Please provide numbers separated by comma to calculate the standard deviation, variance, mean, sum, and margin of error.
Do I buy juice at the sale price of 2 bottles with 32 oz each for $6.00, or do I buy 1 bottle containing 72 oz for $6.99? Now you can use our unit price calculator to calculate the cost per ounce of one or both deals, and quickly figure out which is the better bargain. Shakuntala Devi (4 November 1929 – 21 April 2013) was an Indian mathematician, writer and mental calculator, popularly known as the 'Human Computer'.Devi strove to simplify numerical calculations for students. So, for this example, you would add 01 + 20 + 19 + 72 for a total of 132. This formula will result in either a two digit or three digit number. Step 2 If your result from step one is a two digit number, simply add the two numbers together. For example, if the number you received was 52, you would add 5 + 2 to get 7.
RelatedProbability Calculator | Sample Size Calculator | Statistics CalculatorStandard deviation in statistics, typically denoted by σ, is a measure of variation or dispersion (refers to a distribution's extent of stretching or squeezing) between values in a set of data. The lower the standard deviation, the closer the data points tend to be to the mean (or expected value), μ. Conversely, a higher standard deviation indicates a wider range of values. Similarly to other mathematical and statistical concepts, there are many different situations in which standard deviation can be used, and thus many different equations. In addition to expressing population variability, the standard deviation is also often used to measure statistical results such as the margin of error. When used in this manner, standard deviation is often called the standard error of the mean, or standard error of the estimate with regard to a mean. The calculator above computes population standard deviation and sample standard deviation, as well as confidence interval approximations.
Population Standard Deviation
The population standard deviation, the standard definition of σ, is used when an entire population can be measured, and is the square root of the variance of a given data set. In cases where every member of a population can be sampled, the following equation can be used to find the standard deviation of the entire population:
| Where xi is an individual value μ is the mean/expected value N is the total number of values |
For those unfamiliar with summation notation, the equation above may seem daunting, but when addressed through its individual components, this summation is not particularly complicated. The i=1 in the summation indicates the starting index, i.e. for the data set 1, 3, 4, 7, 8, i=1 would be 1, i=2 would be 3, and so on. Hence the summation notation simply means to perform the operation of (xi - μ2) on each value through N, which in this case is 5 since there are 5 values in this data set.
EX: μ = (1+3+4+7+8) / 5 = 4.6
σ = √[(1 - 4.6)2 + (3 - 4.6)2 + .. + (8 - 4.6)2)]/5
σ = √(12.96 + 2.56 + 0.36 + 5.76 + 11.56)/5 = 2.577
Sample Standard Deviation
In many cases, it is not possible to sample every member within a population, requiring that the above equation be modified so that the standard deviation can be measured through a random sample of the population being studied. A common estimator for σ is the sample standard deviation, typically denoted by s. It is worth noting that there exist many different equations for calculating sample standard deviation since unlike sample mean, sample standard deviation does not have any single estimator that is unbiased, efficient, and has a maximum likelihood. The equation provided below is the 'corrected sample standard deviation.' It is a corrected version of the equation obtained from modifying the population standard deviation equation by using the sample size as the size of the population, which removes some of the bias in the equation. Unbiased estimation of standard deviation however, is highly involved and varies depending on distribution. As such, the 'corrected sample standard deviation' is the most commonly used estimator for population standard deviation, and is generally referred to as simply the 'sample standard deviation.' It is a much better estimate than its uncorrected version, but still has significant bias for small sample sizes (N<10).
| Where xi is one sample value x̄ is the sample mean N is the sample size |
Refer to the 'Population Standard Deviation' section for an example on how to work with summations. The equation is essentially the same excepting the N-1 term in the corrected sample deviation equation, and the use of sample values.
Applications of Standard Deviation
Standard deviation is widely used in experimental and industrial settings to test models against real-world data. An example of this in industrial applications is quality control for some product. Standard deviation can be used to calculate a minimum and maximum value within which some aspect of the product should fall some high percentage of the time. In cases where values fall outside the calculated range, it may be necessary to make changes to the production process to ensure quality control.
