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Measures of Central Tendency and Dispersion

Partition Values or Fractiles

Partition values or fractiles such a quartile, a decile, etc. are the different sides of the same story. In other words, these are values that divide the same set of observations in different ways. So, we can fragment these observations into several equal parts. Let’s dig deeper!

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What is a Quartile?

quartileSource: pixabay

Whenever we have an observation and we wish to divide it, there is a chance to do it in different ways. So, we use the median when a given observation is divided into two parts that are equal.  Likewise, quartiles are values that divide a complete given set of observations into four equal parts.

Basically, there are three types of quartiles, first quartile, second quartile, and third quartile. The other name for the first quartile is lower quartile. The representation of the first quartile is ‘Q1.’ The other name for the second quartile is median. The representation of the second quartile is by ‘Q2 .‘  The other name for the third quartile is the upper quartile. The representation of the third quartile is by ‘Q3.’

  • First Quartile is generally the one-fourth of any sort of observation. However, the point to note here is, this one-fourth value is always less than or equal to ‘Q1.’  Similarly, it goes for the values of ‘Q2‘ and ‘Q3.’

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What are Deciles?

Deciles are those values that divide any set of a given observation into a total of ten equal parts. Therefore, there are a total of nine deciles. These representation of these deciles are as follows – D1,  D2, D3, D4, ……… D9.

Dis the typical peak value for which one-tenth (1/10) of any given observation is either less or equal to D1. However, the remaining nine-tenths(9/10) of the same observation is either greater than or equal to the value of D1.

What do you mean by Percentiles?

Last but not the least, comes the percentiles. The other name for percentiles is centiles. A centile or a percentile basically divide any given observation into a total of 100 equal parts. The representation of these percentiles or centiles is given as – P1,  P2, P3, P4, ……… P99.

Pis the typical peak value for which one-hundredth (1/100) of any given observation is either less or equal to P1. However, the remaining ninety-nine-hundredth(99/100) of the same observation is either greater than or equal to the value of P1. This takes place once all the given observations are arranged in a specific manner i.e. ascending order.

So, in case the data we have doesn’t have a proper classification, then the representation of pth quartile is (n + 1 )pth

Here,

n = total number of observations.
p = 1/4, 2/4, 3/4 for different values of Q1, Q2, and Q3 respectively.
p = 1/10, 2/10, …. 9/10 for different values of D1, D2, …… D9 respectively.
p = 1/100, 2/100, ….. 99/100 for different values of P1, P2, ……… P99 respectively.

Formula

At times, the grouping of frequency distribution takes place. For which, we use the following formula during the computation:

Q = l1 + [(Np– Ni)/(Nu-Ni)] * C

Here,

l1 = lower class boundary of the specific class that contains the median.
N= less than the cumulative frequency in correspondence to l(Post Median Class)
N= less than the cumulative frequency in correspondence to l(Pre Median Class)
C = Length of the median class (l– l1)

The symbol ‘p’ has its usual value. The value of ‘p’ varies completely depending on the type of quartile. There are different ways to find values or quartiles. We use this way in a grouped frequency distribution. The best way to do it is by drawing an ogive for the present frequency distribution.

Hence, all that we need to do to find one specific quartile is, find the point and draw a horizontal axis through the same. This horizontal line must pass through Np. The next step is to draw a perpendicular. The perpendicular comes up from the same point of intersection of the ogive and the horizontal line. Hence, the value of the quartile comes from the value of ‘x’ of the given perpendicular line.

Solved Questions for You!

Question: Here are the wages of some laborers: Rs. 82, Rs. 56, Rs. 120, Rs. 75, Rs. 80, Rs. 75, Rs. 90, Rs. 50, Rs. 130, Rs. 65. Find the values of Q1, D6, and P82.

Solution: The wages in ascending order – Rs. 50, Rs. 56, Rs. 65, Rs. 75, Rs. 75, Rs. 80, Rs. 90, R 82, Rs. 90, Rs. 120, Rs. 130

So,

Q= (n + 1)/4 th value
= 2.75th value where n = 10
= 2nd value + 0.75*difference between the 3rd and 2nd value
= Rs. 62.75

D= (15 + 1) * 6/10 th value
= 6.60th value
= 6th value + 0.60*difference between the 7th and 6th values
= Rs. 81.20

P1(10 + 1) * 82/100th value
= 9.02th value
= 9th value + 0.02*difference b/w the 10th and 9th values
= Rs. 120.20

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