Quartile Deviation

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Quartile Deviation

Quartiles


For the quartile deviation, let us have an idea about the quartiles. We know that the median divides the given observations arranged in ascending or descending order into two equal parts. Similarly, the quartiles divide the given observations into four equal parts after arranging them in ascending or descending order. So there are three quartiles denoted by Q1, Q2 and Q3 known as the first, the second and the third quartile respectively.



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If N be the number of observations of discrete data arranged in the ascending or descending order, then the quartiles are given by the following formula:

Q1 = value of ("N+1" /"4" )^"th" item Q2 = value of ("N+1" /"2" )^"th" item Q3 = value of ("3" 〖"(N+1)" 〗^"th" )/"4" item


In the case of continuous frequency distribution, the quartiles are calculated by the following formula:

Q1 = L + "i" /"f" ("N" /"4" " – cf" ) Q2 = L + "i" /"f" ("N" /"2" " – cf" ) Q3 = L + "i" /"f" ("3N" /"4" " – cf" ) Where, L = lower limit of the quartile class f = frequency of the quartile class cf = cumulative frequency of preceding class i = height of class-interval



Inter Quartiles Range


Inter Quartile Range: The difference between the third and the first quartile is called inter quartile range.


i.e.    Inter Quartile Range = Q3 – Q1

Quartiles Deviations


The average of deviation of the first quartile and the third quartile taken from median (M) is called the quartile deviation or semi-inter quartile range.

i.e. Quartile Deviation = (("Q" _"3" - "M" )" + (M " -〖" Q" 〗_"1" ")" )/"2" = ("Q" _"3" - "Q" _"1" )/"2" ∴ Quartile Deviation, Q.D. = ("Q" _"3" - "Q" _"1" )/"2"


Coefficient of Quartiles


As the unit of quartile deviation is same as that of the given observations, the two distributions with different units cannot be compared by quartile deviation. To facilitate the comparison of the quartiles of two or more than two series, a relative measure, the coefficient of quartile deviation is used, which is calculated as,

Coefficient of Quartile Deviation = (("Q" _"3 " -" " "Q" _"1" )/"2" )/(("Q" _"3" " + " "Q" _"1" )/"2" ) = ("Q" _"3" - "Q" _"1" )/("Q" _"3" + "Q" _"1" ) ∴ Coefficient Q.D. = ("Q" _"3" - "Q" _"1" )/("Q" _"3" + "Q" _"1" )

Worked Out Examples



Example 1: Calculate the quartile deviation and its coefficient of the following data: 24, 32, 46, 48, 39, 42, 28, 25, 26, 24, 38 Solution: Here, Arranging the data in ascending order, 24, 24, 25, 26, 28, 32, 38, 39, 42, 46, 48 No. of data (N) = 11 ∴ First quartile (Q1) = ("N+1" /"4" )^"th" item = ("11+1" /"4" )^"th" item = 3rd item = 25 ∴ Third quartile (Q3) = ("3" 〖"(N+1)" 〗^"th" )/"4" item = ("3" 〖"(11+1)" 〗^"th" )/"4" item = 9th item = 42 ∴ Quartile deviation (Q.D.) = ("Q" _"3" - "Q" _"1" )/"2" = ("42 " - "25" )/"2" = "17" /"2" = 8.5 ∴ Coefficient of Q.D. = ("Q" _"3" - "Q" _"1" )/("Q" _"3" + "Q" _"1" ) = ("42 " - "25" )/"42 + 25" = "17" /"67" = 0.25 Hence, quartile deviation = 8.5 and its coefficient = 0.25.


Example 2: Calculate the quartile deviation and its coefficient from the following data: Ages 12 13 14 15 16 17 18 No. of Students 12 21 15 20 17 10 5 Solution: Here, Ages (in years) No. of Students (f) Cumulative frequency (cf) 12 13 14 15 16 17 18 12 21 15 20 17 10 5 12 33 48 68 85 95 100 N = 100 Here, N = 100 ∴ First quartile (Q1) = ("N+1" /"4" )^"th" item = ("100+1" /"4" )^"th" item = 25.25th item = 13 ∴ Third quartile (Q3) = ("3" 〖"(N+1)" 〗^"th" )/"4" item = ("3" 〖"(100+1)" 〗^"th" )/"4" item = 75.75th item = 16 ∴ Quartile deviation (Q.D.) = ("Q" _"3" - "Q" _"1" )/"2" = ("16 " -" 13" )/"2" = "3" /"2" = 1.5 ∴ Coefficient of Q.D. = ("Q" _"3" - "Q" _"1" )/("Q" _"3" + "Q" _"1" ) = ("16 " - "13" )/"16 + 13" = "3" /"29" = 0.103 Hence, quartile deviation = 1.5 and its coefficient = 0.103.


