Introduction
Statistical concepts can assist in gaining insights from different data sets. Outliers can considerably skew mean, median, and mode, resulting in distorted depictions of data (Warner, 2021a). Identifying and solving such a situation is important for precise statistical analysis. In addition, sample size plays a crucial role in finding confidence intervals, influencing their accuracy and reliability. By determining the impact of outliers and the sum of squares, distinguishing between t and standard normal distributions, and considering the effect of N on CI, researchers can improve their data analysis and have accurate conclusions.
The Effect of Skewness on Measures of Central Tendency
Consider a situation where a class of 20 learners is given a math test. Most of the students perform well by scoring between 70 to 90%. Unfortunately, one of the students unintentionally provided wrong answers and got a score of zero. This outlier, or an extreme value, significantly affects the central tendency calculations.
Additionally, the students’ mean score will be lower due to the outlier (Warner, 2021a). The median will be lower since the outlier reduces the overall distribution of the scores. However, the mode representing the most frequent score will remain unaffected.
To identify such a scenario, an individual can plot a histogram to envisage the distribution of the math test scores. If a clear skewness is noticed, it indicates the existence of outliers (Warner, 2021a). To solve this issue, one way is to eliminate the outlier and recalculate the measures. However, when the outlier is valid and does not result from a data entry mistake, reporting the measures with and without the outlier may be suitable to offer a comprehensive understanding of the data.
Sum of Squares
The sum of squares (SS) is a statistical measure that offers information on the variability or dispersion in a dataset. It talks about the total variance for a set of numbers. The value of SS will be equal to 0 when each number in the data set is equal to the mean (Warner, 2021a). For example, if the data set contains similar numbers, the SS will be zero. However, SS can’t be negative since its formula involves the squares of a value that can be greater than or less than the mean, hence generating a non-negative number.
Sampling Distribution
A sampling distribution is a likelihood distribution constructed from a sample of data. It gives information on the distribution of a specific statistic in a population. σ offers insights into the population’s dispersion or spread of distinct values (Warner, 2021b). Understanding the theoretical sampling distribution’s characteristics and shape is essential for formulating inferences about a population concerning sample statistics. The knowledge of σ gives comprehension of the spread, accuracy, and dependability of the estimations obtained from various samples.
The t and Standard Normal Distribution
Although the t and standard normal distributions are utilized in statistical inference, they vary in some aspects. The t-distribution has a thicker tail compared to the standard normal distribution, showing that it is more probable to have extreme values (Thompson et al., 2020). The number of observations (N) will influence the confidence interval (CI) when using a t-distribution. A wide N will lead to a narrow CI, as more observations give more accurate information about the population.
Conclusion
With a good understanding of the effects of outliers, the attributes of SS, a comparison of t and normal distributions, and accounting for N in CIs, researchers can make informed analyses of their data. An outlier considerably impacts measures of central tendency in a dataset by skewing the outcome towards the end of the distribution. Although SS assists with the analysis of the data dispersion, it cannot offer insights into the shape of the dataset.
References
Thompson, G. Z., Maitra, R., Meeker, W. Q., & Bastawros, A. F. (2020). Classification with the matrix-variate-t distribution. Journal of Computational and Graphical Statistics, 29(3), 668-674. Web.
Warner, R. M. (2021a). Applied statistics I: Basic bivariate techniques (3rd ed.). Thousand Oaks, CA: Sage Publications.
Warner, R. M. (2021b). Applied Statistics II: Multivariable and Multivariate Techniques. Los Angeles, CA: Sage Publications.