Business Education: Basics of Computation and Quantitative Analysis

Calculating Annual Profits from Sales Data
Companies calculate annual profit to ascertain how much money they have brought in through the year. The first step in getting the annual profit is to add up all of the revenues. This may include sales from different products or other sources of income. Next, all of the costs for the year must be added together. This may include many different types of costs.
Companies need to subtract to account for returns. They need to consider all expenses, including but not limited to operating expenses, depreciation expenses, interest, etc. Finally, they will take the total costs and subtract that from the total income. This will give the profit. For instance, if a company has revenues of $100,000 and costs of $50,000, then it made $50,000 in profit. One of the best ways to get all the data for this calculation is from the income statement, which lists revenues and expenses.

Profit Margins
The profit margin statistic makes it possible to compare companies of different sizes because it is expressed as a percentage.
To calculate profit margin, first the raw profit is calculated. This is done by subtracting total costs from sales revenue. For instance, if sales of a product made $10,000 for the company and costs were $8,000, then raw profit will be $10,000–$8,000, or $2,000.

Next, the gross profit is divided by the revenue.
Thus $2,000 would be divided by $10,000 for an answer of 0.2.

The profit margin is this number expressed as a percentage. To get this, a person would multiply the decimal by 100, thus 0.2 is equal to a 20 percent profit margin.

Sale Prices
The sale price of an item is the price minus the discount, which is calculated by the percentage off. To calculate the discount, one should multiply the percentage off and the full price. For instance, to calculate the sale price of an item that is $100 and 20 percent off, first the percentage should be transformed to decimal form, which is done by moving the decimal point to the left by two digits. Thus 20 percent is the same as 0.2, five percent is the same as 0.05, and one percent is the same as 0.01. To calculate the discount, one would multiple 0.2 by 100. This equals $20.

The sale price is calculated by taking the full price and subtracting the discount.
In this example, it is $100 – $20. Thus the sale price is $80.

Commissions
Many employees are paid on commission; thus, it is important to understand how to properly calculate them. First, the employee must understand the amount of money that is eligible for commission. For instance, a salesman might receive commissions when selling certain products but not others. Some people get a commission on sales while others receive profit-based commissions. Next, employees should ascertain the percentage commission they receive. This may vary.

For instance, a salesman might receive a different percentage of commission depending on what he sells. Once he knows the commissionable amount and rate of commission, he will then multiply them together. For instance, if an employee receives a 10 percent commission on
$100, then he will receive $10 in commission (0.1 × $100). The employee will then need to calculate taxes on the commission to ascertain his final pay.

Calculating Hourly Wages from an Annual Full-Time Salary
It is simple to calculate hourly wages from an annual full-time salary. First, the amount of hours worked in the year must be calculated. If the person worked 40 hours a week, then the number of hours worked in the year would be 40 × 52, or 2,080 hours. If they worked 35 hours a week, then it would be 35 × 52, or 1,820. Generally, full-time indicates a
40-hour work week, so 2,080 should be assumed if there is no other information except that it is full-time. To calculate the hourly wage, simply divide the full-time salary by 2,080. Thus if the person earned $41,600, then the hourly salary would be $41,600/2080, or $20 an hour.

Percentages
To calculate a percentage from a decimal, one has to simply multiply it by 100. Thus, 0.15 is 15 percent. To increase a number by a certain percentage (like when a product is marked up by a percentage), a person would first calculate the increase by multiplying the percentage by the cost. For instance, if the cost of a good is $100 and the company wants to mark it up 50 percent, then the markup would be $100 multiplied by 50 percent (which is the same as 0.5), and that would equal $50. Thus, the cost plus the markup would be $150. A person can also use it to find wholesale prices. If the wholesale price is 75 percent of the price of $100, then to get the wholesale price, a person would just multiply $100 by 75% or 0.75 to get a wholesale price of $75.
Percentages are used in many business applications.

Compound Interest
Compound interest is when interest gets added back to the principal and then interest is calculated on that additional amount when the next compounding period happens.

The formula to calculate compound interest is
.

