## Central Tendencies

The mean, the mode, and the median are the three measures of central tendency that are critical in describing a clustering pattern of distribution of a given data. Heiman (61) explains that the mean is a common measure of central tendency for it shows an average value of a variable measured on a continuous scale. A real-world example is the mean sales volume of Samsung’s mobile phones in the United States.

As an international company, Samsung may be interested in determining the mean sales volume of its mobile phones in the American market. In this view, the mean sales volume is essential for it indicates the number of mobile phones that Americans purchase in a week, a month, or a year. The distribution of sales volume over a period provides the basis for Samsung to project its sales and produce a commensurate number of mobile phones to satisfy ensuing demand.

The major advantage of the mean is that it provides an accurate estimate of a certain variable for it considers every value in the distribution spectrum. However, the disadvantage of the mean is that it overlooks the influence of outliers on skewness and kurtosis of data. In essence, the mean gives an accurate measurement of central tendency when the data follow the normal distribution.

The mode is another measure of central tendency, which indicates the most frequent value in the distribution of data. Variables on continuous or categorical scales have the mode. As a real-world example, the model of Samsung’s smartphone model is sold in the United States. Since Samsung produces diverse models of smartphones, the determination of the mode is critical in identifying models that register the highest sales.

The analysis of sales records provides not only the number and the mean of sales volume but also the mode of smartphones that dominate a particular market. The knowledge of the modal smartphone enables marketers and sales agents of Samsung to stock smartphones that sell best in their respective markets. Moreover, the understanding of the modal smartphone allows Samsung to produce certain models of phones than others in response to market demands and trends.

The mode is advantageous for it easily reveals the most frequent value without any calculations (Heiman 62) The disadvantage with the mode is that the value does not reveal a unique attribute of data because more than one value of mode may exist, and thus, complicates the interpretation of data. Other disadvantages are that the mode is very rare in continuous data and does not give an accurate measure of central tendency, especially when the mode lies in the outlier region of the normal distribution.

As a measure of central tendency, the median is the value that is in the middle of the distribution when data is organized in an ordinal manner from the smallest value to the largest one. If a distribution has an odd number of values, the middle value becomes the median but if a distribution has even numbers, the average of the two middle values becomes the median (Heiman 64). For instance, the median income for households is a real-world example of the use of the median in determining the economic status of people.

The median income of households is the level of income that divides people into lower and upper economic classes. The advantage of the median is that outliers do not have a significant influence on their value within the distribution. Nevertheless, the disadvantages of the median are that it is not a unique value and requires extensive computation.

## Continuous Random Variables

The distribution of continuous random variables exhibits the normal probability distribution. The mean and the standard deviation are two parameters that define the distribution of data in the normal probability distribution. A continuous random variable with infinite values follows the normal distribution with the equal values of mean, mode, and median, symmetric curve, and the area under the curve as one.

A real-world example of the data that exhibit normal distribution is the lifetime of smartphones. Depending on numerous factors, such as damage, battery life, usage, and obsolescence, the lifetime of smartphones varies from one person to another. Smartphones have a short lifetime of one to six years because they are fragile, endure constant usage, have weak batteries, and become obsolete due to technological evolution.

Manufactures and customers of smartphones grapple with the challenge of a short lifetime, and thus, they want to prolong the lifespan of their devices. In this view, the normal distribution provides a robust way of depicting the distribution of smartphones based on their lifetime.

As the distribution of smartphones’ lifetime varies from 1-6 years, the normal distribution would indicate that lifetimes of the majority smartphones are between 3-4 years while equal proportions would have lifetimes of 1-2 years and 5-6 years. Fundamentally, smartphones with lifetimes of less than 3 years have a short lifespan while those with lifetimes of more than 4 years have a long lifespan. Additionally, smartphones with between 3 and 4 years have an average lifespan. Comparisons of lifetimes would indicate if a smartphone has a lifetime that is equal, greater, or less than the average lifetime.

There are three instances where the distribution lifetimes of smartphones would exhibit different distribution patterns. In the first instance, the lifetime of smartphones would follow the normal distribution when there is no protection and careful usage. Such a distribution would be a two-tailed because the distribution of lifetimes would be greater or less than the mean, the mode, or the median of the normal distribution.

In the second instance where there are careless usage and poor maintenance of the smartphones, it implies that the distribution would exhibit a positive skew since most smartphones would have lifetimes of less than the average values. When the median is less than the mean, a positive skew occurs (Terrel 387). The skewed distribution is one-tailed because it occurs on the left side of the normal distribution.

Moreover, in the third instance where customers and manufacturers of smartphones devise ways of prolonging lifespan by improving usage and protection of the fragile devices, a lifetime of a smartphone would increase. In the normal distribution, the lifespan would be greater than the average value, which means that the distribution of data would exhibit a negative skew. According to Terrel (387), a negative skew ensues when the median is greater than the mean. Thus, a negative skew is one-tailed for most data would be on the right side of the distribution curve.

Usually, continuous variables have infinite or finite values that depict their distribution when mapped on the normal probability distribution. Depending on the mean, the mode, and the median for data, the distribution of data can exhibit the normal distribution, a negative skew, or a positive skew. In the case of the lifetime of smartphones, it is apparent that the distribution takes different patterns. A two-tailed distribution occurs when there is a normal distribution of data while a one-tailed distribution occurs when there is a skewed distribution. These tailed instances of smartphones’ lifetime distributions show contrasting distribution patterns, which deviate from the normal distribution.

## Works Cited

Heiman, Gary. *Basic Statistics for the Behavioral Sciences.* New York: Cengage Learning, 2013.

Terrell, Steven. *Statistics Translated: A Step-by-Step Guide to Analyzing and **Interpreting Data*. New York: Guilford Press, 2012.