# Statistical Analysis in Finance

## Statistical Distributions

By the end of this section, you will be able to:
1. Construct and interpret a frequency distribution.
2. Apply and evaluate probabilities using the normal distribution.
3. Apply and evaluate probabilities using the exponential distribution.

### Frequency Distributions

A frequency distribution provides a method to organize and summarize a data set. For example, we might be interested in the spread, center, and shape of the data set’s distribution. When a data set has many data values, it can be difficult to see patterns and come to conclusions about important characteristics of the data. A frequency distribution allows us to organize and tabulate the data in a summarized way and also to create graphs to help facilitate an interpretation of the data set.
To create a basic frequency distribution, set up a table with three columns. The first column will show the intervals for the data, and the second column will show the frequency of the data values, or the count of how many data values fall within each interval. A third column can be added to include the relative frequency for each row, which is calculated by taking the frequency for that row and dividing it by the sum of all the frequencies in the table.

### Normal Distribution

The normal probability density function, a continuous distribution, is the most important of all the distributions. The normal distribution is applicable when the frequency of data values decreases with each class above and below the mean. The normal distribution can be applied to many examples from the finance industry, including average returns for mutual funds over a certain time period, portfolio values, and others. The normal distribution has two parameters, or numerical descriptive measures: the mean, , and the standard deviation, . The variable x represents the quantity being measured whose data values have a normal distribution.
The curve in  is symmetric about a vertical line drawn through the mean, . The mean is the same as the median, which is the same as the mode, because the graph is symmetric about . As the notation indicates, the normal distribution depends only on the mean and the standard deviation. Because the area under the curve must equal 1, a change in the standard deviation, , causes a change in the shape of the normal curve; the curve becomes fatter and wider or skinnier and taller depending on . A change in  causes the graph to shift to the left or right. This means there are an infinite number of normal probability distributions.
To determine probabilities associated with the normal distribution, we find specific areas under the normal curve, and this is further discussed in Apply the Normal Distribution in Financial Contexts. For example, suppose that at a financial consulting company, the mean employee salary is $60,000 with a standard deviation of $7,500. A normal curve can be drawn to represent this scenario, in which the mean of $60,000 would be plotted on the horizontal axis, corresponding to the peak of the curve. Then, to find the probability that an employee earns more than $75,000, you would calculate the area under the normal curve to the right of the data value $75,000.
Excel uses the following command to find the area under the normal curve to the left of a specified value:
For example, at the financial consulting company mentioned above, the mean employee salary is $60,000 with a standard deviation of $7,500. To find the probability that a random employee’s salary is less than $55,000 using Excel, this is the command you would use:
Thus, there is a probability of about 25% that a random employee has a salary less than $55,000.

### Exponential Distribution

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, a finance professional might want to model the time to default on payments for company debt holders.
An exponential distribution is one in which there are fewer large values and more small values. For example, marketing studies have shown that the amount of money customers spend in a store follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The random variable for the exponential distribution is continuous and often measures a passage of time, although it can be used in other applications. Typical questions may be, What is the probability that some event will occur between x1 hours and x2 hours? or What is the probability that the event will take more than x1 hours to perform? In these examples, the random variable x equals either the time between events or the passage of time to complete an action (e.g., wait on a customer). The probability density function is given by
where  is the historical average of the values of the random variable (e.g., the historical average waiting time). This probability density function has a mean and standard deviation of .
To determine probabilities associated with the exponential distribution, we find specific areas under the exponential distribution curve. The following formula can be used to calculate the area under the exponential curve to the left of a certain value:

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A frequency distribution provides a method of organizing and summarizing a data set and allows us to organize and tabulate the data in a summarized way. Once a frequency distribution is generated, it can be used to create graphs to help facilitate an interpretation of the data set. The normal distribution has two parameters, or numerical descriptive measures: the mean, , and the standard deviation, . The exponential distribution is often concerned with the amount of time until some specific event occurs.

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### Normal Distribution Stock Return Calculations
