Partition Values

The locations at which the data is divided into equal halves are called partition values. These are what they are:

• Median
• Quartiles
• Deciles
• Percentiles

Quartiles:

Three points can be used to divide the data into four equal sections. These three values are referred to as quartiles (namely, 1st quartile, 2nd quartile, 3rd quartile). After sorting the data into ascending order, the quartiles are represented by Qi, where I = 1, 2, and 3. Qi is the value corresponding to the (iN/4)th observation.

Quartiles For Ungrouped Data:

• The lower half of a data set is the set of values that lies left of the median when arranged in ascending order.
• The upper half of a data set is the set of values that lies right of the median when arranged in ascending order.
• The first quartile which is denoted by Q1 shows the median of the lower half of the data set.
• The third quartile which is denoted by Q3 shows the median of the upper half of the data set.

Example 1: Find the first and third quartiles of the data set {3, 7, 8, 5, 12, 14, 21, 13, 18}.

Arranging the data in ascending order: 3, 5, 7, 8, 12, 13, 14, 18, 21.

Qi=\(\frac{i(N+1)}{4}\), where i=1,2,3

So,

Q1=\(\frac{(N+1)}{4}\)

=\(\frac{10}{4}\)

=2.5

=\(\frac{2nd item+3rd item}{2}\)

=\(\frac{5+7}{2}\)

=6

Similarly,

Q3=\(\frac{3(N+1)}{4}\)

=7.5

=\(\frac{7th item+8th item}{2}\)

=16

Quartiles for grouped data:

Quartiles for grouped data is calculated similarly as median, i.e,

\(Qi= L + \frac{\frac{iN}{4}- c.f.}{f} \times h \), where i =1,2,3

Example 2:

Find Q1,Q2,Q3 from the following frequency distribution table:

Q1=N/4=5 lies on the interval 10-20

Q1=\( L + \frac{\frac{N}{4}- c.f.}{f} \times h \), here i =1

=\(10 + \frac{5- 2}{3} \times 10 \)

=20

Q2=2N/4=10 lies on the interval 20-30

Q2=\( L + \frac{\frac{2N}{4}- c.f.}{f} \times h \), here i =2

=\( 20+ \frac{10- 5}{5} \times 10 \)

=30

Q3=3N/4=15 lies on the interval 40-50

Q3=\( L + \frac{\frac{3N}{4}- c.f.}{f} \times h \), here i =3

=\( 40 + \frac{15- 12}{6} \times 10 \)

=45

Deciles:

You can divide a data set into ten equal sections by nine points. Deciles are the name given to these nine points. Where i=1 to 9, deciles are represented as Di. The formula is as follows for calculating deciles as quartiles:

\(Di= L + \frac{\frac{iN}{10}- c.f.}{f} \times h \), where i =1-9

Example 3:

Find the deciles for the following data. 3, 15, 24, 28, 33, 35, 38, 42, 43, 38, 36, 34, 29, 25, 17, 7, 34, 36, 39, 44, 31, 26, 20, 11, 13, 22, 27, 47, 39, 37, 34, 32, 35, 28, 38, 41, 48, 15, 32, 60, 56, 13.

Given,

Let us first group the data and create a frequency distribution table as follows:

Now ,

D1=\(\frac{N}{10}\)=4.2 lies on the interval 10-17

=\(L + \frac{\frac{N}{10}- c.f.}{f} \times h \), here i =1

=\(10 + \frac{4.2-2.}{5} \times 7\)

=12.828

Similarly the deciles for rest 8 classes is calculated changing the value of i.

Percentile:

Although the term "percentile" is frequently used, it has no one, accepted definition. The most frequent definition of a percentile that is frequently encountered is a figure where a specific proportion of scores fall below the percentile. You may be aware that you received a test score of 70 out of 90. But without you know the percentile you fall into, that number has no actual relevance. If you know that you scored in the 80th percentile, it signifies that you outperformed 80% of test-takers. The partition of data into 100 equal portions using 99 points is another way to determine percentile.

• 1st quartile is equal to 25th percentile.
• 2nd quartile is equal to 50th percentile.
• 3rd quartile is equal to 75th percentile.

The percentile rank is calculated using the formula:

R=\(\frac{p}{100}\)N

Where P is the desired percentile and N is the number of data points.

Use the formula,

R=\(\frac{p}{100}\)N

3=\(\frac{p}{100}\)4

3=\(\frac{p}{25}\)

p=75

Therefore, the score 30 has the 75th percentile

Note that, if the percentile rank R is an integer, the Pth percentile would be the score with rank R When the data points are arranged in ascending order.

If R is, not an integer, then the Pth percentile is calculated as shown.

Let I be the integer part and D be the decimal part of R. Calculate the scores with the ranks I and I+1. Multiply the difference of the scores by the decimal part of R. The Pth percentile is the sum of the product and the score with the rank I.