Circular Descriptive Statistic for Describing Students Sleeping

Time

Cici Suhaeni, M.Si

Dr. I Made Sumertajaya

Bogor Agricultural University

(IPB)

Presented : 10th Okct 2015, ICSMTR 2015

Data

Linear Directional

Circular Spherical

Background

What is circular DATA? • Representated by directions • Directions are measured by angles ranging from 0 to

360 degree. • Periodic or cyclic • Taken from compass and clock as a measuring

instrument • Circular data was taken from clock, must be convert

to angles • Circular data is taken from compass : wind direction

and directions of migrating bird • Circular data is taken from the clock : arrival time

Number line

Linear Data

-∞ ∞ 0 -1 1 ... ...

Circular Data

Linear and Circular Data

Circle

Treat the

circular

data as

linear data

is not

relevant.

What is circular statistics?

D A T A

STATISTICAL ANALYSIS

Circular data Circular statistics

Why MUST CIRCULAR STATISTICS?

Ex :

Data : 30° and 300° Mean =...? Linear : mean = 165 Circular :

Mean direction =−15°

𝛼 0 = arctanS

C

S = sin𝛼𝑖

𝑛

𝑖=1

C = cos𝛼𝑖

𝑛

𝑖=1

Mean ≠ mean direction

Descriptif Statistic

Statistic Linear Circular

Central tendency

Mean Mean direction

Median Median direction

Dispersion Variance Circular Variance Concentration Parameter

Why does students sleeping time?

• The sleep quality of college students seems poor, and going to Participation

Exercises Class turns out to be main way of doing exercise (Chang S.P et al,

2013).

• There is correlation between duration of sleep with academic performance.

The students with adequate sleep have a high academic performance

(Marpaung et al, 2013)

• Amount of sleep per night have positive correlation with GPA of college

students (Lowry, M. et al, 2010)

• Short sleepers have lower overall grade-point averages than did long

sleepers (Kelly, W.E. et al, 2001)

• Hours of sleep is associated with a variety of health-risk behaviors

(McKnight_Eily et al, 2011)

Sleep problem is very important to be discussed

In the previous studies about “sleeping” case,

the measured variable generally is the

duration or quality of sleep. So, make sleeping

time as a variable is a special issue in the

statistical analysis world. This variable is a

different perspective in view of the quality of

one’s sleep.

Studying sleeping time by circular statistics is something interesting and important..

Why does Descriptive Statistics?

•Studies on the circular statistics have not been so much done

•Start the study from the simple method : descriptive statistics

The purpose of this Study

To apply circular descriptive statistics for describing students sleeping time.

To compare circular descriptive statistic and linear descriptive statistic.

Data

Generated from Von Mises Distribution

For small(k=0.05) ,

middle (k=5) ,

big kappa (k=20)

Simulation Data

Student sleeping time

Bachelor students in 5th semester from Dept. of Statistics IPB

Real Data

Convert time to angle

𝛼° =𝑥

𝑘× 360°

1 hour = 15o 1o = 4 minutes

Jam Derajat

0 0

1 15

3 45

6 90

12 180

18 270

21 315

24 360

Result

Description of Simulation Data

Circular data plot and rose diagram for simulation data with mean direction 0 and kappa 0,05, n=100

Circular data plot and rose diagram for simulation data with mean direction 0 and kappa 20, n=100

For small concentration

parameter, data spread around the

circle

For large concentration

parameter, data gather in certain point in the circle

Comparation between linear statistic and circular statistic

→ Mean direction

→ Mean

→ Median direction

→ Median

(i) n=10, 𝜅 = 0.05 (ii) n=10, 𝜅 = 5 (iii) n=10, 𝜅 = 20

(iv) n=100, 𝜅 = 0.05 (v) n=100, 𝜅 = 5 (v) n=100, 𝜅 = 20

Analyze circular data using linear statistics is not very very relevant.

So, use circular statistics to analyze circular data!

Description of Students Sleeping Time

Statistic

All Female Male

Circular Time Circular Time Circular Time

mean

direction 351.103 11:24 pm 351.493 11:26 pm 350.186 11:21 pm

mean 245.176 04:20 pm 240.500 04:02 pm 256.359 05:05 pm

median

direction 345 23.000 345 23.000 345 23.000

concentration

parameter 12.647 11.269 18.062

circular

varians 0.040 0.045 0.028

Graphical presentation for all students data 11:24 pm

04:20 pm

→ Mean direction

→ Mean

Graphical presentation for female

students data

→ Mean direction

→ Mean

11:26 pm

04:02 pm

Graphical presentation for female

students data

→ Mean direction

→ Mean

11: 21 pm

05:05 pm

Conclusion

• Linear descriptive statistic have very different value with circular linear statistic. So, use circular statistics to analyze circular data.

• Mean of sleeping time of 5th semester students from Dept. of Statistics IPB is at 11:24 pm, the female students is at 11:26 pm, and the male students is 11:21 pm.

• Variability of sleeping time data is relative large.

Reference

• Chang, S.P., Huang, S.C., Chen, Y.H., Wright, R., Lung-Chuan, Liao. The relationship between Sleep

Quality and the Exercise Participation Behavior of College Students in the Central Taiwan Region.

International Journal of Sport and Exercise Science (2013), 5(2): 13-18.

• Fisher NI. 1995. Statistical Analysis of Circular Data. Cambridge : Cambridge University Press.

• Jammalamadaka SR, SenGupta A. 2001. Topics in Circular Statistics. Singapore: World Scientific.

• Lowry, M., Dean, K., and Manders, K. The Link Between Sleep Quality and Academic Performance for

the College Student. Sentience : The University of Minnesota Undergraduate Journal of Psychology

(2010), Volume 3.

• Mardia KV, Jupp PE. 2000. Directional Statistics. West Sussex : John Wiley & Sons.

• Marpaung, P.P, Supit, S., and Nancy, J. Gambaran Lama Tidur terhadap Prestasi Belajar Siswa.

Journal e-Biomedik(eBM), Volume 1 Nomor 1, Maret 2013, hlm. 543-549.

• McKnight-Eily, L.R., Eaton, D.K, Lowry, R., Croft, J.B., Presley-Cantrell, L., and Perry, G.S. Relationship

between Hours of Sleep and Health-risk Behaviors in US Adolescent Students. Preventive Medicine

(2011), doi: 10.1016/j.ypmed.2011.06.020.