This document discusses different measures of position including quartiles, deciles, and percentiles. Quartiles divide a distribution into four equal parts at the 25th, 50th, and 75th percentiles. The second quartile is equal to the median. Deciles divide a distribution into ten equal parts. Percentiles divide a distribution into 100 equal parts and are used to characterize what percentage of values fall below each percentile point. Formulas are provided to calculate quartiles, deciles, and percentiles based on class boundaries, cumulative frequencies, and interval sizes.

1. The other measures
of position
QUARTILES, DECILES, AND
PERCENTILES

2. The quartiles
Quartiles are the score points which
divide a distribution into four equal parts.
25 % fall below the first quartile (Q1),
50% below the second quartile (Q2), and
75% are less than the third quartile (Q3).

3. The quartiles
 Note that the second quartile (Q2), is equal to
median.
 We recall that in determining the median class, we
have to multiply n by
1
2
since 50% are below median.
 Logically for the first quartile, we multiply by n by
1
4
,
and for the third quartile, we multiply by
3
4
.

4. The quartiles
 Here’s the formula.
 𝑸𝟏 = 𝑿𝒍𝒃 +
𝒏
𝟒
−𝒄𝒇𝒑
𝑭𝒒
𝐢
Where
 Xlb = lower boundary of the class interval which contains Q1 (first
quartile class)
 Cfp = cumulative frequency for the class interval preceding the first
quartile class.
 Fq = frequency of the first quartile class
 i = interval size

5. The quartiles
 Here’s the formula.
 𝑸𝟑 = 𝑿𝒍𝒃 +
𝟑𝒏
𝟒
−𝒄𝒇𝒑
𝑭𝒒
𝒊
Where
 Xlb = lower boundary of the class interval which contains Q3 (third
quartile class)
 Cfp = cumulative frequency for the class interval preceding the third
quartile class.
 Fq = frequency of the third quartile class
 i = interval size

6. The deciles
 Nine score-points are required to divide a distribution into ten equal parts.
They are called deciles and denoted by D1, D2, D3, …, D9.
The formulas are as follows:
 D1 = Xlb +
1
1𝑜
𝑛 −𝑐𝑓𝑝
𝑓𝑑
𝑖
 D2 = Xlb +
1
5
𝑛 −𝑐𝑓𝑝
𝑓𝑑
𝑖
 D3 = Xlb +
3
1𝑜
𝑛 −𝑐𝑓𝑝
𝑓𝑑
𝑖
 D9 = Xlb +
9
1𝑜
𝑛 −𝑐𝑓𝑝
𝑓𝑑
𝑖

7. The percentile
 Percentiles are the ninety-nine score points which divide a
distribution into 100 equal parts.
 They are generally used to characterize values according to
the percentage below them.
 For example the first percentile (P1) separates the lowest 1%
from the other 99%, the second percentile (P2), separates
the lowest 2% from the other 98% and so on.
 15 % are less than the 15th percentile and 20 % are below
the 20th percentile; the middle 80 % is determined by the
10th and the 90th percentiles.

8. The percentile
 If k% are less than a given percentile, then
Pk = Xlb +
𝑘𝑛
100
−𝑐𝑓𝑝
𝑓𝑝
𝑖
Where
 Xlb = lower boundary of the kth percentile class
 Cfp = cumulative frequency for the class interval preceding the kth class.
 Fq = frequency of the kth percentile class
 i = interval size in the kth percentile class

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