# Statistics: Understanding and Exploring Assumptions

Subject: Sciences 6 1484 7 min College

It is very vital in statistics to ensure that statistics test assumptions are met, different test statistics have diffeassumptionsption wish are based on the test being carried out. The assumptions ascertain that the result obtaobtained isnot biased in any way, this ensures the tests are reliable, objective, and accurate. They also allow the tests to be carried out in controlled experiments, facilitating comparison and the tests they to be carried out, as it is impossible to consider all variables involved in a test objectively (Weinberg, 2008, p. 690). Failure to adhere to the assumption renders the test to be bias, inaccurate, and not being objective toward the experiment being carried out.     Day one hygiene is normally distributed from the bell shape of the normal curve plotted in the histogram. Also from the p-p curvet, the expected cumulative probability is almost equal to the observed cumulative probability, thus concluding the data set is normally distributed.

Day two hygiene data set is not normally distributed as the normal curve plotted in the histogram is skewed to the right. Also from the p-p curvet, the expected cumulative probability varies a lot from the observed cumulative probability.

Day three hygiene data set is not normally distributed as the normal curve plotted in the histogram is skewed to the right. Also from the p-p curve, the expected cumulative probability varies a lot from the observed cumulative probability.

Standard Descriptive Statistics for Hygiene Data Set.

 Hygiene (Day 1 of Download Festival) Hygiene (Day 2 of Download Festival) Hygiene (Day 3 of Download Festival) N Valid 810 264 123 Missing 0 546 687 Mean 1.7934 .9609 .9765 Std. Error of Mean .03319 .04436 .06404 Median 1.7900 .7900 .7600 Mode 2.00 .23 .44(a) Std. Deviation .94449 .72078 .71028 Variance .892 .520 .504 Skewness 8.865 1.095 1.033 Std. Error of Skewness .086 .150 .218 Kurtosis 170.450 .822 .732 Std. Error of Kurtosis .172 .299 .433 Range 20.00 3.44 3.39

Day one hygiene computation results are mean =1.7934, median =1.7900, and mode =2.0, the descriptive statistic are approximately all equal Skewness is computed to be 8.865 implying the data set is asymmetric. Kurtosis is computed to be 170.450 implying a possibility of leptokurtic distribution. There is large variation in the data set; range = 20 and variance =0.892. From the computational result of the data set, normal distribution assumptions are not met, hence the data is not normally distributed

Day two hygiene data has a large variation between the mode, me, and, theme values (mean =0.9609, median =0.7900, mode =0.23). Skewness is computed to be 1.095 implying it is approximately symmetric. Kurtosis is computed to be 0.822 implying a possibility of mesokurtic distribution. There is a small l variation in the data set; range = 3.44 and variance =0.520. From the computational result of the data set normal distribution assumptions are met, hence the data is normally distributed

Day three hygiene data set has a large variation among the mode, mean and Indian values (mean =0.9765, mode =0.7600, median =0.44). Skewness is computed to be 1.033 implying the data set is approximately symmetric. Kurtosis is computed to be 0.732 implying a possibility of mesokurtic distribution. There is a small variation in the data set; range =3.39 and variance =0.504. From the computational result of the data set, normal distribution aassumptionsnsre met, hence concluding the data is normally distributed.

The results from computation are opposite of the observed results, this shows the subjective nature of results derived from observation

Standard Descriptive Statistics for Exam Data.

