The following table gives the distribution of daily wages of 500 families in a certain city.

Daily wages |
No. of families |

Below 100 | 50 |

100 – 200 | 150 |

200 – 300 | 180 |

300 – 400 | 50 |

400 – 500 | 40 |

500 – 600 | 20 |

600 above | 10 |

Draw a ‘less than’ ogive for the above data. Determine the median income and obtain the limits of income of central 50% of the families.

#### Solution

To draw a ogive curve, we construct the less than cumulative frequency table as given below:

Daily wages |
No. of families (f) |
Less than cumulative frequency(c.f.) |

Below 100 | 50 | 50 |

100 – 200 | 150 | 200 |

200 – 300 | 180 | 380 |

300 – 400 | 50 | 430 |

400 – 500 | 40 | 470 |

500 – 600 | 20 | 490 |

600 above | 10 | 500 |

Total |
500 |

The points to be plotted for less than ogive are (100, 50), (200, 200), (300, 380), (400, 430), (500, 470), (600, 490) and (700, 500).

Here, N = 500 = 125

For Q_{1}, `"N"/4=500/4` = 250

For Q_{2}, `"N"/2=500/2`

For Q_{3}, `"3N"/4=(3xx500)/4` = 375

∴ We take the points having Y coordinates 125, 250, and 375 on Y-axis. From these points, we draw lines parallel to X-axis. From the points where these lines intersect the curve, we draw perpendiculars on X-axis. X-Co-ordinates of these points gives the values of Q_{1}, Q_{2}, and Q_{3}.

∴ Q_{1} ≈ 150, Q_{2} ≈ 228, Q_{3} ≈ 297

∴ Median = 228

50% of families lies between Q_{1} and Q_{3}

∴ Limits of income of central 50% of families are from ₹ 150 to ₹ 297.