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Linear Interpolation

Chemistry: IB Diploma Higher Level, topic 7.2.2

Linear Interpolation

Linear interpolation can be used to estimate different values in a set of data such as the median, quartiles and inter-quartile range.

Median piece of data

Median piece of data

  • The median occurs at the position given by this equation:
  • position of median=number of pieces of data2\text{position of median}=\frac{\text{number of pieces of data}}{2}
  • We can then find the class interval that contains the median, which is one that has the lowest cumulative frequency that is greater than this number.
Median piece of data

Median piece of data

Linear interpolation

Linear interpolation

  • To find the median within the class, we assume that the cumulative frequency graph is a straight line between the lower and upper limits of the class.
  • First, calculate the gradient of the straight line within the class interval using the equation:
  • g=y2−y1x2−x1g = \frac{\large y_2 - y_1}{\large x_2 - x_1}
    • Where x1≤x<x2x_1 \leq x \lt x_2 is the class interval.
Linear interpolation 2

Linear interpolation 2

  • Then, equate this to the gradient of the straight line between the median and the lower limit of the class interval:
  • 103−4940−30=63.5−49M−30\frac{\large 103 - 49}{\large 40 - 30} = \frac{\large 63.5 - 49}{\large M - 30}
  • 5.4=14.5M−305.4 = \frac{\large 14.5}{\large M - 30}
  • So the median is 32.7 kg.
Linear interpolation 2

Linear interpolation 2

Percentiles

Percentiles

  • The same method of linear interpolation works to find any percentile.
  • To find out the 30th percentile, we first find the position:
  • 127&times;0.3=38.1
  • Then we use the equation:
  • 49−1730−20=38.1−17P−20\frac{\large 49 - 17}{\large 30 - 20} = \frac{\large 38.1 - 17}{\large P - 20}
  • 3.2=21.1P−203.2 = \frac{\large 21.1}{\large P - 20}
  • So the 30th percentile is 20.1 kg.
Percentiles

Percentiles

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