The Following Data Represents The Marks Of 11 Students In Gr

The Following Data Represents The Marks Of 11 Students In Group A 1

The following data represents the marks of 11 students in group (A): 14, 20, 17, 16, 18, 19, 15, 13, 2, 20, 14, 12. Compute the best central and dispersion measures. If the central and dispersion measures for the marks of the students in group (B) are as follows: n, mean, median, mode, s.d., min, max. Which group is better in the sense of homogeneity, group (A) or group (B)? State the reason for your answer.

Paper For Above instruction

Understanding the distribution of student marks is essential for evaluating the performance and consistency within a group. In this analysis, we examine the marks of Group A and compare their central tendency and dispersion measures with those of Group B to determine which group exhibits more homogeneity.

Analysis of Group A

The given data set for Group A consists of the following marks: 14, 20, 17, 16, 18, 19, 15, 13, 2, 20, 14, 12. First, we need to organize and analyze these marks to calculate key statistical measures.

Descriptive Statistics for Group A

Number of students (n): 12  (note: original data states 11 students, but 12 values are provided; assuming correction for accurate calculation)

Sorted data: 2, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 20

1. Measures of Central Tendency

Mean: The average is calculated by summing all values and dividing by the number of data points.

Sum of marks: 2 + 12 + 13 + 14 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 20 = 168

Mean = 168 / 12 = 14

Median: The middle value(s) when data is ordered.

Since there are 12 data points, median will be the average of the 6th and 7th values.

6th value: 15, 7th value: 16

Median = (15 + 16) / 2 = 15.5

Mode: The most frequently occurring value(s).

Values 14 and 20 occur twice; thus, the dataset is bimodal with modes 14 and 20.

2. Measures of Dispersion

Range: Max – Min = 20 – 2 = 18

Standard Deviation (s.d.): The square root of the variance; measuring the spread around the mean.

Calculations involve finding the squared deviations from the mean, summing these, dividing by n, and taking the square root.

Variance = [(2 - 14)^2 + (12 - 14)^2 + ... + (20 - 14)^2] / 12

Calculating squared deviations:

  • (2 - 14)^2 = 144
  • (12 - 14)^2 = 4
  • (13 - 14)^2 = 1
  • (14 - 14)^2 = 0
  • (14 - 14)^2 = 0
  • (15 - 14)^2 = 1
  • (16 - 14)^2 = 4
  • (17 - 14)^2 = 9
  • (18 - 14)^2 = 16
  • (19 - 14)^2 = 25
  • (20 - 14)^2 = 36
  • (20 - 14)^2 = 36

Sum of squared deviations: 144 + 4 + 1 + 0 + 0 + 1 + 4 + 9 + 16 + 25 + 36 + 36 = 272

Variance = 272 / 12 ≈ 22.67

Standard Deviation = √22.67 ≈ 4.76

Analysis of Group B

The data for Group B (assumed given): n, mean, median, mode, s.d., min, max. Since actual numerical values are not provided, a hypothetical example will be considered for illustration purposes:

  • n = 12
  • Mean = 15.8
  • Median = 16
  • Mode = 14
  • S.d. = 2.5
  • Min = 10
  • Max = 20

Based on these approximate measures, Group B has a slightly higher mean and median than Group A, and a lower standard deviation, indicating less variability and more homogeneity.

Discussion on Homogeneity

Homogeneity within a group reflects how similar the data points are, often measured by the dispersion or variability. A lower standard deviation signifies that marks are closely clustered around the mean, indicating higher homogeneity. Conversely, a higher standard deviation suggests greater spread and less uniformity among students' performance.

In the case of Group A, the standard deviation is approximately 4.76, which indicates a moderate spread around the mean of 14. The presence of outliers like the low score of 2 increases variability and reduces homogeneity. In contrast, if Group B's standard deviation is lower, approximately 2.5, then the marks are more tightly clustered, denoting higher homogeneity.

Considering the measures, Group B would be deemed more homogeneous if its dispersion is lower, illustrating consistent performance among its members. In academic settings, such homogeneity can be valuable for collaborative activities, as it reflects uniform competence.

Conclusion

In summary, the analysis reveals that Group A's marks exhibit moderate variability with a mean of 14 and a standard deviation of approximately 4.76, impacted by the presence of an outlier. If Group B's measures show a lower standard deviation, it indicates more consistent and homogeneous performance. Therefore, in the context of homogeneity, Group B would be considered better, provided its measures indeed reflect less variability. Ultimately, the choice of the better group depends on the measure of dispersion, with lower variability indicating higher homogeneity.

References

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