### Linear Interpolation

The following table shows the marks obtained for mathematics in a certain group:

Class | Frequency |

11 - 20 | 10 |

21 - 30 | 18 |

31 - 40 | 30 |

41 - 50 | 10 |

51 - 60 | 14 |

61 - 70 | 16 |

71 - 80 | 2 |

Now, let's make the corresponding cumulative frequency table for the same data. It is as follows:

Class | Frequency | Cumulative Frequency |

11 - 20 | 10 | 10 |

21 - 30 | 18 | 28 |

31 - 40 | 30 | 58 |

41 - 50 | 10 | 68 |

51 - 60 | 14 | 82 |

61 - 70 | 16 | 98 |

71 - 80 | 2 | 100 |

In order to plot a cumulative frequency graph, we have to plot **cumulative frequency**
against the **upper-class-boundary** of each class. The curve should look like the following:

#### Finding the median

The **median** is the ** n/2 **^{th} value.

100/2 = 50

50^{th} value.

50^{th} value lies in the **31 - 40** class - i.e. anywhere between 30.5 and 40.5. We use **linear interpolation** to find it.

Since we treat the segment of the curve as a straight line in this class - shown in brown colour - the process is called **linear interpolation.** Let's consider the gradient of the line segment as follows:

Gradient = (58-28)/(40.5-30.5)

Gradient = (50-28)/(m-30.5)

So, 30/10 = 22/(m-30.5)

3 = 22/(m-30.5)

3m - 91.5 = 22

3m = 113.5

m = 37.8

median = 37.8

#### Finding the first quartile - Q1

The **first quartile ** is the n/4 ^{th} value.

100/4 = 25

25^{th} value.

25^{th} value lies in the **21 - 30** class - i.e. anywhere between 20.5 and 30.5. We use **linear interpolation** to find it.

Let's consider the gradient of the red line segment as follows:

Gradient = (28-10)/(30.5-20.5)

Gradient = (25-10)/(Q1-20.5)

So, 18/10 = 15/(Q1-20.5)

1.8 = 15/(Q1-20.5)

1.8Q1 - 36.9 = 15

1.8Q1 = 51.9

Q1 = 28.8

First Quartile = 28.8

#### Finding the third quartile - Q3

The **third quartile ** is the 3n/4 ^{th} value.

100 X (3/4) = 75

75^{th} value.

75^{th} value lies in the **51 - 60** class - i.e. anywhere between 50.5 and 60.5. We use **linear interpolation** to find it.

Let's consider the gradient of the yellow line segment as follows:

Gradient = (82-68)/(60.5-50.5)

Gradient = (75-68)/(Q3-50.5)

So, 14/10 = 7/(Q3-50.5)

1.4 = 7/(Q3-50.5)

1.4Q3 - 70.7 = 7

1.4Q3 = 77.7

Q3 = 55.5

Third Quartile = 55.5