 Percentage on SPSS exam Computer literacy Percentage of lectures attended Numeracy N Valid 100 100 100 100 Missing 0 0 0 0 Mean 58.10 50.71 59.765 4.85 Median 60.50(a) 51.43(a) 62.000(a) 4.37(a) Mode 72(b) 54 48.5(b) 4 Std. Deviation 21.316 8.260 21.6848 2.706 Variance 454.354 68.228 470.230 7.321 Skewness -.107 -.174 -.422 .961 Std. Error of Skewness .241 .241 .241 .241 Kurtosis -1.105 .364 -.179 .946 Std. Error of Kurtosis .478 .478 .478 .478 Range 84 46 92.0 13
 University Statistic Std. Error Numeracy Dunce town University Mean 4.12 .292 95% Confidence Interval for Mean Lower Bound 3.53 Upper Bound 4.71 5% Trimmed Mean 4.06 Median 4.00 Variance 4.271 Std. Deviation 2.067 Minimum 1 Maximum 9 Range 8 Inter-quartile Range 3 Skewness .512 .337 Kurtosis -.484 .662 Sussex University Mean 5.58 .434 95% Confidence Interval for Mean Lower Bound 4.71 Upper Bound 6.45 5% Trimmed Mean 5.38 Median 5.00 Variance 9.432 Std. Deviation 3.071 Minimum 1 Maximum 14 Range 13 Inter-quartile Range 5 Skewness .793 .337 Kurtosis .260 .662 Percentage of lectures attended Dunce town University Mean 56.260 3.3619 95% Confidence Interval for Mean Lower Bound 49.504 Upper Bound 63.016 5% Trimmed Mean 56.544 Median 60.500 Variance 565.135 Std. Deviation 23.7726 Minimum 8.0 Maximum 100.0 Range 92.0 Inter-quartile Range 28.8 Skewness -.309 .337 Kurtosis -.383 .662 Sussex University Mean 63.270 2.6827 95% Confidence Interval for Mean Lower Bound 57.879 Upper Bound 68.661 5% Trimmed Mean 63.672 Median 65.750 Variance 359.849 Std. Deviation 18.9697 Minimum 12.5 Maximum 100.0 Range 87.5 Inter-quartile Range 30.0 Skewness -.365 .337 Kurtosis -.221 .662 Computer literacy Dunce town University Mean 50.26 1.141 95% Confidence Interval for Mean Lower Bound 47.97 Upper Bound 52.55 5% Trimmed Mean 50.12 Median 49.00 Variance 65.094 Std. Deviation 8.068 Minimum 35 Maximum 67 Range 32 Inter-quartile Range 12 Skewness .225 .337 Kurtosis -.515 .662 Sussex University Mean 51.16 1.203 95% Confidence Interval for Mean Lower Bound 48.74 Upper Bound 53.58 5% Trimmed Mean 51.36 Median 54.00 Variance 72.341 Std. Deviation 8.505 Minimum 27 Maximum 73 Range 46 Inter-quartile Range 9 Skewness -.538 .337 Kurtosis 1.379 .662 Percentage on SPSS exam Dunce town University Mean 40.18 1.780 95% Confidence Interval for Mean Lower Bound 36.60 Upper Bound 43.76 5% Trimmed Mean 40.06 Median 38.00 Variance 158.477 Std. Deviation 12.589 Minimum 15 Maximum 66 Range 51 Inter-quartile Range 18 Skewness .309 .337 Kurtosis -.567 .662 Sussex University Mean 76.02 1.443 95% Confidence Interval for Mean Lower Bound 73.12 Upper Bound 78.92 5% Trimmed Mean 75.86 Median 75.00 Variance 104.142 Std. Deviation 10.205 Minimum 56 Maximum 99 Range 43 Inter-quartile Range 12 Skewness .272 .337 Kurtosis -.264 .662  The descriptive statistics from the computation of the data set for the computer literacy, percentage of lectures attended and percentage on spss exam have met the normal distribution assumptions. Based on the results of skewness is approximately close to zero, kurtosis is in the range of -3<k<3, and means, mode, and median are approximately equal. The numeracy data set do not meet the normal distribution assumption hence not normally distributed, based on skewness, kurtosis, and central tendency descriptions from computation (Landau, 2004, p. 272).

 Levene’s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Computer literacy Equal variances assumed .064 .801 -.543 98 .588 -.900 1.658 -4.190 2.390 Equal variances not assumed -.543 97.728 .588 -.900 1.658 -4.190 2.390 Percentage of lectures attended Equal variances assumed 1.731 .191 -1.630 98 .106 -7.0100 4.3011 -15.5454 1.5254 Equal variances not assumed -1.630 93.400 .107 -7.0100 4.3011 -15.5507 1.5307

The Levene test significance value is 0.801 for computer literacy, hence the result for assumed equal variance for the groups is used to analyze the results. The t-test significance value is -0.543 and the mean difference confidence interval does not contain a zero, this indicates a significant difference between the two universities. This leads to the conclusion that computer literacy between Duncetown University and Sussex University is significantly different.

The Levene test significance value for lectures attended percentage is 0.191, hence the result for assumed equal variance for the groups is used to analyze the results. The mean difference confidence interval does not contain a zero; this indicates a significant difference between the two universities. This leads to the conclusion that the percentage of the lectures attended between Duncetown University and Sussex University are significantly different.

The assumption of normality assumes that the mean of different samples from the same population are equal and also equal to the mean of the population itself. Homogeneity of variance assumes that when comparing the variance of two populations based on their mean, the variance should be equal or variability between variables in two different populations is the same (Weinberg, 2008, p. 553). If the assumption of the homogeneity of variance is violated, a statistician may turn to other inference drawing techniques about the population mean comparison without impacting on the intended result. Ranked data technique may be used in case normality assumption is not observed without any impact on the desired result (Landau, 2004, p. 301). Degree of freedom correction methods may be used in case assumptions of variance homogeneity are violated without impact on desired results. Alternatively, the course of action is simply written “Proceed with caution”, this declares the statistician proceeds with t-test with caution as the assumption is violated.

## Reference List

Landau, S. (2004). A Handbook of Statistical Analyses Using SPSS. New York. Barnes & Noble.

Weinberg, S. (2008). Statistics using SPSS: An Integrative Approach. California. Wadsworth Publishing.