The objective of the given task is to demonstrate the use of various techniques of descriptive statistics. Unlike inferential statistics which aims at estimating the characteristics of the population based on sample statistics, the descriptive statistics aim to outline the summary of the sample data. Also, descriptive statistics can be used to outline the linear relationship between different variables through the use of tools such as correlation. For the purpose of this report, a primary data of 30 students has been randomly selected through the use of a survey which aims to collect information related to GPA, study time, course and relevant demographic details such as age and sex. It is imperative to note that for the purpose of data collection only the final year students were considered. The data has been analysed in line with the objectives of the study using appropriate of descriptive statistics.
The given section presents the various tasks which need to be implemented with regards to the given task and the collected data.
The following survey has been used for the collection of data from the students.
The data that is being used for the given task has been obtained using the above mentioned survey which was distributed online and responses were solicited online only. Further, no personal details have been recorded as part of this survey so as to keep the anonymity of the respondents.
It is required to draw a frequency distribution table to summary the GPA of the sample information. In order to obtain the same, the various classes need to be defined which have been defined in line with the guideline provided in the question. Considering that the maximum possible GPA at the university is 4 with minimum of 0, the four classes are defined as follows.
1) 0.01 to 1.00
2) 1.01 to 2.00
3) 2.01 to 3.00
4) 3.01 to 4.00
It is noteworthy that class boundaries are also included in the respective classes that have been outlined above. For instance, a GPA value of 2.01 would fall in the 2.01 to 3.00 interval while a value of 2.00 would fall in the 1.01 to 2.00 interval. Taking into consideration the raw data collected, the respective frequency of each of the classes is distributed. The formula for cumulative frequency is indicated below and also it is noteworthy that for the first class interval, the cumulative frequency is equal to the respective frequency of the first class interval (Fehr & Grossman, 2014).
Cumulative frequency of a particular class = Cumulative frequency of previous class + Respective frequency of the given class
The requisite grouped data for GPA is shown below.
|
Class boundaries |
Mid-point |
Frequency |
Cumulative Frequency |
|
0.01 – 1.00 |
0.50 |
1 |
1 |
|
1.01 – 2.0 |
1.50 |
5 |
6 |
|
2.01 – 3.0 |
2.50 |
13 |
19 |
|
3.01 – 4.0 |
3.50 |
11 |
30 |
Mean
The computation of mean GPA based on the grouped data obtained above is indicated as follows (Hillier, 2016).
|
Mid-point (x) |
Frequency (f) |
Mid-point * Frequency (x* f) |
|
0.5 |
1 |
0.5 |
|
1.5 |
5 |
7.5 |
|
2.5 |
13 |
32.5 |
|
3.5 |
11 |
38.5 |
|
Total |
30 |
79 |
Mean =
Hence, the mean GPA of the given sample data would be 2.63.
Median
The computation of median GPA based on the grouped data obtained above is indicated as follows
|
Class boundaries |
Frequency |
|
0.01 – 1.00 |
1 |
|
1.01 – 2.0 |
5 |
|
2.01 – 3.0 |
13 |
|
3.01 – 4.0 |
11 |
Median is termed as middle value which would be (15th value + 16th value)/2. Here, 15th and 16th value would fall in the class boundaries of 2.01 – 3.0. The following formula can be used to compute median of grouped data (Hastie, Tibshirani & Friedman, 2016).
Median =
Where,
L= Lower class boundaries of the median group = 2.01
N = total number of observations =30
B =Cumulative frequency of the group earlier than median group = 6
G = Frequency of the median group = 13
W =Group width = 0.99
Now,
Median =
Hence, the median GPA of the given sample data is 2.70.
Mode
The computation of modal GPA based on the grouped data obtained above is indicated as follows.
|
Class boundaries |
Frequency |
|
0.01 – 1.00 |
1 |
|
1.01 – 2.0 |
5 |
|
2.01 – 3.0 |
13 |
|
3.01 – 4.0 |
11 |
Maximum frequency falls in the range of 2.01 -3 and hence, it would be termed as modal group. The relevant formula for mode computation in grouped data is indicated below (Hair, Wolfinbarger, Money, Samouel & Page, 2015).
