- Miscellaneous
- Speech or Presentation
- May 23, 2018

Application of Functions, Distributions and Probability Concepts to Business Problems Quantitative methods in general have a wide range of uses in helping to make sound business decisions. Examples of the most useful techniques are probability, forecasting, data mining and time analysis. We shall first consider the application of exponential and reciprocal functions to business problems. Then, after briefly explaining frequency distributions and measures of central tendency and dispersion, we shall consider how these have also affected the writers thought processes regarding applying quantitative methods to business problems. Finally, we look at the usefulness of probability concepts, in particular the central limit theorem and in converting data to indexes.

Exponential and Reciprocal Functions

Exponential and reciprocal functions arise in a number of situations. The former occurs in growth situations, hence the term exponential growth in common parlance when there is slow initial growth but eventual rapid growth. In the business context, we come across this situation in relation to finance. For example, a new product would normally take time to become established, so sales are likely to be slow at first. However, as more and more people value the product and buy it (assuming they do), sales figures rise sharply. This function is also often "incorporated into models of failure rates, market sizes and many probability situations" (Curwin & Slater, 2008, p. 500). Thus, if the business problem is to forecast future growth potential so that the necessary resources can be arranged in time, past data can be extrapolated to provide this information.

Reciprocal functions are a reciprocal (or an inverse reflection) of another. For example, the reciprocal of a positive exponential function would be a negative exponential function. In other words, instead of growth, there would a decay. This is also a likely scenario in the case of business failure. For example, if the product referred to above turns out to be a flop, sales figures would rapidly decline as in a negative exponential function. Both exponential and reciprocal functions have affected my thought processes in the way they can aid in visualising information that follow these growth and decay patterns.

Frequency Distributions

A frequency distribution is defined as "the function of the distribution of a sample that corresponds to the probability density function of the given underlying population and tends to it as the sample size increases" (Borowski & Borwein, 1989, p. 233). By presenting a frequency distribution instead of raw data alone, the data becomes more manageable. It is constructed by noting, "the number of observations associated with each score value in a set of data that is quantitative in nature" (Yoder, 2008). A simple frequency distribution is drawn by making a two-column table, one for values of x (the score value), and the other for values of f (the number of observations).

Measures of central tendency (or average) and dispersion (or spread) are used to describe the features of the distribution. The former indicate the average or what is the typical value for all the set of values. The mode is the most frequently occurring value i.e. the one that occurs most often, the median value is the middle value when all the values are arranged in order, and the mean is usually the arithmetical mean figure obtained by summing all the values and dividing by their number. Other types of mean are geometric and harmonic mean. Measures of dispersion indicate the spread of the values, or "the degree to which the values of a frequency distribution are scattered around some central point" (Borowski & Borwein, 1989, p. 171). The range is the absolute difference between the least and greatest values, whereas standard deviation is the square root of the variance. Variance is obtained "by taking the expected value of the square of the difference between the random variable and its mean" (Borowski & Borwein, 1989, p. 623). A standard normal distribution has a mean of zero and a standard deviation of unity.

Businesses have to work with a variety of information and measures of central tendency and dispersion provide useful means of describing important business information. For example, a distribution can be made of the amount customers spend on a particular product per visit. The measure of central tendency would tell us the average amount a typical customer spends, and the measure of dispersion would indicate the range or spread of the amounts. Businesses can also observe trends and make predictions based on the quantitative data gathered. A roughly drawn standard normal distribution curve, described as a bell shape, is shown below.

Fig. 1: A standard normal distribution curve

Probability, Central Limit Theorem and Indexes

Probability has a very important role in statistical theory, quite simply because not everything is absolutely certain. It is central to business problems as well. When we make business decisions, we usually do so in the context of uncertainty, i.e. in between the probability scale from impossible (p=0) to most definite (p=1). The likelihood of things happening, whether in the business world or in general, are not always zero or unity. "If all the affairs of life were as clear-cut as this, statisticians would be out of a job" (Moroney, 1990, p. 5) and making business decisions would be straightforward. The best we can do therefore, is to estimate how sure we are and rely on these sureties to base important business decisions. In most cases, this entails collecting empirical data and analysing it to derive the probability values.

The Central Limit Theorem states "the fundamental statistical result that if a sequence of independent identically distributed random variables each has finite variance, then as their number increases, their sum (or, equivalently, their arithmetic mean) approaches a normally distributed random variable" (Borowski & Borwein, 1989, p. 75). What this means is that if we obtain a sufficient number of samples, then the mean of their values would tend to form a normal distribution as in fig. 1. Furthermore, this shows why it is necessary to take a sufficiently large number of samples that would be representative of the population. For example, a business cannot characterise a typical customer by just asking one or two customers even if they are selected at random. The sample would need to be large enough before we can describe customer behaviour in general, and before we can use this data to target customers better.

When we convert data to indexes, we usually do so because it helps to compare a variable in one period of time to the same variable in another period of time (Haeussler & Paul, 1987, p. 718). The latter period is known as the base period and each set of data forms a time series. For a business, the base period could be the first year of operation. Using indexes would then help to see how well the business is improving over time in various respects.

References

Borowski, E. J. & Borwein, J. M. (1989). Collins Dictionary of Mathematics. HarperCollins Publishers.

Curwin, Jon & Slater, Roger. (2008). Quantitative Methods for Business Decisions. 6th edition. Cengage Learning EMEA.

