Uncertainties

Every measurement that is taken of a star or set of stars comes with an associated uncertainty, which is sometimes called an 'error'. The term uncertainty is perhaps more accurate in the sense that it describes the level of confidence we have in our values, whereas the term error suggests that mistakes have been made in either the data collection or analysis.

Quantifying the uncertainty allows us to place errorbars on our datapoints. The uncertainty is related to a quantity known as signal-to-noise (SNR) which can be calculated easily from the data collected by Makali'i. The SNR is a measure of the quality of the data i.e. how 'loud' the signal is in relation to the background 'noise' and is given by the equation

SNR = square root (counts * gain)

where counts is the value that Makali'i returns for a given star and gain is a measure of the efficiency of the camera in terms of turning incoming photons into electrons. The gain can be found in the FITS header but for the files used in this activity, the value is 2.62. Generally, a value for SNR greater than 100 is considered good for accurate photometry.

From the SNR, the uncertainty can be calculated as 1/SNR. From this, we can see that a SNR of > 100 leads to an uncertainty of < 0.01 magnitudes.

From the example on this page, star 1 returns 18805 counts in V-band.

so SNR = square root (18805*2.62) = 222 and its uncertainty = 0.005 magnitudes

From these equations, it can be seen that shrinking the errorbars can be achieved by increasing the signal-to-noise, which in turn is done by either increasing the gain (i.e. having a better, more efficient and almost certainly more expensive camera !) or by increasing the number of counts. Increasing the counts is most easily achieved by increasing the exposure time, although we can examine other solutions such as using larger telescopes or simply observing brighter objects. The calculations can be seen in Figure 1.

Figure 1: SNR and uncertainty calculations.
Credit: Fraser Lewis
From these data, we can produce a plot such as that in Figure 2.

Figure 2: Data from Figure 1 plotted.
Credit: Fraser Lewis
We can then use the uncertainty column to create errorbars in the y-direction. In Excel, the steps are:

  • Click on a datapoint (any of the blue diamonds)
  • Select 'Error Bars' within 'Layout' in the 'Chart Tools'
  • Click on 'More Error Bars Options ...'
  • In the 'Format Error Bars' pop-up window, select 'Custom' and 'Specify Value' and then select the data in the final column for both the 'Positive Error Value'and 'Negative Error Value'
For many datasets, the errorbars will be very small, perhaps even less then the size of the symbol used to denote the data. In order to better show what your final graph may look like, Figure 3 shows the results of data with very low counts and SNR.
Figure 3: Faint stars plotted to show error bars using Excel.
Credit: Fraser Lewis
Finally, it is also possible to add error bars in the x-direction, though if we are using a calculated value such as B-V, this requires us to calculate our uncertainties in a method known as quadrature.

Plotting your data.

Learn about errors in quadrature.