Distance from a target

Bags of potatoes at a supermarket are labelled with weight 3kg. How close are the actual weights to this target?

Single value
The distance of a single value, x , from a target, k , is called its error,

error   =   (xk)

However if we measure n bags of potatoes, how do we combine the errors to give a single measure of accuracy?

Mean error (bias)
The average of the individual errors is equal to the difference between the sample mean and k ,

This quantity is called the bias and clearly tells us something about the accuracy of the weights of the bags of potatoes.
However even if the bias is zero, individual values may be very different from the target, k .
Mean squared error
One solution to the problem of negative errors is to square them before averaging,
mean squared error   =   
Root mean squared error
The main problem with the mean squared error is that its units are the square of those of the raw data. For example, the weights of the bags of potatoes (and therefore the errors) are measured in kg so the squared errors are 'squared kg' and the mean squared error is also 'squared kg'. How do you interpret a value with these units? The solution is to take the square root to return the value to the original units.
root mean squared error   =   

The root mean squared error is a 'typical' error.


Weights of 3kg bags of potatoes

The diagram below shows the weights of seven bags of potatoes labelled '3 kg'.

A square is drawn for each data value whose sides have length equal to the error for that bag of potatoes.

The area of each square is the squared error for the value.

The root mean squared error is the side length of the square whose area is the average of the areas of the squares. It is shown in red on the diagram.

Drag the crosses to see how the values affect the root mean squared error.


You may notice that an outlier corresponds to a square with a very large area, so it has a disproportionate effect on the root mean squared error.