The procedure for computing the quartiles, deciles etc are same as the median while computing these values in individual and discrete series we add 1 to N whereas in continuous series we do not add 1.

Thus Q₁ = size of [(N + 1)/4]^th item (individual observations and discrete series)

Q₁ = size of N/4 (in continuous series)

Q₃ = size of 3 (N +1) /4^th item (in individual and discrete series)

Q₃ = size of 3N / 4^th item (in continuous series)

D₄ = size of 4(N+1) / 10^th item (in individual and discrete series)

D₄ = size of 4N / 10^th item (in continuous series)

P₆₀ = size of 60 (N +1) / 100^th item (in individual and discrete series)

P₆₀ = size of 60 N / 100^th item (in continuous series)

Illustration:

Calculate the lower and upper quartiles third docile and 20^th percentile from the following data

Solution:

As we are given mid-points we will first do the lower and upper limits of the various classes. The method for finding these limits is to take the difference between the two central values divide it by 2 and then deduct the values so obtained from the lower limit and add it to the upper limit in the given case    7.5 - 2.5 / 2 = 5/2 = 2.5 the first class should be 0-5 second 5-10 etc, 

Calculation of     Q₁    Q₂    D₃    P₂₀

Lower quartile  Q₁ = size of N/4^th item = 100 / 4 = 25^th item

Q₁ lies in the class 5 -10

Q₁ = L+ N/4 – c.f / f xi

L = 5 N/4 = 25, c.f = 7,f = 18 / = 5

Q₁ = 5 + 25 – 7 / 18 X 5 = 10

Upper quartile Q₃ = size of 3 N / 4^th item = 3 X 100 / 4 = 75^th item

Q₃ lies in the class 15 – 20

Q₃ = L + 3N / 4 – c.f / f x i

L + 15, 3N/4 = 75, c.f = 50, f = 30 I + 5

∴  Q₃ = 15 + 75 – 50 / 30 x 5 = 15 + 4.17 = 19.17

Third docile

D₃ = size of 3N/10^th item = 3 x100/10 = 30^th item

D₃ lies in the class 10 – 15

D₃ = L+ 3 N / 10 – c.f / f x I

L = 10, 3N/10 = 30, c.f = 25 f = 25, I = 5

D₃ = 10 + 30 – 25 / 25 x 5 = 10 + 1 + 11

Twentieth percentile

P₂₀ = size of 20N / 100^th item = 20 X 100 / 100 = 20^th item

P₂₀ = L+ 20N/100-c.f / f x i

L = 5, 20 N/ 100 = 20 c.f = 7, f = 18 i=5

P₂₀ = 5 +20 – 7 / 18 x 5+ 5 + 3.61 + 8. 61.