Question

A researcher has obtained the number of hours worked per week during the summer for a sample of fifteen students.

40

25

35

30

20

40

30

20

40

10

30

20

10

5

20

Using this data set, compute the

a.	Median and mode
b.	mean
c.	40th percentile
d.	range
e.	sample variance
f.	standard deviation

Answer 1

a)

Median = 0.5(n+1)th value = 8th value of sorted data = 25

Mode = value that occur the most = 20

b)

Mean, x̅ = Ʃx/n = 375/15 = 25

c)

Step 1. Arrange the data in ascending order: 5, 10, 10, 20, 20, 20, 20, 25, 30, 30, 30, 35, 40, 40, 40

Step 2. Compute the position of the pth percentile (index i):

i = (p / 100) * n), where p = 40 and n = 15

i = (40 / 100) * 15 = 6

Step 3. The index i is an integer ⇒ the 40th percentile is the average of the values in the 5th and 6th positions (20 and 20 respectively)

40th percentile is (20 + 20) / 2 = 20

d)

Range = Max - Min = 40 - 5 = 35

e)

X	Mean, x̅	x-x̅	(x-x̅)²
40	25	15	225
25	25	0	0
35	25	10	100
30	25	5	25
20	25	-5	25
40	25	15	225
30	25	5	25
20	25	-5	25
40	25	15	225
10	25	-15	225
30	25	5	25
20	25	-5	25
10	25	-15	225
5	25	-20	400
20	25	-5	25

∑(x-x̅)² = 1800

Sample Variance, s² = Ʃ(x-x̅)²/(n-1) = 1800/(15-1) = 128.5714

f)

Sample Standard deviation, s = √(Ʃ(x-x̅)²/(n-1)) = √(1800/(15-1)) = 11.3389