# Standard Deviation Calculator

Standard 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 valuesFor 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 = 2.577Sample 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.

Scientific Method

1. Formulate a Question

Define the Question

Review the Literature

Create a Hypothesis

Research starts with a question or assumption you have on a real world phenomenon. Narrow it down to a research question that defines what you want to figure out and review the research and literature already done on that subject. With an understanding of your subject and a well defined question you form an hypothesis that will be tested against an opposite assumption called the null hypothesis.How to define a research problem

How to formulate your research question for a paper

How to write a Hypothesis for your paper

2. Collect Data

Preparation: Make Hypothesis Testable (Operationalization)

Preparation: Design the Study

Conduct the Experiment or Observation

Operationalize the hypothesis to be both testable and falsifiable. Then design a study and construct a test or experiment to collect data. Be aware of validity when choosing variables, especially when studying people. You might not be measuring what you think you are measuring. Qualitative studies tends to have more open questions and hypotheses while quantitative research have an experimental approach focusing more on counting and classifying observations.How to write the methodology you used to gather data for your paper.

How to make scientific observations

3. Test Hypothesis

Organize the Data

Analyse the Results

Check if the Results Support your Hypothesis

Organize the data and analyze it to see if it supports or rejects your hypothesis. The exact type of test used depends upon many things, including the field, the type of data and sample size, among other things. The vast majority of scientific research is ultimately tested by statistical methods, all giving a degree of confidence in the results.How to test an hypothesis

How to write the results in your paper

4. Conclusion

Look for Other Possible Explanations

Generalize to the Real World

Suggestions to Further Research

When looking at your results it is important to be open for other possible explanations. Could the results you got be generalized to the real word? Maybe other variables explain the question better then the ones you chose for your hypothesis? Remember to consider experimental errors and problems with validity and reliability in your conclusion. If your assumption on what you studied was right and your hypothesis was supported by the test, you could consider if it fits in a bigger picture with other research that together could form a theory. If the hypothesis failed you could try to tweak it or make a new hypothesis, corrected with your newly analysed results and test again. Often the conclusion will lead you on to further hypotheses about the phenomenon that suggest a direction for more research by yourself or other scientists.How to draw conclusions

How to scientifically write your conclusion"Science is a verb"

"Homeopaths gets on my nervs with the old; - ‘well, science doesn’t know everything’ ... Science knows it doesn’t know everything ... or it’d stop."Dara O’Briain

You might have come across the expression «science is a verb». This expression is an answer to the misconception of some people that science is mearly accumulation of static knowledge. But what we consider to be knowledge is under constant review and sometimes it changes. Science as a method of research make all theories subject to change with discovery of new evidence. New evidence create new explanations of phenomena, or substanciate the old theories and make us increasingly sure of their soundness.

Why use the Scientific Method?

Relevant pages:

What is Research?

How to define Research?The scientific method is a standard on how to do research that aims to discover new knowledge. Research in the broadest sense of the word includes any gathering of data, information and facts for the advancement of knowledge, but doing science restricts the research to a method that is focused on getting accurate and, most often, narrow conclusions.

Having done research by the scientific method makes it transparent and explorable. All published results from this method is reviewed by scientists against other findings or explanations to see if it is valid, or retested to see if the results were reliable. This is why publishing and review of research is such a big deal for scientists.

“Somewhere, something incredible is waiting to be known.” (Carl Sagan)

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Full reference:

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