The following table gives the frequency distribution of the daily commuting times (in minutes) from home to work for a sample of 25 employees of a company.
Daily Commuting Time (mins) |
Number of Employees f |
Relative Frequency |
Cumulative Frequency |
Midpoint x |
xf |
(x - x?)2 |
0 to less than 10 |
4 |
4/25 |
4 |
5 |
20 |
268.96 |
10 to less than 20 |
9 |
9/25 |
13 |
15 |
135 |
40.96 |
20 to less than 30 |
6 |
6/25 |
19 |
25 |
150 |
12.96 |
30 to less than 40 |
4 |
4/25 |
23 |
35 |
140 |
184.96 |
40 to less than 50 |
2 |
2/25 |
25 |
45 |
90 |
556.96 |
25 |
535 |
1064.80 |
- Construct a relative frequency distribution table for this data.
Relative frequency = frequency divided by total cumulative frequency
?f = 4+9+6+4+2=25
Calculating ?f: 4+9=13, 13+6=19, 19+4=23, 23+2=25
- What proportion of employees has daily commute time less than 30 minutes?
4+9+6=19
- Calculate the median daily commute time of employees.
n = 5, 5/2 = 2.5, (?f)? = 4, L? = 10, C = 20 – 10 = 10, ƒ median = 9
The median class is 10 to less than 20
Me = 10 + (2.5-49)10 = 10 + (-1.5/9) 10 = 10 + (-1/6) 10 = 10 + -1 2/3 = 8 1/3
- Calculate the median daily commute time of employees.
n= 5, 5/2 = 2.5, (?f)? = 4, L? = 10, C = 20 – 10 = 10, ƒ median = 9
The median class is 10 to less than 20
Me = 10 + (2.5-4/9) 10 = 10 + (-1.5/9) 10 = 10 + (-1/6) 10 = 10 + -1 2/3 = 8 1/3
- Calculate the mean daily commute time of employees.
First calculate the midpoint: (0+10)/2=5, (10+20)/2=15, (20+30)/2=25, (30+40)/2=35, (40+50)/2=45
Then the sum of xf: 4*5=20, 9*15=135, 6*25=150, 4*35=140, 2*45=90
Therefore, 20+135+150+140+90=535
Mean = 535/25=21.4
- Calculate the variance for the daily commute time of employees.
Sample Variance = ?(x-x?n-1 )²
x? = 21.4
5-21.4= (-16.4)² = 268.96; 15-21.4= (-6.4)² = 40.96; 25-21.4= (3.6)² = 12.96;
35-21.4= (13.6)² = 184.96; 45-21.4= (23.6)² = 556.96
? ?(x - x?)2 = 286.96+40.96+12.96+184.96+556.96 = 1064.80
n =5; 5-1=4
? 1064.80/4 = 266.20
n 2 Me L C wmedian n 2 Me L C wmedian