What Is the Addition Rule for Probabilities?

The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually exclusive events happening.

The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.

The Formulas for the Addition Rules for Probabilities Is

Mathematically, the probability of two mutually exclusive events is denoted by:


$P(Y \text{ or } Z) = P(Y)+P(Z)$P(Y or Z)=P(Y)+P(Z)

Mathematically, the probability of two non-mutually exclusive events is denoted by:


$P(Y \text{ or } Z) = P(Y) + P(Z) – P(Y \text{ and } Z)$P(Y or Z)=P(Y)+P(Z)−P(Y and Z)

What Does the Addition Rule for Probabilities Tell You?

To illustrate the first rule in the addition rule for probabilities, consider a die with six sides and the chances of rolling either a 3 or a 6. Since the chances of rolling a 3 are 1 in 6 and the chances of rolling a 6 are also 1 in 6, the chance of rolling either a 3 or a 6 is:

1/6 + 1/6 = 2/6 = 1/3

To illustrate the second rule, consider a class in which there are 9 boys and 11 girls. At the end of the term, 5 girls and 4 boys receive a grade of B. If a student is selected by chance, what are the odds that the student will be either a girl or a B student? Since the chances of selecting a girl are 11 in 20, the chances of selecting a B student are 9 in 20 and the chances of selecting a girl who is a B student are 5/20, the chances of picking a girl or a B student are:

11/20 + 9/20 – 5/20 =15/20 = 3/4

In reality, the two rules simplify to just one rule, the second one. That’s because in the first case, the probability of two mutually exclusive events both happening is 0. In the example with the die, it’s impossible to roll both a 3 and a 6 on one roll of a single die. So the two events are mutually exclusive.

Mutual Exclusivity

Mutually exclusive is a statistical term describing two or more events that cannot coincide. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other.  For a basic example, consider the rolling of dice. You cannot roll both a five and a three simultaneously on a single die. Furthermore, getting a three on an initial roll has no impact on whether or not a subsequent roll yields a five. All rolls of a die are independent events.

What Is Analysis of Variance (ANOVA)?

Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.

The t- and z-test methods developed in the 20th century were used for statistical analysis until 1918, when Ronald Fisher created the analysis of variance method. ANOVA is also called the Fisher analysis of variance, and it is the extension of the t- and z-tests. The term became well-known in 1925, after appearing in Fisher’s book, “Statistical Methods for Research Workers.” It was employed in experimental psychology and later expanded to subjects that were more complex.

The Formula for ANOVA is:


$\begin{aligned} &\text{F} = \frac{ \text{MST} }{ \text{MSE} } \\ &\textbf{where:} \\ &\text{F} = \text{ANOVA coefficient} \\ &\text{MST} = \text{Mean sum of squares due to treatment} \\ &\text{MSE} = \text{Mean sum of squares due to error} \\ \end{aligned}$​F=MSEMST​where:F=ANOVA coefficientMST=Mean sum of squares due to treatmentMSE=Mean sum of squares due to error​

What Does the Analysis of Variance Reveal?

The ANOVA test is the initial step in analyzing factors that affect a given data set. Once the test is finished, an analyst performs additional testing on the methodical factors that measurably contribute to the data set’s inconsistency. The analyst utilizes the ANOVA test results in an f-test to generate additional data that aligns with the proposed regression models.

The ANOVA test allows a comparison of more than two groups at the same time to determine whether a relationship exists between them. The result of the ANOVA formula, the F statistic (also called the F-ratio), allows for the analysis of multiple groups of data to determine the variability between samples and within samples.

If no real difference exists between the tested groups, which is called the null hypothesis, the result of the ANOVA’s F-ratio statistic will be close to 1. The distribution of all possible values of the F statistic is the F-distribution. This is actually a group of distribution functions, with two characteristic numbers, called the numerator degrees of freedom and the denominator degrees of freedom.

Example of How to Use ANOVA

A researcher might, for example, test students from multiple colleges to see if students from one of the colleges consistently outperform students from the other colleges. In a business application, an R&D researcher might test two different processes of creating a product to see if one process is better than the other in terms of cost efficiency.

The type of ANOVA test used depends on a number of factors. It is applied when data needs to be experimental. Analysis of variance is employed if there is no access to statistical software resulting in computing ANOVA by hand. It is simple to use and best suited for small samples. With many experimental designs, the sample sizes have to be the same for the various factor level combinations.

ANOVA is helpful for testing three or more variables. It is similar to multiple two-sample t-tests. However, it results in fewer type I errors and is appropriate for a range of issues. ANOVA groups differences by comparing the means of each group and includes spreading out the variance into diverse sources. It is employed with subjects, test groups, between groups and within groups.

One-Way ANOVA Versus Two-Way ANOVA

There are two main types of ANOVA: one-way (or unidirectional) and two-way. There also variations of ANOVA. For example, MANOVA (multivariate ANOVA) differs from ANOVA as the former tests for multiple dependent variables simultaneously while the latter assesses only one dependent variable at a time. One-way or two-way refers to the number of independent variables in your analysis of variance test. A one-way ANOVA evaluates the impact of a sole factor on a sole response variable. It determines whether all the samples are the same. The one-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.

