The Q test

A problem which often arises when making replicate measurements is that one of the measurements may yield a result which differs excessively from all of the others. We are then faced with the problem of deciding whether to retain this questionable result and include in the calculation of the average, or to reject it as being unreliable. This decision can be made with the help of the Q test which is illustrated with the example below:

A student makes five measurement of the density of a solution and obtains the values:

1.053, 1.060, 1.059, 1.070 and 1.058 g/mL. Is 1.070 a "bad point" which should be rejected? We arrange the values in ascending order:

1.053, 1.058, 1.059, 1.060, 1.070 The total spread between the highest and the lowest value is 1.070 - 1.053 = 0.017. The spread between the questionable value and the next value is 1.070 - 1.060 = 0.010. We now define Q_exp as

Q_exp = spread/(total spread) = 0.010/.017 = 0.60

To apply the Q test we use the data in the table below:

Table 1: Critical values for the rejection quotient Q_crit

If Q_exp > Q_crit the questionable result should be rejected. In this case, the number of observations is five and Q_crit = 0.64. Since 0.60 < 0.64 the questionable result should be retained.