Example 3: Calculate the quartile deviation and its coefficient from the following data: Marks 0-20 20-40 40-60 60-80 80-100 No. of Students 12 20 25 18 5 Solution: Here, Marks No. of Students (f) Cumulative frequency (cf) 0-20 20-40 40-60 60-80 80-100 12 20 25 18 5 12 32 57 75 80 N = 80 Here, N = 80 Q1 class = ("N" /"4" )^"th" class = ("80" /"4" )^"th" class = 20th class = 20 – 40 ∴ L= 20, f = 20, cf = 12, i= 20 ∴ Q1 = L + "i" /"f" ("N" /"4" -"cf" ) = 20 + "20" /"20" (20 – 12) = 20 + 8 = 28 Q3 class = ("3N" /"4" )^"th" class = ("3×80" /"4" )^"th" class = 60th class = 60 – 80 ∴ L= 60, f = 18, cf = 57, i= 20 ∴ Q3 = L + "i" /"f" ("3N" /"4" -"cf" ) = 60 + "20" /"18" (60 – 56) = 60 + 3.33 = 63.33 ∴ Quartile deviation (Q.D.) = ("Q" _"3" - "Q" _"1" )/"2" = ("63.33 " -" 28" )/"2" = "35.33" /"2" = 17.67 ∴ Coefficient of Q.D. = ("Q" _"3" - "Q" _"1" )/("Q" _"3" + "Q" _"1" ) = ("63.33 " -" 28" )/"63.33 + 28" = "35.33" /"91.33" = 0.39 Hence, quartile deviation = 17.67 and its coefficient = 0.39.


Do you have any questions regarding the Quartile Deviation?


You can ask your questions or problems here in the comment section below.

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About author: Min Rajbanshi

I am a school math teacher and a blogger since 2012. I do blogging on a variety of topics including math. Over the years, I've explored and written about a wide range of topics, with a particular emphasis on mathematics. My passion for education extends beyond the classroom, as I enjoy sharing knowledge and insights on various subjects through my blog.

10 comments:

  1. UnknownMay 21, 2021 at 9:47 PM

    calculate lower and upper quartiles, when quartile deviation =10 and co-efficient of quartile deviation=0.5

    ReplyDelete
    Replies
    1. Min RajbanshiMay 22, 2021 at 12:55 AM

      Here is the solution to your problems,
      [image src=" https://1.bp.blogspot.com/-sUm0wdOt6QA/YKi3_oi56WI/AAAAAAAAIms/0DJc_EvUlQ8OTOEw1UD9khdnUXtyASZWACLcBGAsYHQ/s16000/Comment%2B1%2Bsolution.png"/]
      Comment here if you have more problems regarding the quartile deviation.

      Delete
  2. Learner Zone NepalJune 22, 2021 at 6:46 AM

    Less than30 5 30-40 4 40-50 3 50-60 3 60-70 5 find quartile deviation

    ReplyDelete
    Replies
    1. Min RajbanshiJune 22, 2021 at 8:55 AM

      Here is the solution to your problems,
      [image src=" https://1.bp.blogspot.com/-PcJa57QPEOM/YNIDJ1zcEKI/AAAAAAAAJSg/vrjKeeO9lVQvOVIq3rfqe-IoUPtKSrzFQCLcBGAsYHQ/s16000/Comment%2B2%2Bsolution.png"/]
      Comment here if you have more problems regarding the quartile deviation.

      Delete
  3. UnknownNovember 29, 2021 at 7:40 AM

    length in hours:700-900 900-100 1100-1300 1300-1500
    samples: 10 16 26 8
    find out standard deviation by step deviation method

    ReplyDelete
    Replies
    1. Min RajbanshiNovember 30, 2021 at 4:48 AM

      Calculation of Standard Deviation by step-deviation method: [image src="https://blogger.googleusercontent.com/img/a/AVvXsEiZGHMpZWSgApdWNIe0fMnb3nOvmAhq8-U0gxqON9jZx9x9JKRnjjr5aSt9uIxANXU4TE2fz9QXoajPLYHyxEjbTua6dTilTVDTLFtZ4NGH-mfOwIiq84II3O6ESeQvj100F6zmlrgbpyxQ_Ky9uKZ9hJXD_7Hdsj642wpVowX3HOJLMKLL4oHX_Bc41Q=s16000 "/]
      You can solve it by yourself using our app Quick Math Solver

      INSTALL NOW
      [image src="https://blogger.googleusercontent.com/img/a/AVvXsEi7LDProuuL2gGTXODRPsEcnVL4HObAoyaclCqjhX_cI9IUktd8ZLC-EYWRAsiqFjSly-QVBCV6XeumlISBKhs3pevbvtlMgO2JPONlQGCeEMN3_3qdIZyjWpl_WqqWrpqqfz0iCh4w4zC4zyJUQ6ORUzo3Cg4iPDZU8E_ryzFOzlx5Y90eOvP37NV0PA=s16000"/]
      INSTALL NOW

      Delete
  4. sangam lamichhaneFebruary 18, 2022 at 9:09 AM

    Please do one question in which ci is given like 0 to 10. 0 to 20. 0 to 40.

    ReplyDelete
    Replies
    1. Min RajbanshiFebruary 19, 2022 at 2:04 AM

      We have to change the class intervals into 0-10, 10-20, 20-30… and filter the frequency values accordingly and then find quartile deviation. You can post your question here.

      Delete
  5. UnknownMarch 1, 2022 at 6:00 AM

    In symmetric distribution 50% of the items are above 60 and 75% of the items are below 75. Therefore, the coefficent of quartile is ---- and the coefficient of skewness is -----

    ReplyDelete
  6. UnknownOctober 6, 2023 at 6:38 AM

    17,2,7,27,15,5,14,8,10,24,48,10,8,7,18,28 pls solve that problem

    ReplyDelete

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