# NCERT Class 9 Solutions: Statistics (Chapter 14) Exercise 14.3-Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Q-9 100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters of the English alphabet in the surnames was found as follows:

Number of letters | Number of surnames |

1 - 4 | 6 |

4 - 6 | 30 |

6 - 8 | 44 |

8 - 12 | 16 |

12 - 20 | 4 |

- Draw a histogram to depict the given information.
- Write the class interval in which the maximum number of surname lies.

Solution (I) :

The problem with this data is that the class intervals are of varying width. The principle behind histograms is that the area of each bar represents the fraction of a frequency (probability) distribution within each bin (class, interval) . Therefore, as the bar in a histogram is spread across bins, it is squished in height (that is, area remains the same) . Here we choose the bin size to be 2 letters. So we divide the frequency (number of surnames names) into bins of 2 letters.

For example, number of surnames with 1 – 4 letters is 6 however since 4 – 1 = 3, this frequency is not spread across 2 letters but 3. The height in frequency representation in histogram is the proportion of the number of surnames per 2 letters interval, that is .

We calculate the histogram heights for other intervals as follows:

Number of letters | Frequency (Number of surnames) | Width of class | Height of rectangle |

1 - 4 | 6 | 3 | |

4 - 6 | 30 | 2 | |

6 - 8 | 44 | 2 | |

8 - 12 | 16 | 4 | |

12 - 20 | 4 | 8 |

Now plot the histogram

On x-axis represent the class intervals of the number of letters and on y-axis the proportion of the number of surnames per 2 letters interval.

Solution (II) :

The maximum number of surnames lies in interval of 6 - 8 and it has 44 surnames.