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Download now on Google PlayThe formula for finding the arithmetic mean using the direct method is given by:

\(\overline X = \frac{\sum fx}{\sum f}\)

Where \(x\) is the midpoint of the class interval and \(f\) is the frequency.

**Steps**:

**1**. Calculate the midpoint of the class interval and name it as \(x\).

**2**. Multiply the midpoints\(x\) with the frequency\(f\) of each class interval and name it as \(fx\).

**3**. Find the values \(\sum fx\) and \(\sum f\).

**4**. Divide \(\sum fx\) by \(\sum f\) to determine the mean of the data.

Example:

The following frequency distribution table shows that the number of trees based on the height in metres. Find the average height of the trees.

Height (in \(m\)) | \(30 - 40\) | \(40 - 50\) | \(50 - 60\) | \(60 - 70\) | \(70 - 80\) |

Number of trees | \(124\) | \(156\) | \(200\) | \(10\) | \(10\) |

**Solution**:

Let us form a frequency distribution table.

Height(in \(m\)) | Number of trees(\(f\)) | Midpoint(\(x\)) | \(fx\) |

\(30 - 40\) | \(124\) | \(35\) | \(4340\) |

\(40 - 50\) | \(156\) | \(45\) | \(7020\) |

\(50 - 60\) | \(200\) | \(55\) | \(11000\) |

\(60 - 70\) | \(10\) | \(65\) | \(650\) |

\(70 - 80\) | \(10\) | \(75\) | \(750\) |

Total | \(\sum f = 500\) | \(\sum fx = 23760\) |

Mean \(\overline X = \frac{23760}{500}\) \(= 47.52\)

Therefore, the average height of the trees is \(47.52\).