Mean Value
- Variance and Standard Deviation
- Standard Deviation of the Mean
- Stating the Result of the Measurement

Suppose you measure the same quantity

This is a very common question in all kinds of scientific measurements.Fortunately, the answer is straightforward:

*Mean Value*

If you have*n*independently measured values of the observable X_{n}, then the mean value of these measurements is:*Example:*

Suppose we measure the temperature within a room five different times and obtain the values 23.1°C, 22.5°C, 21.9°C, 22.8°C, and again 22.5°C. In this example,*n*= 5. X_{1}= 23.1°C, X_{2}= 22.5°C, and so on. The mean value of these temperature measurements is then:(23.1°C+22.5°C+21.9°C+22.8°C+22.5°C) / 5 = 22.56°C

*Variance and Standard Deviation*

Now we want to know how uncertain our answer is, that is to say how close the mean value of our independent measurements is likely to be to the true answer. In order to find out, we first calculate the standard deviation,The standard deviation measures the width of the distribution of the individual measurements X

_{i}. (The square of the standard deviation is also known as the variance).

*Example:*

For our five measurements of the temperature above the variance is[(1/4)·{(23.1-22.56) ^{2}+(22.5-22.56)^{2}+(21.9-22.56)^{2}+(22.8-22.56)^{2}+(22.5-22.56)^{2}}]^{1/2 }°C=0.445°C

*Standard Deviation of the Mean*

The standard deviation does not really give us the information of the uncertainty in our measurements. For this, one introduces the standard deviation of the mean,which we simply obtain from the standard deviation by division by the square root of

*n*. This standard deviation of the mean is then equal to the error, dX which we can quote for our measurement.

*Example:*

For our temperature measument, the standard deviation of the mean is then0.445°C / 5 ^{1/2}= 0.199°C*Stating the Result of the Measurement*

The result of the measurement is finaly given asThus the combined result of performing

*n*independent measurement of the same physical quantity is the mean plus/minus the standard deviation of the mean.

*Example:*

For our temperature measurement we will finally obtain as the answer:T = 22.6°C +- 0.2°C Note that we have rounded the quoted error to the first significant digit and then also rounded the quoted mean value to the same accuracy.