The Following Are The Grades Which 40 Students Obtained In A Certa
Q2. The following are the grades which 40 students obtained in a certain course in 1997 E.C. here in Mekelle University of the Arid Campus. a. Construct an absolute frequency distribution. b. Convert the distribution obtained in (a) into a Relative & Percentage distribution. c. Convert the distribution in (a) into a “Less than” & a “More than” cumulative distribution d. Construct a histogram, frequency polygon and ogive curve Q3. Consider the following three data sets A, B and C. A = {9,10,11,7,13} B = {10,10,10,10,10} C = {1,1,10,19,19} Calculate: The mean of each data set. The standard deviation of each data set.
Introduction
Analyzing student grades provides valuable insights into performance distribution and overall achievement levels within a course. By constructing frequency distributions, converting to relative and cumulative formats, and visualizing data through histograms, polygons, and ogives, educators can identify patterns, strengths, and areas needing improvement. Additionally, understanding basic statistical measures such as mean and standard deviation applied to different data sets offers a quantitative perspective on data variability and central tendency.
Part 1: Grade Data Analysis for 40 Students in Mekelle University
Constructing an Absolute Frequency Distribution
The first step involves organizing the grades into a clear frequency distribution. Suppose the grades obtained by the 40 students are aggregated into classes or intervals. For illustration, assume the grades fall within the following ranges and frequencies (this data is hypothetical for demonstration, as actual grades are not provided):
60-69: 8 students
70-79: 15 students
80-89: 10 students
90-100: 7 students
This absolute frequency distribution categorizes the data, providing a straightforward count of how many students fall into each grade interval.
Converting into Relative & Percentage Distributions
To obtain the relative frequency, divide each class frequency by the total number of students (40):
60-69: 8/40 = 0.20
70-79: 15/40 = 0.375
80-89: 10/40 = 0.25
90-100: 7/40 = 0.175
Percentage distribution multiplies these relative frequencies by 100:
60-69: 20%
70-79: 37.5%
80-89: 25%
90-100: 17.5%
Converting into “Less than” and “More than” Cumulative Distributions
The “less than” cumulative distribution sums the frequencies starting from the lowest class:
< 70: 8
< 80: 8 + 15 = 23
< 90: 23 + 10 = 33
< 101: 33 + 7 = 40
The “more than” cumulative distribution starts from the highest class:
>= 90: 7
>= 80: 10 + 7 = 17
>= 70: 15 + 10 + 7 = 32
>= 60: 40
Constructing a Histogram, Frequency Polygon, and Ogive Curve
Using the frequency data, one can plot:
Histogram:
bars representing the frequency of each grade interval.
Frequency Polygon:
points plotted at class midpoints connected by straight lines.
Ogive Curve:
cumulative frequencies plotted against class boundaries to visualize cumulative distribution.
Part 2: Analyzing Data Sets with Statistical Measures
Data Sets
Set A: {9,10,11,7,13}
Set B: {10,10,10,10,10}
Set C: {1,1,10,19,19}
Calculations of Means
The mean (average) is calculated by summing all data points and dividing by the number of points:
Set A:
(9 + 10 + 11 + 7 + 13) / 5 = 50 / 5 = 10
Set B:
(10 + 10 + 10 + 10 + 10) / 5 = 50 / 5 = 10
Set C:
(1 + 1 + 10 + 19 + 19) / 5 = 50 / 5 = 10
Calculations of Standard Deviations
The standard deviation measures the dispersion of data points around the mean. It is computed as: Standard deviation (s) = √[ Σ (xi - µ)² / (n - 1) ]
Where µ is the mean, xi are data points, and n is the number of points.
Set A
Deviations: (9 - 10)² = 1, (10 - 10)² = 0, (11 - 10)² = 1, (7 - 10)² = 9, (13 - 10)² = 9
Sum of squared deviations = 1 + 0 + 1 + 9 + 9 = 20
Variance = 20 / (5 - 1) = 20 / 4 = 5
Standard deviation = √5 ≈ 2.24
Set B
Deviations: all zeros, since all values are 10
Standard deviation = 0
Set C
Deviations: (1 - 10)² = 81, (1 - 10)² = 81, (10 - 10)² = 0, (19 - 10)² = 81, (19 - 10)² = 81
Sum of squared deviations = 81 + 81 + 0 + 81 + 81 = 324
Variance = 324 / (5 - 1) = 81
Standard deviation = √81 = 9
Conclusion
The analysis of the grades and the statistical measures on the data sets reveals several critical insights. Constructing an absolute frequency distribution helps categorize student performance, while converting to relative and percentage distributions allows for easier comparison and interpretation. The cumulative “less than” and “more than” distributions facilitate understanding of data accumulation, which is vital for creating visual representations like histograms and ogives.
Furthermore, calculating the mean and standard deviation of the datasets illustrates how data is centered and dispersed. Notably, while Sets A and B share the same mean, their variability differs markedly, as shown by the standard deviation. Set C demonstrates a much higher spread, emphasizing the diversity in data points.
References
Freund, J. E. (2010).
Modern Elementary Statistics
. Pearson Education.
Dalgaard, P. (2008).
Introductory Statistics with R . Springer.
Levitan, M. (2000). Practical Data Analysis. SIAM.
Devore, J. L. (2011).
Probability and Statistics for Engineering and the Sciences . Brooks Cole.
Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
ISO/IEC 25012:2008. Software engineering Software product Quality Requirements and Evaluation (SQuaRE) — Data quality model.
Moore, D. S., & McCabe, G. P. (2009).
Introduction to the Practice of Statistics
. W. H. Freeman.
Messick, S. (1989). Validity. In R. L. Linn (Ed.), Educational Measurement (3rd ed.). American Council on Education/Macmillan. Glen, S. (2010).
Introduction to descriptive statistics . Statistics How To.
McClave, J. T., & Sincich, T. (2012).