Statistical Analysis of Results - Research Paper Example

Statistical Analysis of Research Results [Insert Here] [Insert Here] [Insert April 2009 Statistical Analysis of Research Results Percentiles indicate the position of a value with respect to the entire data set. That is, a student in the pth percentile performed better than approximately p-percent of the class. They are also useful in comparing an individual student's performance to the rest of the class or particular group that they belong to ("What is percentile rank, and what does it mean", 2008). Thus, prior to any statistical analysis, computing for the corresponding percentile rank of each grade in the dataset would give an idea of the grade distribution for the population. Below is the outline of the steps to take when computing for the percentile rank: 1) Group data into fixed intervals (here, we compute for the percentile for each score so this step is not necessary) 2) Count the frequency (f), or the number of times that a particular score appears in the data set. 3) Compute for the cumulative frequency (cf). 4) Percentile Rank is then computed as: Where: p = Percentile Rank for the ith score Cf = Cumulative Frequency of the ith score f = Frequency for the ith score N = Total Number of observations Computations for Grade 11 Math Assessment Score and Grade 11 SAT scores for Math follow on the next page. Table1: Grade 11 Math Assessment Score Frequency Table 2: Grade 11 SAT Score (Math) Frequency Distributions Distributions and Percentile Ranks and Percentile Ranks Grade 11 Math Score Frequency Cumulative Frequency Percentile Rank SAT Math Score Frequency Cumulative Frequency Percentile Rank 0 1 1 1 280 1 1 1 177 1 2 1 300 3 4 3 185 1 3 2 330 1 5 3 192 2 5 3 340 2 7 4 202 3 8 5 380 2 9 5 205 2 10 5 390 2 11 6 208 1 11 6 400 1 12 6 213 1 12 6 410 2 14 7 214 3 15 8 420 5 19 11 217 2 17 9 430 3 22 12 219 1 18 9 440 1 23 12 221 2 20 10 450 2 25 13 222 2 22 11 460 3 28 15 224 2 24 12 470 5 33 18 227 4 28 15 480 6 39 21 228 1 29 15 490 7 46 25 230 4 33 17 500 10 56 30 231 2 35 18 510 7 63 33 232 2 37 19 520 7 70 37 234 5 42 22 530 9 79 42 235 3 45 23 540 3 82 42 237 1 46 23 550 3 85 43 238 5 51 26 560 4 89 45 239 5 56 29 570 8 97 50 241 2 58 29 580 11 108 56 242 3 61 31 590 13 121 63 243 5 66 34 600 9 130 67 245 2 68 34 610 8 138 71 246 7 75 39 620 10 148 76 247 6 81 42 630 6 154 78 248 4 85 43 640 7 161 82 250 7 92 47 650 12 173 89 251 3 95 48 660 9 182 93 252 6 101 51 670 2 184 92 253 8 109 56 680 1 185 92 255 9 118 61 690 5 190 96 256 6 124 63 710 4 194 98 257 9 133 68 720 1 195 97 258 7 140 71 730 1 196 98 259 6 146 74 760 1 197 98 261 5 151 76 770 1 198 99 262 4 155 78 780 2 200 100 263 5 160 80 790 1 201 100 264 2 162 81 265 7 169 85 266 8 177 90 267 6 183 92 269 2 185 92 270 12 197 100 271 2 199 99 300 2 201 100 Data from Table 1 shows that 51% of the students scored below the Grade 11 Math Score of 252 indicating that half of the populations may be having difficulty with the subject. However we see from Table 2 that the median SAT Math Score of the students is 570. Since SAT scores range from 200-800, a score of 570 for Math may be considered as above average and meets the minimum requirement of some major public universities in the country. With this result, it is of further interest to see consider some factors which have effect on SAT scores. The dataset contains 7 variables namely: Student Identifier Number, GPA, Track Rank, Grade 8 Science Assessment Score, Grade 8 Language Assessment Score, Grade 11 Math Assessment Score and Grade 11 SAT Score for Math. It is of interest to determine the how these variables affect each other. Pearson's correlation, often denoted as r, can be used to compute for the degree of relationship two variables have with each other. The closer the value is to 1, the more associated the two variables are. Values for the Pearson's correlation coefficient may be either positive or negative. A negative value for Pearson's correlation indicates that as the value of one variable goes up, the value for its associated variables goes down or vice versa. A positive value for Pearson's correlation indicates that the values for both variables increase and decrease together. It is computed as follows: However, having high degrees of association does not necessarily mean that the relationship between these variables are statistically significant. A t-test normally accompanies the Pearson's Correlation coefficient to test for the statistical significance of the relationship. The null hypothesis for the test is: There is no significant relationship between the two variables. The t-value to compute the significance of a relationship is as follows: Where n = number of paired observations R = Pearsons correlation coefficient The null hypothesis is rejected only if the t-value is less than the critical value. P-values may also be used to determine the significance of the relationship. The smaller the p=value, the more significant the relationship between two variables will be. Table 3: Table of Sums and Sum of Squares by variable GPA 640.0467 2071.5089 TRACK RANK 582.20938 1753.3595 GRADE 8 ASSESSMENT SCORE (SCIENCE) 48760 11871276 GRADE 8 ASSESSMENT SCORE (LANGUAGE) 46754 10925254 GRADE 11 ASSESSMENT SCORE (MATH) 49644 12394950 Grade-11 SAT (Math section) Score 112940 65338200 Table 4: Cross Product Sum of Squares GPA TRACK RANK GRADE 8 ASSESSMENT SCORE (SCIENCE) GRADE 8 ASSESSMENT SCORE (LANGUAGE) GRADE 11 ASSESSMENT SCORE (MATH) GRADE 11 SAT (Math section) Score GPA 2071.51 1882.3 155350.43 148900.86 158472.9 363784.86 TRACK RANK 1882.3 1753.36 142244.04 136012.06 144820.46 334462.96 GRADE 8 ASSESSMENT SCORE (SCIENCE) 155350.43 142244.04 11871276 11317037 12050671 27565400 GRADE 8 ASSESSMENT SCORE (LANGUAGE) 148900.86 136012.06 11317037 10925254 11519336 26269990 GRADE 11 ASSESSMENT SCORE (MATH) 158472.9 144820.46 12050671 11519336 12394950 28218470 GRADE 11 SAT (Math section) Score 363784.86 334462.96 27565400 26269990 28218470 65338200 Making use of the formula for Pearson's Correlation coefficient a final table displaying the degree of relationship for all the variables in the data set is shown below. P-values were generated through the help of statistical software. Table 5: Pearson Correlation Coefficients and corresponding p-values GPA TRACK RANK GRADE 8 ASSESSMENT SCORE (SCIENCE) GRADE 8 ASSESSMENT SCORE (LANGUAGE) GRADE 11 ASSESSMENT SCORE (MATH) GRADE 11 SAT (Math section) Score GPA 1 0.66 Read More