Answer to Question #91039 in Statistics and Probability for ABU

Question

The following table shows the levels of retirement benefits given to a group of
workers in a given establishment.

Retirement benefits £ ‘000 No of retirees (f)
20 – 29 50
30 – 39 69
40 – 49 70

Required:-
Calculate the percentile coefficient of Kurtosis and briefly comment on the value
obtained
50 – 59 90
60 – 69 52
70 – 79 40
80 – 89 11

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c} Class \ Intervals & Frequencies\ (f_i) & Cumulative\ Frequencies \\ \hline 20-29 & 50 & 50 \\ 30-39 & 69 & 119 \\ 40-49 & 70 & 189 \\ 50-59 & 90 & 279 \\ 60-69 & 52 & 331 \\ 70-79 & 40 & 371 \\ 80-89 & 11 & 382 \\ \end{array}$$ $$\sum f_i=382$$

The percentile coefficient of Kurtosis, denoted as $K_p,$ is defined in terms of quartiles and percentiles as

$$K_p={Q_3-Q_1 \over 2(P_{90}-P_{10})}$$

$$Q_1=L_1+\bigg[{\sum f_i\times 1/4-C_f \over f_1}\bigg]\cdot C_1$$

$$\sum f_i\times 1/4=382\times 1/4=95.5, L_1=30,$$ $$C_f=50, C_1=10, f_1=69$$

$$Q_1=30+\bigg[{95.5-50 \over 69}\bigg]\cdot 10=36.5942$$

$$Q_3=L_3+\bigg[{\sum f_i\times 3/4-C_f \over f_3}\bigg]\cdot C_3$$

$$\sum f_i\times 3/4=382\times 3/4=286.5, L_3=60,$$ $$C_f=279, C_3=10, f_3=52$$

$$Q_3=60+\bigg[{286.5-279 \over 52}\bigg]\cdot 10=61.4423$$

$$P_k=L_1+\bigg[{\sum f_i\times (k/100)-C_f \over f_k}\bigg]\cdot C_k$$

$$k=90, \sum f_i\times (k/100)=382\times (90/100)=343.8,$$ $$L_1=70, C_f=331, C_{90}=10, f_{90}=40$$

$$P_{90}=70+\bigg[{343.8-331 \over 40}\bigg]\cdot 10=73.20$$

$$k=10, \sum f_i\times (k/100)=382\times (10/100)=38.2,$$ $$L_1=20, C_f=0, C_{10}=10, f_{10}=50$$

$$P_{10}=10+\bigg[{38.2-0 \over 50}\bigg]\cdot 10=17.64$$

The percentile coefficient of Kurtosis

$$K_p={61.4423-36.5942 \over 2(73.20-17.64)}=0.2236$$

Since $K_p=0.2236<0.263$, the curve is leptokurtic or thin.