Subject: Compulsory Mathematics

If three numbers divide the series into four equal parts then, these three numbers are called quartiles.

If three numbers divide the series into four equal parts then, these three numbers are called quartiles.

Here 9, 13 and 27 divide the series into 4 equal parts. Therefore, 9, 13 and 27 are called quartiles. 9 is called the first quartile. (Q_{1}). 25% of the items are less than Q_{1 }and 75% of the items are more thanQ_{1}. 13 is called the second quartile. 50% of the items are below and above it. Similarly, 27 is called the third quartile. 75% of the items are less than Q_{3}and 25% are more than Q3.

Therefore, quartiles are also the positional value of the items.

**Calculation of quartiles in continuous series**Steps:

a. Construct the cumulative frequency table.

b. Use the formula to find the quartiles class.

The first quartile, Q_{1} = (\(\frac{N}{4}\))^{th} class

The third quartile, Q_{3} = (\(\frac{3N}{4}\))^{th} class

c. Use the formula to find the exact value of quartiles.

\( Q_1 = L + \frac {\frac{N}{4} - C.f }{f} \times h \)

\( Q_3 = L + \frac {\frac{3N}{4} - C.f }{f} \times h \)

Where,

L = lower limit of the class

c.f. = cumulative frequency of preceding class

f = frequency of class

h = width of class-interval

Ogive is a graphical representation of the distribution of a continuous data. While drawing ogive, points are plotted with cumulative frequency along with y-axis and its corresponding variables along with x-axis.There are two types of the ogive. They are less than ogive and more than ogive.

\( Q_1 = L + \frac {\frac{N}{4} - C.f }{f} \times h \)

\( Q_3 = L + \frac {\frac{3N}{4} - C.f }{f} \times h \)

Where,

L = lower limit of the class

c.f. = cumulative frequency of preceding class

f = frequency of class

h = width of class-interval

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Find the upper quartile of the given data:

50, 40, 55, 60, 61, 70,49

**Solution:**

The given data write in ascending order: 40, 49, 50, 55, 60, 61, 70 where N = 7

\begin{align*} Position \: of\: Q_3 &= \frac{3(N + 1)^{th}}{4}term \\ &= \frac{3(7+1)^{th}}{4}\\ &= 6^{th} \: term \\ 6^{th} \: term \: &represent \: 61 \\ \therefore Upper \: quartile \: &(Q_3) = 61 \end{align*}

Find third quartile (Q_{3}) from the following data:

45, 30, 31, 37, 42, 43, 40, 48

**Solution:**

The given data write in ascending order

30, 31, 37,40, 42, 43, 45, 48, where N = 8

\begin{align*} Position \: of \: Q_3 &= \frac{3(N + 1)^{th}}{4}term \\ &= \frac{3(8 + 1)^{th}}{4} term \\ &= 3 \times 2.25^{th} \: term \\ &= 6.75^{th} \: term \\ \: \\ Q_3 = 6^{th}\:term + &(7^{th}term - 6^{th} term) \times 0.75\\ Q_3 =43 + (45 - &43) \times 0.75 \\ Q_3 = 43 + 1.5 \\ \therefore \: \: Q_3 = 44.5 \: \: _{Ans} \end{align*}

2x + 1, 3x - 1, 3x + 5, 5x - 7, 51, 63 and 70are in ascending order. If the first quartile is 20. What will be the value of x.

**Solution:**

N = 7

\begin{align*} Position \: of \: Q_1 &= \frac{N + 1^{th}}{4}term \\ &= \frac{7 + 1^{th}}{4}term \\ &= 2^{nd} \: term \\ \: \\ 2^{th}\: term \: repr&esent \: 3x -1 \\ \: \\ Q_1 &= 3x -1 \\ 20 &= 3x -1 \\ or, 3x&= 20 +1 \\ or, x &= \frac{21}{3}\\ \therefore x &= 7 \: _{Ans} \end{align*}

Find the lower quartile from the following data 17, 25, 22, 18, 12, 14, 19, 11

**Solution:**

The given data arranging in ascending order.

