Understanding Statistics and Measures of Shape - Essay Example

It appears that the average mean score given by peers to students from private schools is higher than that of students from public schools as far as values are concerned. However, further comments can only be made after the application of advanced statistical tools like hypothesis testing which may help to reveal if there is any significant statistical difference between the average score from two populations (public and private schools). According to Jain and Sandhu, skewness is the measure of asymmetry in the distribution of population and tells if the mean, mode, and median are equal and quartiles are at equal distance from the median (3.2). The distribution pattern of the data as seen from the histogram revealed that scores given by peers to both the public schools and private schools are moderately positively skewed having skewness of 0.934 and 0.754 respectively.

This suggests that the data is not evenly distributed on both sides of the mean for both public and private schools. The positive skewness confirmed that the majority of the students both from public and private schools were given a mean score by peers less than or on the average score and very few were given scores higher than the average. Looking at the distribution pattern between the two histograms, we see that number of students that scored higher than the average score is greater for private schools than the public schools.

The box plot for the mean score given by peers revealed that there were four outliers for public schools and three outliers from private schools towards the upper side of the box plot. This suggested that a nearly equal number of students from public and private were given an outstanding score by peers (closer to 5).
First Quartile SAT (Combined)
According to Cook and Upton, median divides the order population data into two groups whereas quartiles divide the population into four groups with first quartile (lowest 25%), second quartile (lowest 50%), and the third quartile indicating the lowest 75% of the population group (53).

In this section, a statistical comparison of the first quartile of SAT Combined score for students from public and private schools is carried out. We see that the mean for SATQ1 for public and private schools is 943.39 and 1000.4 respectively. Again, as far as the values are concerned it appears that the students in the first quartile of the population from private schools scored better on SAT (Combined) than those from public schools. However, advanced tools like hypothesis testing can be used to see if there is a significant statistical difference between the average score of two populations.

Also, we see that data distribution for both public and private schools has skewness values of 0.418 and 0.449 respectively. A value of skewness between -0.5 and +0.5 suggests distribution be approximately symmetric (TC3.com). Also, the variance for public and private schools is 13140.55 and 26215.29 respectively. These values indicate that the data for private schools have more widespread than that for public schools. Based on this two deductions can be made; firstly, there are slightly less number of students in the first quartile from both public and private schools who scored on SAT (Combined) higher than the mean value, secondly, the data of SATQ1 for private schools have higher spread from mean value than the data of SATQ1 for public schools.

Although positively skewed, the scores of SAT (Combined) from public schools remained closer to its mean value of 943.39 as compared to scores from private schools. Also, the box plot for SATQ1 scores revealed two outliers each on the upper side of the boxplot for both public and private schools. This suggests that there were two students each in the first quartile of public and private schools that score extraordinarily than the other students in their population.