# Mean Values and Error Estimates

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Suppose you measure the same quantity n times independently, obtaining n values of the observable: X1, X2, ... , Xn. How can you state your answer for the combined result of these measurements scientifically?

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

1. Mean Value
If you have n independently measured values of the observable Xn, 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. X1 = 23.1°C, X2 = 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

2. 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 Xi. (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

3. 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 then 0.445°C / 51/2 = 0.199°C

4. Stating the Result of the Measurement
The result of the measurement is finaly given as

Thus 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.