Subject: Optional Mathematics

The measures of any set of data show the range of the data or the amount of variation is known as variability or dispersion.

It is already mentioned earlier that if the values of data in a set are widely dispersed then the different measures of central tendency cannot represent the values of the data accurately. Likewise, it has also been mentioned that if the data are close together then the measures of central tendency may represent each value of data in the set more precisely. While working on various tasks using statistics, it is necessary to find the different measures of dispersion. Here, the following four measures of dispersion are included:

- Range
- Mean deviation
- Quartile Deviation
- Standard Deviation

The difference between the highest and the lowest values of the variable of any set of data is called the range. The range tells us how far the values of data are spread out.

In any variation,

If, the Largest item = L

the smallest item = S, then

Range (R) = L - S

**Coefficient of Range**A range is an absolute value. Hence, to compare variations between two distributions we generally use the relative value which is calculated using the following formula. This relative value is called the coefficient of range.

Coefficient of Range = \(\frac{L - S}{L + S}\)

**Merits of range **

- Rigidly identified
- Simple understand and easy to estimate
- Required minimum time for variability

** Demerits of range**

- Changed by variation of sampling
- Changed by extreme value
- Cannot be used for open-end distributions
- Not suitable for more mathematical treatment

The difference between the first quartile (Q_{1}) and the third quartile (Q_{3}) is known as the interquartile range. The half of the interquartile range is called semi- interquartile range or quartile deviation.

Thus, if Q_{1}and Q_{3} represent the first and the third quartile respectively,

(Q.D.) = \(\frac {Q_3 - Q_1}2\)

To find the quartile deviation according to this formula, we need to find Q_{1}and Q_{3}. For this, let us recall the formulas to find Q_{1} and Q_{3}.

For individual and discrete series:

Q_{1} = (\(\frac{N+1}{4}\))^{th} term

Q_{3} = [\(\frac {3(N+1)}{4}\)]^{th} term

where, N = no. of terms in the series.

Similarly,

for continuous series, the quartiles are found as follows:

Q_{1} = (\(\frac N4\))^{th} term

Q_{3} = (\(\frac {3N}4\))^{th} term

From this the class interval in which Q_{1} and Q_{3} lies can be found. To find the quartiles the following formulas are used:

Q_{1} = L + \(\frac {\frac N4- c.f.}f\) × i

where, L = lower limit of Q_{1} class

c.f. = cumulative frequency of the class preceding the class containing Q_{1}f = frequency of the class

i = class size

Similarly,

Q_{3} = L + \(\frac {\frac {3N}4- c.f.}f\) × i

**Coefficient of Quartile Deviation**As the quartile deviation is an absolute value, we need to find the coefficient of quartile deviation to compare this with other data.

Coefficient of Quartile Deviation = \(\frac {Q_3 - Q_1}{Q_3 + Q_1}\)

**Merits of Quartile Deviation**

- Rigidly described
- Simple to understand and easy to estimate
- Not changed by ultimate values
- Useful to study dispersion in open-end series

**Demerits of Quartile Deviation**

- Not based on all observation
- Affected by the fluctuations of sampling
- Not competent of extra mathematical practice

- The measures of any set of data show the range of the data or the amount of variation is known as variability or dispersion.
- The difference between the highest and the lowest values of the variable of any set of data is called the range.
- The difference between the first quartile (Q
_{1}) and the third quartile (Q_{3}) is known as the interquartile range. The half of the interquartile range is called semi- interquartile range or quartile deviation. - Coefficient of Range = \(\frac{L - S}{L + S}\)
- Quartile deviation (Q.D.) = \(\frac {Q_3 - Q_1}2\)

- 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 range of the following data:

32, 28, 35, 41, 42, 38, 32, 34, 35, 32.

Soln**:** Here, Largest item (L) = 42

Smallest item (S) = 28

∴ Range(R) = L - S = 42 kg - 28 kg. Ans.

Find the range of the following data:

400, 100, 200, 300, 250, 150.

Soln:

Here, Largest item (L) = Rs. 400

Smallest item (S) = Rs. 100

∴ Range(R) = L - S = Rs. 400 - Rs. 100 = Rs. 300 Ans.

Find the range of the following data:

The temperature of a certain place taken every two hours of a day are recorded as below:

8^{o}C, 7^{o}C, 6^{o}C, 8^{o}C, 10^{o}C, 12^{o}C,13^{o}C, 15^{o}C, 14^{o}C, 11^{o}C, 10^{o}C,9^{o}C and so on.

Soln:

Here, Largest item(L) = 15^{o}C

Smallest item (S) = 6^{o}C

∴ Range(R) = L -S = 15^{o}C - 6^{o}C = 9^{o}C.

Find the range:

Marks Obtained | 40 | 45 | 50 | 55 | 60 |

No of students | 2 | 2 | 5 | 3 | 2 |

Soln:

Here, Largest mark (L) = 60

Smallest mark (S) = 40

∴ Range (R) = L - S = 60 - 40 = 20 Ans.