Hence, the mode of the sample GPA would be 2.80.
The suitable chart summarising the information indicated in each of the given variables is as exhibited below.
The relevant graph to summarise the age distribution is indicated below.
From the above graph, it is apparent that the distribution of respondents across age is quite similar. This is confirmed from the fact that the minimum representation of a given age is 5 respondents while the maximum is 7. This ensures that the data is not skewed with regards to a particular age. Having the representation of higher ages is imperative as it may have significant relationship with academic performance.
The relevant graph to summarise the gender distribution is indicated below.
It is apparent from the above pie chart that both the genders are quite well represented in the survey. This ensures that an accurate representation is presented considering the fact that gender may have a significant influence on the academic performance and also the self-study time. Thereby, it was necessary to have a healthy distribution of both genders.
The relevant graph to summarise the program that respondents have enrolled for is indicated below.
It is evident from the above graph that the representation of the educational program across the respondents is also quite uniform. This has been deliberately done so as to ensure that difficulty across the programs should not lead to skewing of the sample data.
The relevant graph to summarise the self-study time for respondents on average daily is indicated below.
It is apparent from the above chart that maximum respondents tend to lie in the daily self-study time group on an average of 2.8 hours – 3.8 hours. Representation is thin at the extremes, which is on expected lines. This is indicative that majority of the students tend to devote on an average 1.8 hours to 3.8 hours on self-study daily.
The histogram of GPA of the survey respondents is indicated below.
From the above histogram, it is apparent that the shape of the GPA distribution is not symmetric since the left hand tail is not equal to the right hand tail. Infact, the tail on the left side is significantly greater than the corresponding tail on the right side. This is indicative of the presence of left or negative skew. Considering that skew is present, hence it can be concluded that the distribution of GPA for the given respondents is not normal (Eriksson & Kovalainen, 2015). Majority of the respondents tend to have a high GPA and only a few respondents tend to have a GPA lower than 2.
The correlation between the given two variables i.e. GPA and self-study related time has been analysed using the correlation coefficient which has been computed using EXCEL as the enabling tool.
It is apparent from the above output that the correlation between the given two variables is 0.1947. The correlation coefficient is positive which implies that a better GPA would be scored by the student who tends to devote a higher daily hours on average for self-study (Flick, 2015). However, the magnitude of the correlation coefficient is quite small which implies that the correlation between the given variables is weak and thus would not be of much significance (Hillier, 2016). This is not surprising considering there are other variables such as attendance in class, underlying IQ, cultural background that would also tend to have an impact on the academic performance of students.
Conclusion
It is apparent from above that a survey has been used for data collection which is then used for grouping of the raw data GPA. This grouped data has four classes across which 30 students GPA is distributed. Using the grouped data regarding the GPA, the mean, median and mode have been computed as 2.63, 2.70 and 2.80 respectively. Besides, using the suitable graphs the summary of the different variables in the sample data has been represented. An interesting aspect is that the underlying sample is quite heterogeneous. Based on the GPA histogram, it can be concluded that the sample GPA distribution is asymmetric and negative skew is present. Also, the correlation between the time spent daily on self-study on GPA is positive but indicative of weak correlation owing to low value of 0.1947. This is indicative of other factors being present on which the GPA would depend.
References
Eriksson, P. & Kovalainen, A. (2015). Quantitative methods in business research (3rd ed.). London: Sage Publications.
Fehr, F. H., & Grossman, G. (2014). An introduction to sets, probability and hypothesis testing (3rd ed.). Ohio: Heath.
Flick, U. (2015). Introducing research methodology: A beginner’s guide to doing a research project (4th ed.). New York: Sage Publications.
Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., & Page, M. J. (2015). Essentials of business research methods (2nd ed.). New York: Routledge.
Hastie, T., Tibshirani, R. & Friedman, J. (2016). The Elements of Statistical Learning (4th ed.). New York: Springer Publications.
Hillier, F. (2016). Introduction to Operations Research. (6th ed.). New York: McGraw Hill Publications.
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