A two-way ANOVA is an extension of the one-way ANOVA. With a one-way, you have one independent variable affecting a dependent variable. With a two-way ANOVA, there are two independents. For example, a two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as salary and skill set. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time.

What Is an Average Collection Period?

Average collection period refers to the amount of time it takes for a business to receive payments owed by its clients in terms of accounts receivable (AR). Companies use the average collection period to make sure they have enough cash on hand to meet their financial obligations. The average collection period is an indicator of the effectiveness of a firm’s AR management practices and is an important metric for companies that rely heavily on receivables for their cash flows.

How Average Collection Periods Work

Accounts receivable is a business term used to describe money that entities owe to a company when they purchase goods and/or services. Companies normally make these sales to their customers on credit. AR is listed on corporations’ balance sheets as current assets and measures their liquidity. As such, they indicate their ability to pay off their short-term debts without the need to rely on additional cash flows.

The average collection period is an accounting metric used to represent the average number of days between a credit sale date and the date when the purchaser remits payment. A company’s average collection period is indicative of the effectiveness of its AR management practices. Businesses must be able to manage their average collection period to operate smoothly.

A lower average collection period is generally more favorable than a higher one. A low average collection period indicates that the organization collects payments faster. However, this may mean that the company’s credit terms are too strict. Customers who don’t find their creditors’ terms very friendly may choose to seek suppliers or service providers with more lenient payment terms.

Formula for Average Collection Period

Average collection period is calculated by dividing a company’s average accounts receivable balance by its net credit sales for a specific period, then multiplying the quotient by 365 days.

Average Collection Period = 365 Days * (Average Accounts Receivables / Net Credit Sales)

Alternatively and more commonly, the average collection period is denoted as the number of days of a period divided by the receivables turnover ratio. The formula below is also used referred to as the days sales receivable ratio.

Average Collection Period = 365 Days / Receivables Turnover Ratio

The average receivables turnover is simply the average accounts receivable balance divided by net credit sales; the formula below is simply a more concise way of writing the formula.

Average Accounts Receivables

For the formulas above, average accounts receivable is calculated by taking the average of the beginning and ending balances of a given period. More sophisticated accounting reporting tools may be able to automate a company’s average accounts receivable over a given period by factoring in daily ending balances.

When analyzing average collection period, be mindful of the seasonality of the accounts receivable balances. For example, analyzing a peak month to a slow month by result in a very inconsistent average accounts receivable balance that may skew the calculated amount.

Net Credit Sales

Average collection period also relies on net credit sales for a period. This metric should exclude cash sales (as those are not made on credit and therefore do not have a collection period).

In addition to being limited to only credit sales, net credit sales exclude residual transactions that impact and often reduce sales amounts. This includes any discounts awarded to customers, product recalls or returns, or items re-issued under warranty.

Importance of Average Collection Period

Average collection period boils down to a single number; however, it has many different uses and communicates a variety of important information.

• It tells how efficiently debts are collected. This is important because a credit sale is not fully completed until the company has been paid. Until cash has been collected, a company is yet to reap the full benefit of the transaction.
• It tells how strict credit terms are. This is important as strict credit terms may scare clients away; on the other hand, credit terms that are too loose may attract customers looking to take advantage of lenient payment terms.
• It tells how competitors are performing. This is important because all figures needed to calculate the average collection period are available for public companies. This gives deeper insight into what other companies are doing and how a company’s operations compare.
• It tells early signals of bad allowances. This is important because as the average collection period increases, more clients are taking longer to pay. This metric can be used to signal to management to review its outstanding receivables at risk of being uncollected to ensure clients are being monitored and communicated with.
• It tells of a company’s short-term financial health. This is important because without cash collections, a company will go insolvent and lack the liquidity to pay its short-term bills.

How to Use Average Collection Period

The average collection period does not hold much value as a stand-alone figure. Instead, you can get more out of its value by using it as a comparative tool.

The best way that a company can benefit is by consistently calculating its average collection period and using it over time to search for trends within its own business. The average collection period may also be used to compare one company with its competitors, either individually or grouped together. Similar companies should produce similar financial metrics, so the average collection period can be used as a benchmark against another company’s performance.

Companies may also compare the average collection period with the credit terms extended to customers. For example, an average collection period of 25 days isn’t as concerning if invoices are issued with a net 30 due date. However, an ongoing evaluation of the outstanding collection period directly affects the organization’s cash flows.

Example of Average Collection Period

As noted above, the average collection period is calculated by dividing the average balance of AR by total net credit sales for the period, then multiplying the quotient by the number of days in the period.

Let’s say a company has an average AR balance for the year of $10,000. The total net sales that the company recorded during this period was $100,000. We would use the following average collection period formula to calculate the period:

($10,000 ÷ $100,000) × 365 = Average Collection Period

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The average collection period, therefore, would be 36.5 days. This is not a bad figure, considering most companies collect within 30 days. Collecting its receivables in a relatively short and reasonable period of time gives the company time to pay off its obligations.

