Title: CENTRAL LOCATION AND DISPERSION

1. How can using the arithmetic mean help you measure the central location?

Central tendency is the central value of a frequency distribution. In most cases, it is regarded as the central number in a set of data. On the other hand, arithmetic mean is the most used measure of central tendency as compared to other measures of central tendency like median and mode. Arithmetic mean is also referred to as the average or the mean and is defined as the total of numeric values of all observations and the total is divided by the total number of observation (McCullough and Deborah, 2007). For instance, if a class had a total of five students and they had 23, 45,40,36 and 30 marks, then the arithmetic mean is the sum of their marks divided by 5 ( Learning Express, 2005) as shown below;

AVERAGE= {23+45+40+36+30} ÷5 = 34.8

Therefore, the arithmetic mean for the students is 34.8 which is the central value of the frequency distribution and the value is the central one for the student’s marks.

2. Why is it important that the central limit theorem has a minimum requirement for sample sizes?

The distribution of the total value of a huge number of independent variables will be normal not considering the initial distribution if the variables are identically distributed (Adams and William, 1974). The law as stated above sets a minimum requirement for sample space of 30 is required in the central limit theorem because if the size is small then the power of such a test may be quite low meaning that it will be unlikely to get a significant result statistically even if the theory is correct. It is also important because with the central limit theorem it is hard to overstate thus the working of many statistical procedures is made possible (Fischer and Hans, 2011).

For example, if Y= (Y1, Y2….) is a set of numbers with common probability density function f, the mean is µ and its variance is α, it is assumed that 0<α<∞0<∞ so that the random variables are random and not constants if it is composed of independent and identically distributed random variables.

References

Adams, William J. The Life and Times of the Central Limit Theorem. New York: Kaedmon Pub. Co, 1974.

Fischer, Hans. History of the Central Limit Theorem: From Classical to Modern Probability Theory.NewYork:Springer,2011.

LearningExpress (Organization). Math Skills Success Course. 4, 4. New York, N.Y.: LearningExpress, 2005.

MacCullough, Deborah L. A Study of Experts’ Understanding of Arithmetic Mean. 2007.