In this formula, A represents the investment's future value (with interest). P is the principal or initial investment, r is the annual interest rate expressed as a decimal, n represents how many times the interested is compounded annually, and T is how long the money has been invested (in years). For instance, if an amount of $10,000 was invested, the interest rate was 0.1 and it is compounded monthly, then after 20 years, the investment value would be
,which equals
$73,280.74.

Frequency Distributions
Frequency distributions are tables that show how often specific results emerged. For instance, a teacher might do a frequency table showing how often the students got an A, B, C, D or F. The teacher would create the table listing the grades on the left and then put the number of students that achieved each one to the right. To find out how many earned a grade, the person could look at the table.
Frequency tables are used often in business. For instance, if one needed to find out how often sales reached a certain level, he (or she) could put the data into a frequency table and see how often that occurred. Some frequency tables will also list a cumulative total; this is good if the person wanted to see how often sales reached at least a certain level.

Mean, Mode, Median, and Standard Deviation
Mean is simply the average of a group of numbers. This is calculated by adding all the numbers and dividing by how many numbers there are. For instance, if the numbers are 3, 2, 3, and 4, one would add them to get 12 and then divide by 4 to get 3 as the mean. The mode is simply the number that appears the most. In the group 3, 2, 3, and 4, the number 3 is the mode because it appears twice. The median is the number that would appear in the middle if the numbers were put in order.
If the numbers are 7, 3, 9, 1, 5, the numbers would be put in order to 1, 3, 5, 7, 9, and the mode would be 5 because that is in the middle. To get the standard deviation, the mean should be calculated first. Take each number and subtract the mean, then get the squared difference by squaring the result.
Finally, take all the squared differences in the group and find their average.
That is the standard deviation.

Sampling Techniques and Selection
There are different methods of sampling used when doing research. The basic categories are probability sampling, in which every individual has a chance of getting selected in a known probability; and nonprobability sampling, in which some units may not have a chance of selection, or their probability isn't known. Simple random sampling is the best probability method, but is often very difficult to achieve. Every individual in the population has the same chance of being selected. Systemic sampling (also known as Nth name selection technique) takes every Nth individual from the population. With stratified sampling, the population is divided into stratums, or groups with a shared characteristic such as gender or age. These are matched against how many actually reside in the population. Then the researcher uses random sampling to take the number from each stratum that represents its presence in the population. With convenience sampling, the researcher finds a convenient sample. This is not as accurate as other methods. With judgment sampling, the researcher uses his judgment to find the sample. Quote sampling starts out like stratified sampling but, instead of random sampling, they use judgment or convenience sampling to pick the individuals from the stratums.

Line Graphs
A line graph shows the connection between information. It is often used to show changes over time. For instance, a company may want to show how sales changed from month to month. On the vertical axis, they would label the different sales levels, including both the lowest and highest possible amount. On the horizontal axis, the months would be listed. For every month, a dot would be placed on the sales level. Once all the dots are posted for the different months, a line would be drawn to connect the dots to more easily show the relationship. In this case, mangers could see whether sales went up, went down, stayed stagnant, or had no general pattern as the months went by. Line graphs can be useful visual aids for businesses.

Bar Graphs
A bar graph can be used to show data in a way that is clear and easy to understand. The first step is the title. For instance, a baseball coach may create a bar graph of hits made by players and call it 'Hits ade By Players.' Next, a label for each axis must be given. There will be a grouped data axis representing the groups such as 'Baseball Players.' This can be either the horizontal or vertical axis. In this case, it will be at the bottom of the graph. A label should be created for the frequency axis such as 'Number of Hits.' This should be on the left side of the graph. A range also needs to be produced. If the best hitter had 18 hits, then the scale might go from 0-20. The scale interval should be considered. There could be one mark for each hit or a marker to signify every five hits. Now the data is used to draw in bars to the correct height. For instance, the player that made 10 hits will have a bar that rises to the 10 marker. This will be repeated for all the players.

