When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns. For example, assume an investor had to choose between two stocks.

Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20 percentage points pp and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional two percentage points of return is not worth the additional 10 pp standard deviation greater risk or uncertainty of the expected return. Stock B is likely to fall short of the initial investment but also to exceed the initial investment more often than Stock A under the same circumstances, and is estimated to return only two percent more on average.

Calculating the average or arithmetic mean of the return of a security over a given period will generate the expected return of the asset. For each period, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and taking the average gives the overall variance of the return of the asset.

The larger the variance, the greater risk the security carries.

## Basic Biostatistics

Finding the square root of this variance will give the standard deviation of the investment tool in question. Population standard deviation is used to set the width of Bollinger Bands , a widely adopted technical analysis tool. The most commonly used value for n is 2; there is about a five percent chance of going outside, assuming a normal distribution of returns. Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical tools that now have a valid basis from which to work.

To gain some geometric insights and clarification, we will start with a population of three values, x 1 , x 2 , x 3. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and P would lie on L. So it is not unreasonable to assume that the standard deviation is related to the distance of P to L. That is indeed the case. To move orthogonally from L to the point P , one begins at the point:. An observation is rarely more than a few standard deviations away from the mean.

Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table. The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constant.

If a data distribution is approximately normal, then the proportion of data values within z standard deviations of the mean is defined by:. The proportion that is less than or equal to a number, x , is given by the cumulative distribution function :. This is known as the The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean.

This is because the standard deviation from the mean is smaller than from any other point. The precise statement is the following: suppose x 1 , Variability can also be measured by the coefficient of variation , which is the ratio of the standard deviation to the mean. It is a dimensionless number. Often, we want some information about the precision of the mean we obtained.

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We can obtain this by determining the standard deviation of the sampled mean. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by:. This can easily be proven with see basic properties of the variance :.

However, in most applications this parameter is unknown.

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For example, if a series of 10 measurements of a previously unknown quantity is performed in a laboratory, it is possible to calculate the resulting sample mean and sample standard deviation, but it is impossible to calculate the standard deviation of the mean. The following two formulas can represent a running repeatedly updated standard deviation. A set of two power sums s 1 and s 2 are computed over a set of N values of x , denoted as x 1 , Given the results of these running summations, the values N , s 1 , s 2 can be used at any time to compute the current value of the running standard deviation:.

Where N, as mentioned above, is the size of the set of values or can also be regarded as s 0. In a computer implementation, as the three s j sums become large, we need to consider round-off error , arithmetic overflow , and arithmetic underflow. The method below calculates the running sums method with reduced rounding errors. Applying this method to a time series will result in successive values of standard deviation corresponding to n data points as n grows larger with each new sample, rather than a constant-width sliding window calculation.

When the values x i are weighted with unequal weights w i , the power sums s 0 , s 1 , s 2 are each computed as:. And the standard deviation equations remain unchanged. Note that s 0 is now the sum of the weights and not the number of samples N.

## Normal Distribution Problems with Solutions

The incremental method with reduced rounding errors can also be applied, with some additional complexity. The above formulas become equal to the simpler formulas given above if weights are taken as equal to one. The term standard deviation was first used [17] in writing by Karl Pearson [18] in , following his use of it in lectures. This was as a replacement for earlier alternative names for the same idea: for example, Gauss used mean error. From Wikipedia, the free encyclopedia. For other uses, see Standard deviation disambiguation.

See also: Sample variance. Main article: Unbiased estimation of standard deviation. Further information: Prediction interval and Confidence interval.

Main article: Chebyshev's inequality. Main article: Standard error of the mean. See also: Algorithms for calculating variance.

### Probability less than a z-value

Statistics portal. Studies in the History of the Statistical Method. Teaching Statistics. The American Statistician. Retrieved 5 February Retrieved 30 May Retrieved 29 October Fundamentals of Probability 2nd ed. New Jersey: Prentice Hall. Retrieved 30 September The Oxford Dictionary of Statistical Terms. Oxford University Press. Philosophical Transactions of the Royal Society A.

This test-statistic is then compared with a critical value and if it is found to be greater than the critical value the hypothesis is rejected. The critical values are the boundaries of the critical region. Critical values can be used to do hypothesis testing in following way.

Calculate critical values based on significance level alpha.

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Compare test statistic with critical values. If the test statistic is lower than the critical value, accept the hypothesis or else reject the hypothesis. For checking out how to calculate a critical value in detail please do check. Before we move forward with different statistical tests it is imperative to understand the difference between a sample and a population.

For our example above, it will be a small group of people selected randomly from some parts of the earth. For instance, in our example above if we select people randomly from all regions Asia, America, Europe, Africa etc. In such cases, a population is assumed to be of some type of a distribution. The most common forms of distributions are Binomial, Poisson and Discrete. However, there are many other types which are mentioned in detail at. Now, when we are clear on population, sample, and distribution we can move forward to understand different kinds of test and the distribution types for which they are used.

As we know critical value is a point beyond which we reject the null hypothesis. P-value on the other hand is defined as the probability to the right of respective statistic Z, T or chi. The benefit of using p-value is that it calculates a probability estimate, we can test at any desired level of significance by comparing this probability directly with the significance level.

For e. However, if we calculate p-value for 1. Important point to note here is that there is no double calculation required. In addition to this page which completes computation so you may verify your work, blank statistics paper has been provided.

Of course, sometimes the desired area may not be computed directly. Addition or subtraction of two areas many be required to find the desired area or percent or probability. Given a normal distribution with a mean of State the z-score which matches a score of State the probability a score is greater than 25 if the distribution is normal, the mean is 20, and the standard deviation is 5.

Terry and Tony each got an 88 on stat tests but they were different but equivalent tests. The teachers were willing to state the mean and standard deviation for each test. Below are the statistics.