Data Analysis
The following are the final scores in Probability and Statistics for 40 selected Year 1 Computer Science students during the academic year 2023:
30, 83, 90, 83, 75, 45, 90, 90, 68, 83, 58, 83, 73, 78, 90, 83, 53, 70, 55, 35,
31, 45, 64, 73, 65, 45, 80, 80, 68, 73, 48, 73, 73, 78, 80, 63, 43, 60, 45, 55.
(a) Organize the data into an appropriate table and create a corresponding graph.
(b) Identify the mode of the data set.
(c) Compute the following summary measures:
Arithmetic mean
Median
Variance
Standard deviation
The correct answer and explanation is :
(a) Frequency Distribution Table and Graph
Frequency Distribution Table:
| Score Range | Frequency (f) |
|---|---|
| 30 – 39 | 2 |
| 40 – 49 | 5 |
| 50 – 59 | 5 |
| 60 – 69 | 5 |
| 70 – 79 | 9 |
| 80 – 89 | 7 |
| 90 – 99 | 7 |
Histogram:
(Imagine a bar chart with frequency on the y-axis and score ranges on the x-axis.)
(b) Mode
The mode is the most frequently occurring score. From the dataset:
- 83 appears 5 times
- 73 appears 5 times
- 90 appears 4 times
- 45 appears 4 times
Thus, the dataset is bimodal with modes 73 and 83.
(c) Summary Measures
Arithmetic Mean (𝜇)
[
\mu = \frac{\sum X}{N} = \frac{30+83+90+…+55}{40} = \frac{2615}{40} = 65.375
]
Median
Arranging the data in ascending order:
30, 31, 35, 43, 45, 45, 45, 45, 48, 53, 55, 55, 58, 60, 63, 64, 65, 68, 68, 70, 73, 73, 73, 73, 73, 73, 75, 78, 78, 80, 80, 80, 83, 83, 83, 83, 83, 90, 90, 90
For 40 numbers, the median is the average of the 20th and 21st values:
[
\frac{70 + 73}{2} = 71.5
]
Variance (σ²)
Using:
[
\sigma^2 = \frac{\sum (X – \mu)^2}{N}
]
[
\sigma^2 = \frac{(30-65.375)^2 + (83-65.375)^2 + \dots + (55-65.375)^2}{40} = 354.2
]
Standard Deviation (σ)
[
\sigma = \sqrt{354.2} = 18.83
]
Explanation (300 Words)
The analysis of the final scores of 40 students in Probability and Statistics reveals key insights about their performance. Organizing the data into a frequency distribution table and histogram helps in visualizing how scores are distributed. Most students scored between 70 and 79, with relatively fewer students scoring below 50.
The mode, which represents the most frequently occurring scores, is both 73 and 83, making this dataset bimodal. This suggests that students’ performance clustered around these two scores.
The mean (65.375) is a measure of central tendency, giving an idea of the average score of all students. However, the median (71.5) is slightly higher, indicating a right-skewed distribution, where a few lower scores pull the mean down.
To measure score variability, the variance (354.2) and standard deviation (18.83) were computed. A high standard deviation indicates that the scores are widely spread out from the mean, showing performance disparity among students.
Overall, this statistical analysis helps educators understand student performance, identify common score ranges, and assess whether interventions are needed to support students struggling in Probability and Statistics.