Estimation of Errors

The mean value

To calculate the mean, we need to add all measured values of x and divide them by the number of values we took. The formula to calculate the mean is:

\[\text{mean} = \frac{x_1 + x_2 + x_3 + x_4 + ...+x_n}{n}\]

Estimation of errors

Here, we are going to distinguish between estimating the absolute error, the relative error, and the percentage error.

Estimating the absolute error

To estimate the absolute error, we need to calculate the difference between the measured value x0 and the expected value or standard x_ref:

\[\text{Absolute error} = |x_0 - x_{ref}|\]

Estimating the relative error

To estimate the relative error, we need to calculate the difference between the measured value x0 and the standard value x_ref and divide it by the total magnitude of the standard value x_ref:

\[\text{Relative error} = \frac{|x_0 - x_{ref}|}{|x_{ref}|}\]

Estimating the percentage error

To estimate the percentage error, we need to calculate the relative error and multiply it by one hundred. The percentage error is expressed as ‘error value’%. This error tells us the deviation percentage caused by the error.

\[\text{Percentage error} = \frac{|x_0 - x_{ref}|}{|x_{ref}|} \cdot 100 \%\]

What is the line of best fit?

The line of best fit is used when plotting data where one variable depends on another one. By its nature, a variable changes value, and we can measure the changes by plotting them on a graph against another variable such as time. The relationship between two variables will often be linear. The line of best fit is the line that is closest to all the plotted values.

Some values might be far away from the line of best fit. These are called outliers. However, the line of best fit is not a useful method for all data, so we need to know how and when to use it.

Obtaining the line of best fit

To obtain the line of best fit, we need to plot the points as in the example below:

Here, many of our points are dispersed. However, despite this data dispersion, they appear to follow a linear progression. The line that is closest to all those points is the line of best fit.

When to use the line of best fit

To be able to use the line of best fit, the data need to follow some patterns:

1. The relationship between the measurements and the data must be linear.
2. The dispersion of the values can be large, but the trend must be clear.
3. The line must pass close to all values.

Data outliers

Sometimes in a plot, there are values outside the normal range. These are called outliers. If the outliers are fewer in number than the data points following the line, the outliers can be ignored. However, outliers are often linked to errors in the measurements. In the image below, the red point is an outlier.

Drawing the line of best fit

To draw the line of best fit, we need to draw a line passing through the points of our measurements. If the line intersects with the y-axis before the x-axis, the value of y will be our minimum value when we measure.

The inclination or slope of the line is the direct relationship between x and y, and the larger the slope, the more vertical it will be. A large slope means that the data changes very fast as x increases. A gentle slope indicates a very slow change of the data.

Calculating uncertainty in a plot

In a plot or a graph with error bars, there can be many lines passing between the bars. We can calculate the uncertainty of the data using the error bars and the lines passing between them. See the following example of three lines passing between values with error bars:

• Calculate two lines of best fit.
• The first line (the green one in the image above) goes from the highest value of the first error bar to the lowest value of the last error bar.
• The second line (red) goes from the lowest value of the first error bar to the highest value of the last error bar.
• Calculate the slope m of the lines using the formula below.

\[m = \frac{y_2 - y_1}{x_2-x_1}\]

• For the first line, y2 is the value of the point minus its uncertainty, while y1 is the value of the point plus its uncertainty. The values x2 and x1 are the values on the x-axis.
• For the second line, y2 is the value of the point plus its uncertainty, while y1 is the value of the point minus its uncertainty. The values x2 and x1 are the values on the x-axis.
• You add both results and divide them by two:

  \[\text{Uncertainty} = \frac{m_{red}-m_{green}}{2}\]

Let’s look at an example of this, using temperature vs time data.