Standard deviation is also used in weather to determine differences in regional climate. Imagine two cities, one on the coast and one deep inland, that have the same mean temperature of 75°F. While this may prompt the belief that the temperatures of these two cities are virtually the same, the reality could be masked if only the mean is addressed and the standard deviation ignored. Coastal cities tend to have far more stable temperatures due to regulation by large bodies of water, since water has a higher heat capacity than land; essentially, this makes water far less susceptible to changes in temperature, and coastal areas remain warmer in winter, and cooler in summer due to the amount of energy required to change the temperature of water. Hence, while the coastal city may have temperature ranges between 60°F and 85°F over a given period of time to result in a mean of 75°F, an inland city could have temperatures ranging from 30°F to 110°F to result in the same mean.
Another area in which standard deviation is largely used is finance, where it is often used to measure the associated risk in price fluctuations of some asset or portfolio of assets. The use of standard deviation in these cases provides an estimate of the uncertainty of future returns on a given investment. For example, in comparing stock A that has an average return of 7% with a standard deviation of 10% against stock B, that has the same average return but a standard deviation of 50%, the first stock would clearly be the safer option, since standard deviation of stock B is significantly larger, for the exact same return. That is not to say that stock A is definitively a better investment option in this scenario, since standard deviation can skew the mean in either direction. While Stock A has a higher probability of an average return closer to 7%, Stock B can potentially provide a significantly larger return (or loss).
These are only a few examples of how one might use standard deviation, but many more exist. Generally, calculating standard deviation is valuable any time it is desired to know how far from the mean a typical value from a distribution can be.
The Compound Interest Calculator below can be used to compare or convert the interest rates of different compound periods. Please use our Interest Calculator to do actual calculations on compound interest.
RelatedInterest Calculator | Investment Calculator | Auto Loan Calculator
What is Compound Interest?
Simple interest refers to interest earned only on the principal, usually denoted as a fixed percentage of the principal. Determining a single interest payment is as simple as multiplying the interest rate with the principal. Simple interest is seldom ever used in real world applications of interest.
On the other hand, compound interest is interest earned on both the principal and on the accumulated interest. Because interest is also earned on interest, earnings compound over time like an exponentially-growing, avalanching snowball. Compound interest is widely used for interest calculations on many things including mortgages, auto loans, banking, and much more.
In order to determine whether interest is compounded or not in the U.S., the Truth in Lending Act (TILA) requires that lenders disclose all pertinent loan information to borrowers, including whether interest accrues simply or in compounded fashion. Another way to determine whether interest is simple or compounded is to look at the repayment schedule for the loan. In the case of simple interest, each year's interest payment and the total amount owed will be the same. If the interest is compounded, each year's interest payment will be different.
Practical Ways to Use Compound Interest
To start with, any form of savings that doesn't earn interest, such as cash or many checking accounts, will not benefit from compound interest. Common funds that benefit from compound interest include savings accounts, stocks (with reinvested dividends), and some of the more common retirement plans such as 401(k)s and IRAs.
Compound interest can be highly financially rewarding. The longer that interest is allowed to compound for any investment, the greater the growth. While this is true for all investments, retirement investments are the main financial instruments that people use to take full advantage of compound interest. As a simple example, a person at age 19 decides to invest $2,000 every year for eight years at an 8% interest rate. Suddenly, they decide to halt annual payments, but allow the funds to grow uninterrupted until they reach the age of 65. With an initial investment of only $16,000 over eight years, their funds will have grown to almost $430,000 for use in retirement! And all this without paying a single cent for 39 years. This is due in large part to the nature of compound interest.
While compound interest is very effective at growing wealth, it can also work against you if you have any debt that is subject to compound interest. This is why compound interest can be described by some as a double-edged sword. Putting off or prolonging outstanding debt will increase the total interest owed. As such, it is as important to ensure that debts are paid off quickly as it is to put money into a retirement account early to allow it the maximum amount of time to grow.
Factors that Work Against Compound Interest
Tax—If any taxation is to be applied, the rate and timing of taxation will affect the magnitude of compounding interest. The less that taxation is involved, the greater the magnitude of compounding because of fewer reductions in the balance of the investment.