11, 12, 14, 17, 18, 19, 22, 25 where N = 8.

\begin{align*} Position \: of \: first \: quartile \: (Q_1) &= \frac{N+1^{th}}{4}term \\ &= \frac{8+1}{4}term \\ &= 2.25^{th} \: term \\ \: \\ Q_1 = 2^{nd} term + (3^{rd} - &2^{nd}) term \times 0.25 \\ Q_1 = 12 + (14-12) \times &0.25 \\ Q_1 = 12 + 2 \times 0.25\\ \therefore Q_1 = 12.5 \: \: _{Ans} \end{align*}

Find the class of lower quartile (Q_{1}) from the following data.

Marks | 50 | 60 | 70 | 80 | 90 | 100 |

No. of students | 3 | 4 | 7 | 5 | 2 | 9 |

**Solution:**

Calculating lower quartile (Q_{1})

Marks (X) | Frequency (f) | Cumulative frequency (cf) |

50 | 3 | 3 |

60 | 4 | 7 |

70 | 7 | 14 |

80 | 5 | 19 |

90 | 2 | 21 |

100 | 9 | 30 |

N = 30 |

\begin{align*}Position \: of \: lower \: quartile (Q_1) &= \frac{N + 1^{th}}{4}term \\ &= \frac{30 + 1^{th}}{4}term \\ &= 7.75^{th} term \end{align*}

7.75^{th} term represent cf value 14.

\(\therefore\) lower quartile (Q_{1}) = 70 \(_{Ans}\)

1, 5, 7, 2x - 4, x + 7, 2x + 1 and 3x + 2 are in ascending order. If the third quartile is 15. What will be the value of x.

**Solution:**

The given data write in ascending order

\(1,5,7,2x-4,x+7,2x+1,3x+2\) where N = 7

\begin{align*} Position \: of \: Q_3 &= \frac{3(N + 1)^{th}}{4}term \\ &= \frac{3(7+1)^{th}}{4}term \\ &= 3 \times 2^{th} \: term \\ &= 6^{th} term \\ 6^{th} term \: repr&esent \: 2x + 1 \end{align*}

\begin{align*} Q_1 &= 2x+1 \\ 2x + 1 &= 15 \\ or, 2x &= 15 -1 \\ \therefore x &= \frac{14}{2} =7 \: _{Ans} \end{align*}

Calculate the class of third quartile from the given data.

Marks | 0 - 10 | 10 - 20 | 20 - 30 | 30 - 40 |

No. of students | 4 | 8 | 12 | 4 |

**Solution:**

Calculating the Q_{3} class

Marks (x) | No. of students(f) | Cumulative frequenc (cf) |

0 - 10 | 4 | 4 |

10 - 20 | 8 | 12 |

20 - 30 | 12 | 24 |

30 - 40 | 4 | 28 |

N = 28 |

\begin{align*} Position \: of \: Q_3 \: class &= \frac{3N^{th}}{4}term \\ &= \frac{3 \times 28^{th}}{4} term \\ &= 21^{th} \: term \\ Class \: of \: Q_3 &= 20 - 30 \: _{Ans}\end{align*}

Find the upper quartile from the following data.

Marks | 10 | 20 | 30 | 40 | 50 |

No. of students | 5 | 4 | 5 | 6 | 7 |

**solution:**

Calculating the Q_{3} class

Marks (x) | No. of students(f) | Cumulative frequenc (cf) |

10 | 5 | 5 |

20 | 4 | 9 |

30 | 5 | 14 |

40 | 6 | 20 |

50 | 7 | 27 |

N = 27 |

\begin{align*} Position \: of \: upper\:quartile (Q_3) &= \frac{3(N+1)^{th}}{4}term \\ &= \frac{3(27 + 1)^{th}}{4}term \\ &= 3 \times 7^{th} \: term \\ &=21 ^{th} \: term\\ \: \\ 21^{th} \: term \: represent \: c.f \: 27 \\ \therefore Upper \: quartile \: (Q_3) &= 50 \: \: _{Ans} \end{align*}

From the first quartile class from the given graph.

**Solution:**

From graph,

Total number of boys (N) = 60

\begin{align*} Position \: of \: first \: quartile &= \frac{N^{th}}{4}term \\ &= \frac{60^{th}}{4}term \\ &= 15^{th} term \\ Class \: of \: first \: quartile &= 5 - 10 \: _{Ans} \end{align*}

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