Field inspection manual—volumetric measuring devices

Appendix II—Linear interpolation

There are occasions when an inspector may need to interpolate values between two known values. This is common when assessing percentage tolerances, or applying various correction factors to a measured quantity. While not difficult to calculate, it is important that the interpolated value be determined carefully and correctly.

The formula for linear interpolation is:

(B mid) = [[(B upper) − (B lower)] [(A mid) − (A lower)]] ÷ [(A upper) − (A lower)] + (B lower)

Where:

  • Aupper = Upper Known Value
  • Alower = Lower Known Value
  • Bupper = Upper Corresponding Value
  • Blower = Lower Corresponding Value
  • Amid = Mid Known Value
  • Bmid = Mid Unknown Corresponding Value

The concept may be best described by example:

Example:

Assume you are taking a temperature measurement with a certified thermometer.

The thermometer is accompanied with a calibration certificate which lists 'Indicated' and 'True' temperatures. The temperature that you observe (26.50 °C) falls between two adjacent indicated values (20.00 °C & 30.00 °C) on the calibration certificate. How do you find the corresponding 'True' temperature?

Interpolation of the observed temperature value
Indicated temp True temperature
20.00 °C (A lower) 20.20 °C (B lower)
26.50 °C (A mid) B mid
30.00 °C (A upper) 30.25 °C (B upper)

What is the true temperature for an indicated temperature of 26.5 °C?

Bmid = [(30.25 − 20.20) (26.50 − 20.0)] ÷ (30.00 − 20.00) + 20.20

Bmid = [(10.05)(6.50) ÷ 10.00] + 20.20

Bmid = [65.325 ÷ 10.00] + 20.20

Bmid = [6.5325] + 20.20

Bmid = 26.7325 Bmid. 26.73 °C

This formula is useful for setting up a spreadsheet or a small program in a laptop, programmable calculator or PDA. If the interpolation must be calculated manually, the following simplified explanation may make it clearer.

Using a simplified approach:

Figure 1

Figure 1
Description of the figure 1

An example of linear interpolation where an observed indicated temperature (26.50) falls between two indicated temperatures (20.00 and 30.00) for which the corresponding true temperatures are known (20.20 and 30.25). The true temperature corresponding to the observed indicated temperature is unknown (Bmid).

The difference between the upper and lower indicated temperatures is 10.0 and the difference between the observed indicated temperature and the lower indicated temperature is 6.5; the difference between the corresponding true temperatures for the upper and lower indicated temperatures is 10.05 and the difference between the true temperature for the observed indicated temperature and the lower true temperature is unknown (x).

The difference between the true temperature for the observed indicated temperature and the lower true temperature (x) is determined by cross-multiplying the ratio of the difference between the upper and lower indicated temperatures to the difference between the observed indicated temperature and the lower indicated temperature (10.00 over 6.5) by the ratio of the difference between the corresponding true temperatures for the upper and lower indicated temperatures to the difference between the true temperature for the observed indicated temperature and the lower true temperature (10.05 over x), giving x = [10.05 × 6.5] ÷ 10.00 = 6.5325.

The true temperature for the observed indicated temperature (Bmid) is then determined by adding the difference between the true temperature for the observed indicated temperature and the lower true temperature (x) to the lower true temperature, giving Bmid = 6.5325 + 20.20 = 26.7325).

Reduced to two decimal points of precision, the true temperature corresponding to the observed indicated temperature is 26.73 °C.

Linear extrapolation

Either of these two approaches may also be used for linear extrapolation (finding a value not contained within, but rather larger or smaller than the data set), although extreme care must be taken to ensure that the extrapolated value is in fact representative and valid.

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