When the data is grouped in the form of class interval then the mean can be calculated by three methods.

**1. Direct Method**

In this method, we use a midpoint which represents the whole class. It is called the class mark. It is the average of the upper limit and the lower limit.

**Example**

A teacher marks the test result of the class of 55 students for mathematics. Find the mean for the given group.

Marks of Students | 0 – 10 | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 | 50 – 60 |

Frequency | 27 | 10 | 7 | 5 | 4 | 2 |

To find the mean we need to find the mid-point or class mark for each class interval which will be the x and then by multiplying frequency and midpoint we get fx.

Marks of students | Frequency(f) | Midpoint(x) | fx |

0 – 10 | 27 | 5 | 135 |

10 – 20 | 10 | 15 | 150 |

20 – 30 | 7 | 25 | 175 |

30 – 40 | 5 | 35 | 175 |

40 – 50 | 4 | 45 | 180 |

50 – 60 | 2 | 55 | 110 |

f = 55 | fx = 925 |

marks

**2. Deviation or Assumed Mean Method**

If we have to calculate the large numbers then we can use this method to make our calculations easy. In this method, we choose one of the x’s as assumed mean and let it as “a”. Then we find the deviation which is the difference of assumed mean and each of the x. The rest of the method is the same as the direct method.

**Example**

If we have the table of the expenditure of the company’s workers in the household, then what will be the mean of their expenses?

Expense(Rs.) | 100 – 150 | 150 – 200 | 200 – 250 | 250 – 300 | 300 – 350 | 350 – 400 |

Frequency | 24 | 40 | 33 | 28 | 30 | 22 |

**Solution**

Here we take 275 as the assumed mean.

Expenses(Rs.) | Frequency(f) | Mid value(x) | d = x – 275 | fd |

100 – 150 | 24 | 125 | – 150 | – 3600 |

150 – 200 | 40 | 175 | – 100 | – 4000 |

200 – 250 | 36 | 225 | – 50 | -1650 |

250 – 300 | 28 | 275 | 0 | 0 |

300 – 350 | 30 | 325 | 50 | 1500 |

350 – 400 | 22 | 375 | 100 | 2200 |

f = 180 | fd = – 5550 |

**3. Step Deviation Method**

In this method, we divide the values of d with a number “h” to make our calculations easier.

**Example**

The wages of the workers are given in the table. Find the mean by step deviation method.

Wages | 20 – 30 | 20 – 30 | 30 – 40 | 40 – 50 | 50 – 60 |

No. of workers | 8 | 9 | 12 | 11 | 6 |

**Solution**

Wages | No. of workers (f) | Mid-point(x) | Assume mean (a) = 35, d = x – a | h = 10, u = (x – a)/h | fu |

10 – 20 | 8 | 15 | -20 | -2 | -16 |

20 – 30 | 9 | 25 | -10 | -1 | -9 |

30 – 40 | 12 | 35 | 0 | 0 | 0 |

40 – 50 | 11 | 45 | 10 | 1 | 11 |

50 – 60 | 6 | 55 | 20 | 2 | 12 |

f = 46 | fu = -2 |

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