Haeussler, Ernest F. & Paul, Richard S. (1987). Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences. 5th edition. Prentice Hall.

Moroney, M. J. (1990). Facts From Figures. Penguin Books.

Yoder, Marcel S. (2008). Frequency Distributions. Retrieved 25 April 2010 from http://otel.uis.edu/yoder/freq_h.htm.

Exponential and Reciprocal Functions

Exponential and reciprocal functions arise in a number of situations. The former occurs in growth situations, hence the term exponential growth in common parlance when there is slow initial growth but eventual rapid growth. In the business context, we come across this situation in relation to finance. For example, a new product would normally take time to become established, so sales are likely to be slow at first. However, as more and more people value the product and buy it (assuming they do), sales figures rise sharply. This function is also often "incorporated into models of failure rates, market sizes and many probability situations" (Curwin & Slater, 2008, p. 500). Thus, if the business problem is to forecast future growth potential so that the necessary resources can be arranged in time, past data can be extrapolated to provide this information.

Reciprocal functions are a reciprocal (or an inverse reflection) of another. For example, the reciprocal of a positive exponential function would be a negative exponential function. In other words, instead of growth, there would a decay. This is also a likely scenario in the case of business failure. For example, if the product referred to above turns out to be a flop, sales figures would rapidly decline as in a negative exponential function. Both exponential and reciprocal functions have affected my thought processes in the way they can aid in visualising information that follow these growth and decay patterns.

Frequency Distributions

A frequency distribution is defined as "the function of the distribution of a sample that corresponds to the probability density function of the given underlying population and tends to it as the sample size increases" (Borowski & Borwein, 1989, p. 233). By presenting a frequency distribution instead of raw data alone, the data becomes more manageable. It is constructed by noting, "the number of observations associated with each score value in a set of data that is quantitative in nature" (Yoder, 2008). A simple frequency distribution is drawn by making a two-column table, one for values of x (the score value), and the other for values of f (the number of observations).

Measures of central tendency (or average) and dispersion (or spread) are used to describe the features of the distribution. The former indicate the average or what is the typical value for all the set of values. The mode is the most frequently occurring value i.e. the one that occurs most often, the median value is the middle value when all the values are arranged in order, and the mean is usually the arithmetical mean figure obtained by summing all the values and dividing by their number. Other types of mean are geometric and harmonic mean. Measures of dispersion indicate the spread of the values, or "the degree to which the values of a frequency distribution are scattered around some central point" (Borowski & Borwein, 1989, p. 171). The range is the absolute difference between the least and greatest values, whereas standard deviation is the square root of the variance. Variance is obtained "by taking the expected value of the square of the difference between the random variable and its mean" (Borowski & Borwein, 1989, p. 623). A standard normal distribution has a mean of zero and a standard deviation of unity.

Businesses have to work with a variety of information and measures of central tendency and dispersion provide useful means of describing important business information. For example, a distribution can be made of the amount customers spend on a particular product per visit. The measure of central tendency would tell us the average amount a typical customer spends, and the measure of dispersion would indicate the range or spread of the amounts. Businesses can also observe trends and make predictions based on the quantitative data gathered. A roughly drawn standard normal distribution curve, described as a bell shape, is shown below.

Fig. 1: A standard normal distribution curve

Probability, Central Limit Theorem and Indexes

Probability has a very important role in statistical theory, quite simply because not everything is absolutely certain. It is central to business problems as well. When we make business decisions, we usually do so in the context of uncertainty, i.e. in between the probability scale from impossible (p=0) to most definite (p=1). The likelihood of things happening, whether in the business world or in general, are not always zero or unity. "If all the affairs of life were as clear-cut as this, statisticians would be out of a job" (Moroney, 1990, p. 5) and making business decisions would be straightforward. The best we can do therefore, is to estimate how sure we are and rely on these sureties to base important business decisions. In most cases, this entails collecting empirical data and analysing it to derive the probability values.

The Central Limit Theorem states "the fundamental statistical result that if a sequence of independent identically distributed random variables each has finite variance, then as their number increases, their sum (or, equivalently, their arithmetic mean) approaches a normally distributed random variable" (Borowski & Borwein, 1989, p. 75). What this means is that if we obtain a sufficient number of samples, then the mean of their values would tend to form a normal distribution as in fig. 1. Furthermore, this shows why it is necessary to take a sufficiently large number of samples that would be representative of the population. For example, a business cannot characterise a typical customer by just asking one or two customers even if they are selected at random. The sample would need to be large enough before we can describe customer behaviour in general, and before we can use this data to target customers better.

When we convert data to indexes, we usually do so because it helps to compare a variable in one period of time to the same variable in another period of time (Haeussler & Paul, 1987, p. 718). The latter period is known as the base period and each set of data forms a time series. For a business, the base period could be the first year of operation. Using indexes would then help to see how well the business is improving over time in various respects.

References

Borowski, E. J. & Borwein, J. M. (1989). Collins Dictionary of Mathematics. HarperCollins Publishers.

Curwin, Jon & Slater, Roger. (2008). Quantitative Methods for Business Decisions. 6th edition. Cengage Learning EMEA.

Haeussler, Ernest F. & Paul, Richard S. (1987). Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences. 5th edition. Prentice Hall.

Moroney, M. J. (1990). Facts From Figures. Penguin Books.

Yoder, Marcel S. (2008). Frequency Distributions. Retrieved 25 April 2010 from http://otel.uis.edu/yoder/freq_h.htm.