Find the range:

Number | 10 | 11 | 12 | 13 | 14 |

Frequency | 3 | 12 | 13 | 15 | 1 |

Soln:

Here, Largest number (L) = 14

Smallest number (S) = 10

∴ Range (R) = L - S = 14 - 10 = 4 Ans.

Find the range of the following grouped data:

Age(yrs) | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |

No of students | 6 | 10 | 16 | 14 | 10 | 5 |

Soln:

Here, Largest C.I. (80 - 90) and

Smallest C.I (30 -40)

So, Range (R) = upper limit of largest class - lower limit of smallest class

= 90 - 30 = 60 Ans.

Find the range of following data:

Marks obtained | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |

No of students | 10 | 20 | 18 | 32 | 21 |

Soln:

Here, Largest C.I (40-50)

Smallest C.I(0 - 10)

So, Range (R) = upper limit of largest class - lower limit of smallest class

= 50 - 0 = 50 Ans.

Find the range and the coefficient of the following data:

The yearly average rainfall (mm) of a town is as follow:

150, 160, 172, 185, 198, 210

Soln:

Here, Largest term (L) = 210mm

Smallest term (S) = 150mm.

∴ Range (R) = L - S = 210mm - 150mm = 60mm.

∴Cofficient of range

=\(\frac{L - S}{L + S}\) =\(\frac{210kg - 150kg}{210kg + 150kg}\) =\(\frac{60kg}{360kg}\) =\(\frac{1}{6}\) Ans.

Find the range and the coefficient of follwing data:

Weight(kg) | 30 | 32 | 35 | 38 | 32 | 38 | 41 |

No. of Students | 5 | 5 | 6 | 10 | 8 | 4 | 2 |

Soln:

Here, Largest weight (L) = 41kg

Smallest weight (S) = 30kg

∴ Range (R) = L - S = 41kg. 30kg. = 11kg.

∴ Coefficient of Range:

\(\frac{L - S}{L + S}\) =\(\frac{41kg - 30kg}{41kg + 30kg}\) = \(\frac{11kg}{71kg}\) =\(\frac{11}{71}\)

Find the range and its coefficient:

Marks obtained | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |

No of students | 1 | 3 | 5 | 20 | 3 |

Soln:

Here, Largest C. I = 40 - 50

Smallest C. I = 0 - 10

So, Range (R) = Upper limit of largest class - Lower limit of smallest class.

= 50 - 0 = 50 Ans.

∴ Coeff. of Range =\(\frac{L - S}{L + S}\) =\(\frac{50 - 0}{50 + 0}\) = 1 Ans.

Find the quartile deviation and the coefficient of quartic deviation:

22, 26, 14, 30, 18, 17, 35, 41, 12, 32, 34.

Soln:

Here, arranging the given data into ascending order:

12, 14, 17, 18, 22, 26, 30, 32, 34, 35, 41.

No. of terms (n) = 11

∴ First quartile (Q_{1}) =\(\frac{n + 1}{4}\) n^{th} term =\(\frac{11 + 1}{4}\)n^{th} term.

= 3^{th} term = third term = 17

Third quartile (Q_{3}) =\(\frac{3}{4}\)(n + 1) n^{th} term =\(\frac{3}{4}\)(11 + 1)n^{th} term

= 9 n^{th}term = 34

∴ Quartile deviation (Q. D) =\(\frac{Q_{3} - Q_{1}}{2}\) =\(\frac{34 - 17}{2}\) =\(\frac{17}{2}\) = 8.5

∴Coeff. of Q. D. =\(\frac{Q_{3} - Q_{1}}{Q_{3} + Q_{1}}\) =\(\frac{34 - 17}{34 + 17}\) =\(\frac{17}{51}\) =\(\frac{1}{3}\). Ans.

Find the quartile deviation and the coefficient of quartile deviation:

20, 30, 45, 60, 80, 90, 110, 115, 118, 120.

Soln:

Here, no of terms (n) = 10

∴ First quartile (Q_{1}) =\(\frac{n + 1}{4}\)n^{th} term =\(\frac{10 + 1}{4}\)n^{th} term

=\(\frac{11}{4}\)n^{th} term = 2.75 n^{th} term

= 2nd term + (3rd term - 2nd term)× 0.75

=30 +(45 - 30)×0.75 = 30 +15× 0.75

= 41.25

∴ Third quartile (Q_{3}) =\(\frac{3}{4}\)(n + 1) n^{th} term =\(\frac{3}{4}\) (10 + 1) n^{th} term

=\(\frac{33}{4}\)n^{th} term = 8.25 n^{th} term.

= 8^{th} term + (9^{th} term - 8^{th} term)× 0.25

= 115 + (118 - 115)× 0.25

= 115 + 3× 0.25

= 115 + 0.75

= 115.75

∴ Quartile deviation (Q. D) =\(\frac{Q_3 - Q_1}{2}\) =\(\frac{115.75 - 41.25}{2}\) = 37.25 Ans.

∴ Coeff of Q. D.

