Calculating Precision in Parts Per Thousand

General Procedure

Laboratories from CHM 111 and CHM 112 sometimes require students to calculate the ``parts per thousand''. This is performed as follows. If one has several mesasurements m₁, m₂, ..., m_n, the average ($\hat{X}$) is computed as in equation 1.  

$$\hat{X}=\frac{m_1 + m_2 + \cdots + m_n}{n}$$ (1)

The absolute deviation (d) of each of these points from the average is computed as the difference between the average and the measured quantity as indicated in equation 2  

$$d_i=\hat{X}-m_i$$ (2)

The average deviation ($\hat{d}$) is then computed by simply dividing the sum of the absolute values of the individual deviations (d) by the number of samples.

$$\hat{d}=\sum_{i=1}^n \frac{\left\vert d_i\right\vert}{n} = ... ...\vert d_2\right\vert + \cdots + \left\vert d_n\right\vert }{n}$$ (3)

The parts per thousand (ppt) is then computed by dividing the average deviation ($\hat{d}$) by the average value ($\hat{X}$) and multiplying by 1000.

$${\mathrm ppt}=\frac{\hat{d}}{\hat{X}}\times \mathrm{1000}$$ (4)

Example

Assume you have six measurements: 43.5, 42.6, 44.1, 43.0, 42.7, and 42.9. The average of these is:

$$\hat{X}=\frac{43.5 + 42.6 + 44.1 + 43.0 + 42.7 + 42.9}{6}=43.1$$ (5)

The absolute deviations are computed as follows:

The average deviation is therefore:

$$\hat{d} = \frac{\left\vert-0.4\right\vert + \left\vert-0.5\r... ...+ \left\vert.4\right\vert + \left\vert.2\right\vert}{6} = 0.43$$ (6)

Finally, this value is divided by the average and multiplied by 1000.

$${\mathrm ppt}=\frac{\hat{d}}{\hat{X}}\times {\mathrm 1000}= \frac{0.43}{43.1}\times \mathrm{1000}=\mathrm{10\ ppt}$$ (7)

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