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Mathematics

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Subject
Mathematics
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Homework
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1.
Considering the data given by the table:
Length (mm)
150-
154
155-
159
160-
164
165-
169
170-
174
175-
179
180-
184
185-
189
Frequency
5
2
6
8
9
11
6
3
Medium of each
interval (mm)
152
157
162
167
172
177
182
187
The mean value is calculated by considering the formula:
π‘€π‘’π‘Žπ‘› =
βˆ‘
πΉπ‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦
(
𝑖
)
βˆ— π‘€π‘’π‘‘π‘–π‘’π‘š π‘œπ‘“ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™ (𝑖)
βˆ‘
πΉπ‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ (𝑖)
Hence,
π‘€π‘’π‘Žπ‘› =
5 βˆ— 152 + 2 βˆ— 157 + 6 βˆ— 162 + 8 βˆ— 167 + 9 βˆ— 172 + 11 βˆ— 177 + 6 βˆ— 182 + 3 βˆ— 187
5 + 2 + 6 + 8 + 9 + 11 + 6 + 3
π‘€π‘’π‘Žπ‘› =
8530
50
= 170.6
The median corresponds to the data value situated such that when ordered from the lowest to
the highest values it lies in the center of the list, meaning that 50% of the data are lower than
him and 50% of the data are higher than him. Considering that we have a total of 50 data
(calculated by adding the different frequencies for each length interval), we would need to
locate the length corresponding to the data situated in position 26. To do so, we simply have to
count….
Interval 150-154 has a frequency of 5, meaning it has the first 5 data lower than the median
Interval 155-159 has a frequency of 2, meaning it has the 6
th
and 7
th
data lower than the median
Interval 160-164 has a frequency of 6, meaning it has the 8
th
- 13
th
data lower than the median
Interval 165-169 has a frequency of 8, meaning it has the 14
th
– 21
st
data lower than the median
Interval 170-174 has a frequency of 9, meaning it has the 22
nd
- 30
th
data in the ordered list.
Since we were looking for the data located in the 25
th
position, the median would be in the
interval 170-174.
Finally, the mode is calculated by considering that it should present the highest frequency of
the given intervals. Thus, taking a look at the table we see that the highest frequency value is
of 11, which corresponds to the interval 175-179, such that the mode will be 175-179.

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2.
If we consider the data given by the table:
41
17
83
63
54
92
60
58
70
06
63
82
33
44
57
49
34
73
54
63
36
52
32
75
60
33
09
79
28
30
42
93
43
80
03
50
57
67
24
64
63
11
35
82
10
23
00
40
To prepare a frequency distribution of intervals we simply have to count the amount of data lying in
each of the different interval (0-9, 10-19, …). Thus, the frequency distribution would be given by the
table:
10-19
20-29
30-39
40-49
50-59
60-69
70-79
90-99
3
3
7
6
7
8
4
2
3.
To solve this exercise we will take the data from the frequencies table we calculated in exercise
2. That way, you will see a much more straightforward of a representation of a frequency
distribution through a histogram.
Thus, for example, we will consider that the numbers in the different intervals correspond to
the distance travelled from our employees to come to work…
Distance (km)
0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
Frequency
4
3
3
7
6
7
8
4
4
2
If we represent these data as a bar graph, we obtain:

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1. Considering the data given by the table: Length (mm) 150154 5 152 Frequency Medium of each interval (mm) 155159 2 157 160164 6 162 165169 8 167 170174 9 172 175179 11 177 180184 6 182 185189 3 187 The mean value is calculated by considering the formula: π‘€π‘’π‘Žπ‘› = βˆ‘ πΉπ‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ (𝑖) βˆ— π‘€π‘’π‘‘π‘–π‘’π‘š π‘œπ‘“ π‘–π‘›π‘‘π‘’π‘Ÿπ‘£π‘Žπ‘™ (𝑖) βˆ‘ πΉπ‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ (𝑖) Hence, π‘€π‘’π‘Žπ‘› = 5 βˆ— 152 + 2 βˆ— 157 + 6 βˆ— 162 + 8 βˆ— 167 + 9 βˆ— 172 + 11 βˆ— 177 + 6 βˆ— 182 + 3 βˆ— 187 5 + 2 + 6 + 8 + 9 + 11 + 6 + 3 π‘€π‘’π‘Žπ‘› = 8530 = 170.6 50 The median corresponds to the data value situated such that when ordered from the lowest to the highest values it lies in the center of the list, meaning that 50% of the data are lower than him and 50% of the data are higher than him. Considering that we have a total of 50 data (calculated by adding the different frequencies for each length interval), we would need to locate the length corresponding to the data situated in position 26. To do so, we simply have to count…. Interval 150-154 has a frequency of 5, meaning it has the first 5 data ...
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