\(\frac{Q_3 -Q_1}{Q_3 +Q_1}\)=\(\frac{115.75-41.25}{115.75+41.25}\) =\(\frac{74.5}{157}\) =\(\frac{149}{314}\). ans

Find the quartile deviation and the coefficient of quartile deviation of following data:

The ages of students of class 7 of a certain school are recorded as below:

12, 13, 14, 13, 12, 15, 14, 15, 12, 13, 15, 12, 14, 15, 18, 13, 14, 15, 12, 15, 14, 13, 16.

Soln:

Frequency table:

Age(years) (X) | Frequency (f) No. of student | Cumulative frequency (c. f) |

12 | 5 | 5 |

13 | 5 | 5+5=10 |

14 | 5 | 10+5=15 |

15 | 6 | 15+6=21 |

16 | 1 | 21+1=22 |

18 | 1 | 22+1=23 |

N = 23 |

No. of term N = 6

First quartile (Q_{1}) = n^{th} term \(\frac{N + 1}{4}\) th term =\(\frac{23 + 1}{4}\) th term =\(\frac{24}{4}\) th term = 6 th term = 13.

∴ Third quartile (Q_{3}) =\(\frac{3}{4}\) (N + 1) th term =\(\frac{3}{4}\)× 24 th term = 18 th term = 15.

∴ Quartile deviation (Q. D) =\(\frac{Q_3 - Q_1}{2}\) =\(\frac{15 - 13}{2}\) =\(\frac{2}{2}\) = 1.

∴ Coeff. of Q .D =\(\frac{Q_3 - Q_1}{Q_3 + Q_1}\) =\(\frac{15 -13}{15 +13}\) =\(\frac{2}{28}\) =\(\frac{1}{14}\) = 0.07 Ans.

The marks obtained by 45 students of class 9 of a school in the Compulsory Mathematics Examination are recorded as below. Find the quartile deviation.

Marks obtained | 50 | 60 | 75 | 82 | 90 | 91 |

No. of Students | 16 | 12 | 8 | 5 | 3 | 1 |

Soln:

From the given data:

Marks (x) | No. of students (f) | Cumulative frequency (c. f) |

50 | 16 | 16 |

60 | 12 | 28 |

75 | 8 | 36 |

82 | 5 | 41 |

90 | 3 | 44 |

91 | 1 | 45 |

N = 45 |

No. of terms N = 45

First quartile(Q_{3}) =\(\frac{N + 1}{4}\) th term =\(\frac{45 + 1}{4}\) th term =\(\frac{23}{4}\) th term = 11.5 th term = 50

[ Here of greater then 11.5 in 16 which mark is 50.]

Third quartile (Q_{3}) =\(\frac{3}{4}\) (N + 1) th term =\(\frac{3}{4}\) (45 + 1) th term

=\(\frac{69}{2}\) th term = 34.5 th term = 75.

∴ Quartile deviation (Q. D) =\(\frac{Q_3 - Q _1}{2}\) =\(\frac{75 - 50}{2}\) =\(\frac{25}{2}\) = 12.5 Ans.

Find the quartile deviation from following data:

Marks Obtained | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

No. of students | 5 | 15 | 10 | 8 | 6 | 2 |

Soln:

Marks (x) | No. of students (f) | Cumulative frequency (c. f) |

20-30 | 5 | 5 |

30-40 | 15 | 20 |

40-50 | 10 | 30 |

50-60 | 8 | 38 |

60-70 | 6 | 44 |

70-80 | 2 | 46 |

N = 46 |

C. I lying Q_{1} is

=\(\frac{N}{4}\) th term =\(\frac{46}{4}\) th term = 11.5 th term

= 30-40

[ Here C. F. just greater than 11.5 is 20 which is in 30-40]

Using formula,

Q_{1} =L +\(\frac{\frac{N}{4} - c.f.}{f}\)× i

Here, L = 30, \(\frac{N}{4}\) = 11.5, c. f. = 5, f = 15 and i = 10

∴ Q_{1} = 30 +\(\frac{11.5 - 5}{15}\)× 10 = 30 + 4.33 = 34.33

C. I. lying Q_{3} is

=\(\frac{3}{4}\) N th term =\(\frac{3}{4}\)× 46 th term =\(\frac{69}{2}\) th term = 34.5 th term = 50-60

Using formula, Q_{3} = L +\(\frac{\frac{N}{4} - c.f.}{f}\)× i

Here, L = 50, \(\frac{3}{4}\)×N =\(\frac{3}{4}\)× 46 =\(\frac{69}{2}\)

c. f. = 30, f = 8, i = 10

∴Q_{3} = 50 +\(\frac{\frac{69}{2} - 30}{8}\)× 10 = 50 +\(\frac{9}{16}\)× 10 =50 +\(\frac{9}{8}\)×5 =\(\frac{445}{8}\) = 55.625

∴ Quartile deviation (Q. D) =\(\frac{Q_3 - Q_1}{2}\) =\(\frac{55.62 -34.33}{2}\) =\(\frac{21.29}{2}\) = 10.645 Ans.

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