Compute percentile coefficient of kurtosis from the following data and interpret the result.

Hourly wages (Rs) 23-27 28-32 33-37 38-42 43-47 48-52
Number of workers 22 16 9 4 3 1

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Solution:

x f cf
23-27 22 22
28-32 16 38
33-37 9 47
38-42 4 51
43-47 3 54
48-52 1 55
N = 55

Now,

Calculating P75:

P75 = \(75 \left ( \frac{N}{100} \right )^{th} item\) = \(75 \left ( \frac{55}{100} \right )^{th}\) = 41.25th

In cf, the value just greater than 41.25 is 4. so, class  = 33 – 37

Now,

P75 = \(L + \frac{\frac{75 N}{100} – cf}{f} \times i\)

= \(33 + \frac{41.25 – 38}{9} \times 4\)

= 34.45

Calculating P25:

P25 = \(25 \left ( \frac{N}{100} \right )^{th} item\) = \(25 \left ( \frac{55}{100} \right )^{th}\) = 13.75th

In cf, the value just greater than 13.755 is 22. so, class  = 23 – 27

Now,

P25 = \(L + \frac{\frac{25 N}{100} – cf}{f} \times i\)

= \(23 + \frac{13.75 – 0}{22} \times 4\)

= 23.625

Calculating P90:

P90 = \(90 \left ( \frac{N}{100} \right )^{th} item\) = \(90 \left ( \frac{55}{100} \right )^{th}\) = 49.5th

In cf, the value just greater than 49.5 is 51. so, class  = 38 – 42

Now,

P90 = \(L + \frac{\frac{90 N}{100} – cf}{f} \times i\)

= \(23 + \frac{49.5 – 47}{4} \times 4\)

= 40.5

Calculating P10:

P10 = \(10 \left ( \frac{N}{100} \right )^{th} item\) = \(10 \left ( \frac{55}{100} \right )^{th}\) = 5.5th

In cf, the value just greater than 5.5 is 21. so, class  = 23-27

Now,

P10 = \(L + \frac{\frac{10 N}{100} – cf}{f} \times i\)

= \(23 + \frac{5.5 – 0}{22} \times 4\)

= 24

Now,

Percentile cofficient of Kurtosis

= \(\frac{P_{75} – P_{25}}{2(P_{90} – P_{10})}\)

= \(\frac{34.45 – 23.625}{2(40.5 – 24)}\)

= \(\frac{10.825}{33}\)

= 0.33

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