Download SOLITIONS ON UNIVARIATE DESCRIPTIVE ANALYSIS (FIRST

SOLITIONS ON UNIVARIATE DESCRIPTIVE ANALYSIS
(FIRST PART OF THE COURSE)
EXERCISE 1


The variable Evaluation of instructor is a qualitative variable.
Frequency distribution table.
Evaluation of instructor
n
E
A
A
B
P
6
5
3
4
2
Sum



20
Relative frequencies and percentages for all categories.
Evaluation of instructor
ni
fi
%
E
AA
A
B
P
Sum
6
5
3
4
2
20
0.3
0.25
0.15
0.2
0.1
1
30%
25%
15%
20%
10%
100%
What percentage of these students ranked this instructor as above average? 25%
Draw a bar graph for the percentage distribution.
35%
30%
25%
20%
15%
10%
5%
0%
E
AA
A
B
P
The two possible measures of central tendency are the mode and the median. Compute the possible
measures of central tendency.
a.
Mode = E
b.
Median.
Middle position: (20+1)/2 = 10.5
Median = AA
EXERCISE 2
The following data give the numbers of computer keyboards assembled at the Twentieth Century Electronics
Company for a sample of 25 days.
45

52
43

48
52
41
50
56
54
46
47
44
44
42
47
48
50
53
49
51
48
Numbers of computer
keyboards assembled
n
41-44
45-48
49-52
53-56
Sum
5
9
7
4
25
53
51 48 46
Calculate the percentage distribution, using the classes.
Numbers of computer
keyboards assembled
41-44
45-48
49-52
53-56
Sum
ni
fi
%
5
9
7
4
25
0.2
0.36
0.28
0.16
1
20%
36%
28%
16%
100%
What percentage of days corresponds to the numbers of computer keyboards assembled between
45 and 48? 36%

Calculate the cumulative percentage distribution.
Numbers of computer
keyboards assembled
41-44
45-48
49-52
53-56
Sum
ni
Cumulative freq
Cumulative
relative freq
Cumulative%
freq
5
9
7
4
25
5
14
21
25
-
0.2
0.56
0.84
1
-
20%
56%
84%
100%
-
What percentage of days corresponds to the numbers of computer keyboards assembled between 41 and 48?
56%
40%
35%
30%
25%
20%
15%
10%
5%
0%
41-44
45-48
49-52
53-56
EXERCISE 3

The variable Amount of telephone Bill is a continuous quantitative variable.
Amount of
Telephone Bill
Number of Families
(ni)
Cumulative
freq.
cumulative
relative freq.
% percentage
freq.
8
13
17
9
3
50
8
21
38
47
50
-
0.16
0.42
0.76
0.94
1
-
16%
42%
76%
94%
100%
-
20 -| 40
40 -| 60
60 -| 80
80 -| 100
100 -| 120
Sum
How many families in this sample had a bill of $80 or less? 38
What is the corresponding percentage? 76%

Amount of
Number of Families
Telephone Bill
(ni)
(dollars)
20 -| 40
8
40 -| 60
13
60 -| 80
17
80 -| 100
9
100 -| 120
3
Sum
50
Modal-class interval = 60 -|80
EXERCISE 4


Defines the elements and the variable in this data set.
The elements are the cities, the variable is hours spent annually in gridlock.
Construct a frequency distribution table. Take the classes as 44-50, 51-57, 58-64, 65-71, 72-78,
79-85.
Hours
44 -| 50
51 -| 57
58 -| 64
65 -| 71
72 -| 78
79 -| 85
Sum
Cities
(ni)
6
5
4
3
1
1
20
%
30
25
20
15
5
5
100

Construct a histogram for the percentage distribution.
35
30
25
20
15
10
5
0
44 -| 50

58 -| 64
65 -| 71
72 -| 78
79 -| 85
In what percentage of these cities do the motorists spend an average of 64 hours or less annually in
gridlock? 75%
Hours
Cities
(ni)
6
5
4
3
1
1
20
44 -| 50
51 -| 57
58 -| 64
65 -| 71
72 -| 78
79 -| 85
Sum

51 -| 57
%
30
25
20
15
5
5
100
Cumulative
percentage
30
55
75
90
95
100
Starting from the frequency distribution table, compute the mode.
Modal-class interval = 44 -|50
EXERCISE 5
Median.
Ranked data:
0.20 0.80
4.30
5.00
5.25
6.00
Middle position: ((6/2)+(6/2+1))/2 = 3.5
0.20 0.80
4.30
5.00
5.25
6.00
Median = (4.30 + 5.00)/2 = 4.65
Mean.
Mean = (0.20 + 0.80 + 4.30 + 5.00 + 5.25 + 6.00)/6 = 3.59
There is no mode in this data set. Each value in this data set occurs only once.
EXERCISE 6

Compute the mode, the median and the mean.
Numbers of children
Number of Families
0
1
2
3
10
15
21
4
Mode = 2
Median
Middle position: ((50/2)+ ((50/2)+1))/2 = 25.5 Median = (1 + 2)/2 = 1.5
Mean = (10*0 + 15*1 + 2*21 + 3*4)/50 = 1.38

Compute the range, variance and standard deviation.
Numbers of children
xi
Number of Families
ni
xi*ni
xi2
xi2*ni
0
1
2
3
Sum
10
15
21
4
50
0
15
42
12
69
0
1
4
9
0
15
84
36
135
Range = 3 – 0 = 3
Variance = s2= 1/50(135) – 1.90 = 0.80
Standard deviation = s = 0.89
EXERCISE 7
Compute the mode and the mean.
Computers sold
Number of Store
ni
Midpoint
mi
mi*ni
4-12
13-21
22-30
31-39
40-48
Sum
6
9
14
7
4
40
8
17
26
35
44
48
153
364
245
176
986
Modal-class interval = 22-30
Mean = (986)/40 = 24.65
EXERCISE 8

Calculate the mean and the median for these data.
Mean=(17.487+5189+3045+2616+2298+1630+1604+1410)/8= 4409.875
Median :
1. Rank the data:
1410 1604 1630 2298 2616 3045 5189 17487
2. Compute the middle position: ((8/2)+((8/2)+1))/2=4.5 Median=(2298+2616)/2=2457

Do they contain an outlier? Yes, the value 17487

Drop the outlier and recalculate the mean and the median.
Mean=(5189+3045+2616+2298+1630+1604+1410)/8=2541.714
Median Middle position=(7+1)/2=4 Median= 2298
EXERCISE 9


Range = 15 – 2 = 13
Mean = (6+3+7+11+4+3+8+7+2+6+9+15)/12 = 6.75
Variance = s2= 1/12(36+9+49+121+16+9+64+49+4+36+81+225) - 45.56 = 12.69

Standard deviation = s = 3.56
EXERCISE 10
Compute the variance and standard deviation.
No. of days
absent
0-2
3-5
6-8
9-11
12-14
Sum
Number of
employees
ni
10
14
9
4
3
40

Mean = 208/40 = 5.2
Variance = 2= 1/40(1582) – 27.04 = 12.51

Standard deviation =  = 3.54
midpoint
mi
mi*ni
mi2
mi2*ni
1
4
7
10
13
10
56
63
40
39
208
1
16
49
100
169
10
224
441
400
507
1582