If this company’s average collection period was longer—say, more than 60 days— then it would need to adopt a more aggressive collection policy to shorten that time frame. Otherwise, it may find itself falling short when it comes to paying its own debts.

Accounts Receivable (AR) Turnover

The average collection period is closely related to the accounts turnover ratio, which is calculated by dividing total net sales by the average AR balance.

Using the previous example, the AR turnover is 10 ($100,000 ÷ $10,000). The average collection period can also be calculated by dividing the number of days in the period by the AR turnover. In this example, the average collection period is the same as before: 36.5 days.

365 days ÷ 10 = Average Collection Period

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Collections by Industries

Not all businesses deal with credit and cash in the same way. Although cash on hand is important to every business, some rely more on their cash flow than others.

For example, the banking sector relies heavily on receivables because of the loans and mortgages that it offers to consumers. As it relies on income generated from these products, banks must have a short turnaround time for receivables. If they have lax collection procedures and policies in place, then income would drop, causing financial harm.

Real estate and construction companies also rely on steady cash flows to pay for labor, services, and supplies. These industries don’t necessarily generate income as readily as banks, so it’s important that those working in these industries bill at appropriate intervals, as sales and construction take time and may be subject to delays.

The Bottom Line

The average collection period is the average number of days it takes for a credit sale to be collected. During this period, the company is awarding its customer a very short-term “loan”; the sooner the client can collect the loan, the earlier it will have the capital to use to grow its company or pay its invoices. While a shorter average collection period is often better, too strict of credit terms may scare customers away.

What Is Additional Paid-in Capital (APIC)?

Additional paid-in capital (APIC) is an accounting term referring to money an investor pays above and beyond the par value price of a stock.

Often referred to as “contributed capital in excess of par,” APIC occurs when an investor buys newly-issued shares directly from a company during its initial public offering (IPO) stage. APIC, which is itemized under the shareholder equity (SE) section of a balance sheet, is viewed as a profit opportunity for companies as it results in them receiving excess cash from stockholders.

How Additional Paid-in Capital (APIC) Works

During its IPO, a firm is entitled to set any price for its stock that it sees fit. Meanwhile, investors may elect to pay any amount above this declared par value of a share price, which generates the APIC.

Let us assume that during its IPO phase the XYZ Widget Company issues one million shares of stock, with a par value of $1 per share, and that investors bid on shares for $2, $4, and $10 above the par value. Let us further assume that those shares ultimately sell for $11, consequently making the company $11 million. In this instance, the APIC is $10 million ($11 million minus the par value of $1 million). Therefore, the company’s balance sheet itemizes $1 million as “paid-in capital,” and $10 million as “additional paid-in capital.”

Once a stock trades in the secondary market, an investor may pay whatever the market will bear. When investors buy shares directly from a given company, that corporation receives and retains the funds as paid-in capital. But after that time, when investors buy shares in the open market, the generated funds go directly into the pockets of the investors selling off their positions.

Special Considerations

APIC is generally booked in the SE section of the balance sheet. When a company issues stock, there are two entries that take place in the equity section: common stock and APIC. The total cash generated by the IPO is recorded as a debit in the equity section, and the common stock and APIC are recorded as credits.

The APIC formula is:

APIC = (Issue Price – Par Value) x Number of Shares Acquired by Investors.

Par Value

Due to the fact that APIC represents money paid to the company above the par value of a security, it is essential to understand what par actually means. Simply put, “par” signifies the value a company assigns to stock at the time of its IPO, before there is even a market for the security. Issuers traditionally set stock par values deliberately low—in some cases as little as a penny per share—in order to preemptively avoid any potential legal liability, which might occur if the stock dips below its par value.

Market Value

Market value is the actual price a financial instrument is worth at any given time. The stock market determines the real value of a stock, which shifts continuously as shares are bought and sold throughout the trading day. Thus, investors make money on the changing value of a stock over time, based on company performance and investor sentiment.

Additional Paid-in Capital vs. Paid-in Capital

Paid-in capital, or contributed capital, is the full amount of cash or other assets that shareholders have given a company in exchange for stock. Paid-in capital includes the par value of both common and preferred stock plus any amount paid in excess.

Additional paid-in capital, as the name implies, includes only the amount paid in excess of the par value of stock issued during a company’s IPO.

Both of these items are included next to one another in the SE section of the balance sheet.

Benefits of Additional Paid-in Capital

For common stock, paid-in capital consists of a stock’s par value and APIC, the latter of which may provide a substantial portion of a company’s equity capital, before retained earnings begin to accumulate. This capital provides a layer of defense against potential losses, in the event that retained earnings begin to show a deficit. 

Another huge advantage for a company issuing shares is that it does not raise the fixed cost of the company. The company doesn’t have to make any payment to the investor; even dividends are not required. Furthermore, investors do not have any claim on the company’s existing assets.

After issuing stock to shareholders, the company is free to use the funds generated any way it chooses, whether that means paying off loans, purchasing an asset, or any other action that may benefit the company.

Correction—March 29, 2022: A previous version of this article inaccurately represented where APIC appears on the balance sheet.