In: Statistics and Probability
The table below shows marks scored by 42 students in a French
test
35 49 69 57
58 75 48
40 46 86 47
81 67 63
56 80 36 62
49 46 26
41 58 68 73
65 59 72
64 70 64 54
74 33 51
73 25 41 61
56 57 28
I. Group the data into 7 classes and make a
frequency distribution
(3mks)
II. Calculate:
(i) Mean mark
(2mks)
(ii) Median mark
(2mks)
(iii) Mode
(2mks)
(iv) Standard deviation
(3mks)
III. Comment on the distribution
(1mk)
For the data, we have the minimum value is 25 and the maximum value is 86. The difference or the range is
86-25=61. We need to have 7 classes which implies that the class interval is 61/7=8.714~9. Therefore, we shall make class interval 9 with seven classes starting from 25.
I. The frequency distribution is :
The frequency distribution is therefore
Class Interval | Frequency |
25-34 | 4 |
35-44 | 5 |
45-54 | 8 |
55-64 | 12 |
65-74 | 9 |
75-84 | 3 |
85-94 | 1 |
Total | 42 |
II. For calculating various quantities, we make the following table:
Marks | Marks^2 | |
35 | 1225 | |
40 | 1600 | |
56 | 3136 | |
41 | 1681 | |
64 | 4096 | |
73 | 5329 | |
49 | 2401 | |
46 | 2116 | |
80 | 6400 | |
58 | 3364 | |
70 | 4900 | |
25 | 625 | |
69 | 4761 | |
86 | 7396 | |
36 | 1296 | |
68 | 4624 | |
64 | 4096 | |
41 | 1681 | |
57 | 3249 | |
47 | 2209 | |
62 | 3844 | |
73 | 5329 | |
54 | 2916 | |
61 | 3721 | |
58 | 3364 | |
81 | 6561 | |
49 | 2401 | |
65 | 4225 | |
74 | 5476 | |
56 | 3136 | |
75 | 5625 | |
67 | 4489 | |
46 | 2116 | |
59 | 3481 | |
33 | 1089 | |
57 | 3249 | |
48 | 2304 | |
63 | 3969 | |
26 | 676 | |
72 | 5184 | |
51 | 2601 | |
28 | 784 | |
Total | 2363 | 142725 |
i). Mean mark is given by
ii). Median mark:It is the average of 21st and 22nd observations in ascending order. ie
iii). Mode: Is that value in the data that occurs the most in the data.
In the given data, the values that occured most is 56, 57 and 58 which occurred 2 each respectively and the first number is 56.
iv). Standard deviation:
The variance is calculated by
Therefore the standard deviation
III.
The frequency distribution is given by
From the chart we can see that the distribution is fairly symmetrical around the mean and the median is also close to the mean.