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Fees—In the case of long-term investments such as a retirement account, even a fee as low as 1% will have a significant impact on the end result. 1% vs 0.5% may not feel like much over the course of 1 or 2 years, but when saving for retirement, it can mean the difference between retiring at different ages.
Different Compounding Frequencies
Interest can be compounded on any given frequency schedule, and the calculator allows the conversion between compounding frequencies of daily, bi-weekly, semi-monthly, monthly, quarterly, semi-annually, annually, and continuously (infinitely many number of periods). The interest rates of savings accounts and Certificate of Deposits (CD) tend to be compounded annually. Home mortgage loans, home equity loans, and credit card accounts tend to be compounded monthly.
History of Compound Interest
There is evidence from ancient texts that compound interest was first used 4400 years ago by the Babylonians and Sumerians, two of the earliest civilizations in human history. However, their application of compound interest was quite different from what is widely used today. In their application, 20% of the principal amount was accumulated until the interest was equal to the principal, which was then added to the principal. Historically, simple interest was mostly considered legal. However, certain societies didn't grant the same legality to compound interest, labeling it as usury. For example, it was severely condemned by Roman law, and both Christian and Islamic texts have described it as a sin. Nevertheless, compound interest has been in use ever since.
The equation for continuously compounding interest, which is the mathematical limit that compound interest can reach, utilizes something called Euler's Constant, also known as e. Although e is widely used today in many areas, it was discovered when Jacob Bernoulli was studying compound interest in 1683. The mathematician understood that, within a specified finite time period, the more compounding periods involved, the faster the compounding principal was able to grow. It didn't matter whether it was in intervals of years, months, days, hours, minutes, seconds, or nanoseconds, each additional period generated higher returns (for the lender). Bernoulli discerned that this sequence eventually approached a limit he defined as e, which describes the relationship between the plateau and the interest rate when compounding.
Compound Interest Formulas
Basic Compound Interest
Basic formula for compound interest:
| At = A0(1+r)n |
At : amount after time t
r : interest rate
n : number of compounding periods, usually expressed in years
The following is an example of $1,000 in a savings account for two years advertised at 6% APY compounded once a year. Use the equation above to find the total due at maturity:
| At = $1,000 × (1 + 6%)2 = $1,123.60 |
For other compounding frequencies (such as monthly, weekly, or daily), the situation calls for the formula below.
| At = A0 × (1 + r/n)nt |
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where:At : amount after time t
n : number of compounding periods in a year
r : interest rate
t : number of years
Assume that the $1,000 in the savings account in the previous example comes with a 6% rate with interest accumulated daily. The interest earned every day is:
| 6% ÷ 365 = 0.0164384% |
Using the formula above, it is possible to find the value at the end.
| At = $1,000 × (1 + 0.0164384%)(365 × 2) At = $1,000 × 1.12749 At = $1,127.49 |
$1,127.49 will be the end value of a 2-year savings account containing $1,000 that has a 6% interest rate compounded daily.
Rule of 72
The Rule of 72 is a shortcut to determine how long it'll take for a specific amount of money to double, given a fixed return rate that is compounded annually. It can be used for any investment, as long as there is a fixed rate that involves compound interest. Simply divide the number 72 by the annual rate of return and the result of this is how many years it'll take. As an example, $100 with a fixed rate of return of 8% will take around 9 (72 divided by 8) years to become $200. Note that '8' is used to denote 8%, not '0.08'. Keep in mind that the Rule of 72 disregards any investment fees, management fees, and trading commissions, and doesn't account for losses incurred from taxes paid on investment gains. It is best used as a rough guideline.
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Continuous Compound Interest
Continuously compounding interest represents the mathematical limit that compound interest can reach within a specified time period. The continuous compound equation is as follows:
| At = A0ert |
At : amount after time t
r : interest rate
t : number of years
e : mathematical constant e, ~2.718
Say for instance, we wanted to find the maximum interest that could possibly be earned on the $1,000 savings account in two years.
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| At = $1,000e(6% × 2) At = $1,000e0.12 At = $1,127.50 |
From the 3 examples provided it can be seen that the shorter the compounding frequency, ceteris paribus, the higher the interest earned. It can be seen however, that above a certain compounding frequency, the interest gained is marginal, particularly on smaller principals.