Circle Graphs
Circle graphs are useful visual aids that show a variety of data. First, a circle is formed. Then the circle is divided into different sectors to represent the different elements that the graph is representing. The sectors should be the size of their percentage. For instance, a company might make a circle graph representing sales of different colors of a product like a shirt. If half of consumers bought the blue shirt, then half the circle would be a segment representing the blue shirt. If a quarter bought the red shirt, then a quarter of the circle would be a sector representing the red.
Once all the sectors are created, they should be labeled with what they represent and the corresponding percentage. The business can then view the graph to easily get an idea of which colors sold the best.

Scatter Plots
A scatter plot shows how two variables are related to each other. A company can plot the different values of the variables to see how they relate. For instance, a company may want to see how sales are at different price points. On the vertical axis they can put sales while on the horizontal one they can put price. They can then adjust the price of their product and figure out the sales level for one day of each price. They will then plot the numbers on the graph and look for a relationship. Depending on the variables, there may or may not be one. In this case, the expected relationship will be that the higher the price, the lower the sales. They can use the graph to work out what is the best price point to make the best profits. Scatter plots can give a good visual representation of the relationship between two variables and are often used in business.

Random Variable for a Quantity of Interest
In order to graph a random variable and analyze its distribution, it's helpful to quantify it, i.e., to assign a numerical value to each possible event. This is more useful when there is some number that naturally corresponds to the event, rather than just arbitrary assignments. If the random variable represents the colors of the cars passing by, there's little gained by defining 1 to correspond to blue, 2 to red, and so on. However, if you want to quantify a student's attendance in a class, for example, one can derive a suitable number by adding together the number of days the student has been absent plus one fourth the number of days he has been tardy. This then gives a numerical value that can be graphed and otherwise analyzed.

Probability Distribution
A probability distribution is a set of values of a random variable, with a probability assigned to each one. All possible values of the random variable should be represented in the distribution, and the sum of all their probabilities must equal 1, or 100%. For instance, suppose our random variable represents the number of tails that come up if we flip a coin five times. The probability that no tails will come up is
, the probability of one tails is
, the probability of two or three tails is each
, the probability of four tails is
, and the probability that the coin will come up heads all five times is
. Note that as required, the sum of all these probabilities is equal to 1:


Probability in Terms of Relative Frequency for a Finite Distribution
Probability is expressed as a relative frequency. For a discrete distribution, the probability that a variable X has certain values is defined as the sum of relative frequencies for each of those certain values of X,

where
 has the property that

and
 is the relative frequency that
 occurs in the population, and
 if
 is one of the values of X whose relative frequency you wish to determine, otherwise

Example: If two dice are tossed, the variable X is the outcome (sum of the two dice),

Calculate the relative frequency of each outcome,

Calculate the probability, or sum of the relative frequencies, that X<8,


Probability in Terms of Relative Frequency for a Continuous Distribution
For a continuous distribution, the probability is the integral of the probability distribution function (pdf) over the range of values for the variable whose probability one wishes to determine.

Due to the definition of the integral, the relative frequency of a specific value for  is zero because,
 

However, the relative frequency or probability is generally non-zero when integrated over a range of possible values for  

The pdf has the property that when integrated over all possible values of 

Example: The probability that variable X is greater than A and less than B is 1/2. This is written,

Example: The probability that variable X is less than A and greater than B is 1/2. The domain for
 is [0,  This is written,


Graphing a Probability Distribution of a Random Variable
A probability distribution of a random variable may be graphed similarly to a data set. You can plot the value of the variable on the x axis and the probability corresponding to that outcome on the y axis.

Since the probabilities add to 1, the individual probabilities are likely to be small, and the axis should be scaled accordingly.

For example, consider the random variable corresponding to the sum of the numbers on three fair dice. The probability that this sum will be 3 is
 ≈ 0.0046. The probability that the dice will sum to 4 is
 ≈ 0.014, and so on; the largest probabilities arise in the center, where we reach a probability of
 = 0.125 that the sum of the dice will be 10, and the same probability for 11.

If we plot the probabilities of each possible outcome on a graph, we get the following:


Expected Value of a Random Variable for Which the Probabilities of Various Outcomes are Given
The expected value of a random variable is the sum of the possible values of the variable weighted by the probability of that value occurring.

