Physics and Engineering of Radiation Detection

9.1. Measures of Centrality 527

wherewiis the weight or importance of each data point. We will see later that this
weighting mean is the most commonly used method of computing the average.
Mean is easy to calculate but also very easy to get misleading results. If, for
example the output of a series of measurements contains large excursions due to
any reason related to the measurement process, the average of all the data will not
be a faithful representation of the parameter unless those excursions are excluded
from the calculations. This may or may not be possible, depending on the volume
of data and the available computing time and power. In such a situation, there is
another quantity that can be used instead of the mean, that is themedian.Median
is simply the middle number of the sample. To determine median, data is arranged
in ascending or descending order and the middle value is picked. This eliminates
the erroneous data points from the calculation of average (see example below). If
there are two values in the middle, a simple mean of the two values is taken as the
median.
Mode is the most frequently occurring value in the data. It is rarely used in data
analysis.

Example:

A parallel plate ionization chamber is used to measure the intensity of x-rays

coming from an x-ray machine with constant output. The data is amplified,

shaped, digitized by an 8-bit ADC, and stored in the computer memory. The

following is a sample of the ADC counts recorded.

34, 30, 28, 33, 29, 30, 31, 255, 27, 35, 29, 255, 33, 32, 28,30

Compute the measures of central tendency from the data.

Solution:

The out-of-bound values (255) at two points should be excluded from the

measurement of mean. However to see how these two values would affect the

computation of all the three measures of central tendency, let us compute these

quantities with and without these erroneous data points. Using equation 9.1.1

̄x =58. 7 with all values

̄x =30. 6 with two erroneous data points excluded

For median we write the data in ascending order.

27, 28, 28, 29, 29, 30, 30, 30, 31, 32, 33, 33, 34, 35, 255, 255

Since there are two central values therefore the median will be their mean.

Median =

30 + 31

2

=30. 5

If we exclude the two erroneous data points then the median will be

Median =

30 + 33

2

=30.

Mode is the highest occurring value, which in both cases is

Mode = 30