Mathematically, it can be expressed as
.
For instance, suppose you have a weighted die that has a probability
 of coming up 6, a probability
 of coming up 5, and a probability
 of coming up 1, 2, 3, or 4. The expected value of the die's result is therefore
.
Note that, as seen in this example, the expected value is not necessarily a value that you would actually expect to see; the weighted die will never show 4.08. Rather, it plays a role similar to the mean of a data set, providing a measure of central tendency of the variable's distribution.

Finding the Mean of a Probability Distribution
The mean of a probability distribution is synonymous with the expected value of a random variable.  It can be calculated by summing all possible values of the variable weighted by their respective probabilities, 
 , the more familiar formula for the mean of a data set. The more general formula for the mean of a probability distribution, however, holds for arbitrary distributions that may not be uniform.
The mean serves a similar role to the mean of a data set; it provides a measure of central tendency, and a rough idea of what a typical data point looks like. However, it also suffers from some of the same limitations as the mean of a data set, and does not by itself provide a complete picture of the distribution.

Confidence Interval
A confidence interval consists of three components: a statistic, a margin of error, and a confidence level.
The format is
 

The margin of error is proportional to the square root of the estimate of the variance is calculated from sample data.
When the distribution of the variable is known, the confidence level may be calculated by the area under the probability distribution function (pdf) and between the upper and lower limits of the confidence interval of the variable. This area is listed in the table for the distribution function, for example the standard normal table.
There is no single way of constructing the confidence interval of an unobservable population parameter, .
It depends on the statistical variable whose interval is to be estimated and the distribution of that statistical variable (if any exist). The first rule of constructing an interval is that the upper and lower end-points,
 and
, are both functions of the statistical variable
. In other words, the confidence interval will be
. The probability
 that the confidence interval
 includes the unobservable population parameter
 at the
 confidence level is:


How Large Must the Sample Size Be to Calculate the Confidence Interval with No More Than 10% Error?
When sample size is too small, the confidence interval at the
 confidence level

must be replaced with

So, for the
 terms to contribute less than 10% error in the interval for a
 confidence level,

Note: For a 95% confidence level, after rounding up to nearest integer,

Example: What is the minimum sample size required to use the -Table to calculate the 95% confidence interval so that there is no more than 10% error in the interval when



Note: More samples are needed when
 is very low or very high.

Meaning of a Confidence Level (i.e., 95% Confidence Level)
A confidence level of say 95% means that if the same sampling method were repeated a total of 100 times and the confidence interval was calculated each time, the true population parameter would fall inside 95 of the calculated confidence intervals.

Example: A normally distributed variable X is sampled 10 times. The standard deviation and 95% confidence interval is calculated. This process is repeated 10 times.

For a normal distribution, 95% of the values are between  and . The results for the 10 estimates of the 95% confidence interval are,



It turns out that  , therefore, the true population mean does not fall in the confidence interval for sample #8, but it does fall within 9 of the 10 confidence intervals which is reasonably close to the 95% confidence level.

Processes Associated with the Binomial and Geometric Distributions
The process associated with a binomial distribution may be thought of as
 independent trials each with probability
 of a success,
 is the number of successes, and

The binomial distribution function has two fixed parameters
 and
, and one variable
.

The process associated with a geometric distribution may be thought of as independent trials each with probability
 of a success,
 is the number of trials until the first success, and 
The geometric distribution function has one fixed parameter  and one variable .


Processes Associated with the Geometric and Negative Binomial Distributions
The process associated with a geometric distribution may be thought of as independent trials each with probability  of a success,
 is the number of trials until the first success, and negative binomial distribution may be thought of as independent trials each with probability
 of a success,  is the number of failures before  successes, and

The negative binomial distribution function has two fixed parameters
 and
, and one variable
.


Probability of a True Population Statistic Being Within a Confidence Interval Equal to
 and Estimate of a Statistic
 Margin of Error at
 Confidence Level
.
 at 68.26%
. B. contrast, the margin of error of the statistic can be defined for an arbitrary confidence level,
,

 at
 
.

If the true population variance of the statistic is known, its square root is the standard deviation of the statistic, and the following equation applies,

 at
 
.