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Standard deviation is a word used in statistics and probability theory to quantify the degree of dispersion in a numerical data set, or how distant the data points of interest are from the normal (average). “Standard deviation,” abbreviated as SD or StDev, is symbolised by the Greek letter sigma σ when referring to a population estimate based on a sample and the small Latin letter s when referring to sample standard deviation that is determined directly.
The variance is the average of the squared departures from the arithmetic mean, and the standard deviation is the square root of the variance. We square the differences so that larger deviations from the mean are punished more harshly, and it also has the side benefit of treating both positive and negative deviations equally. When describing statistical data, the standard deviation is favoured over the variance since it is expressed in the same unit as the data values. The variance is also calculated by our stdev calculator.
For continuous outcome variables, the entire raw dataset is required, whereas for binomial data such as proportions, conversion rates, recovery rates, survival rates, and so on, the variance and standard deviation can be calculated using only two summary statistics: the number of observations and the rate of events of interest. Both continuous and binomial data are supported by our standard deviation calculator.
A low standard deviation σ implies that the data points are grouped around the sample mean, whereas a large SD shows that the data set spans a wide range of values. The following graph illustrates the idea by contrasting two 18-element distributions with different standard deviations (2.26 and 8.94):
You should use one of two formulas depending on whether you’re computing the standard deviation based on a sample from a population or the full population.
If the data is from a sample, use the following formula to get the sample standard deviation:
Use the following calculation if the data set represents the complete population of interest:
In the population standard deviation calculation above, x is a data point, x̄(read “x bar”) is the arithmetic mean, and n is the number of items in the data set (count). The formula is for a sum of i=1 to i=n. In both cases, the standard deviation equals the square root of the variance, as previously mentioned.
With the flick of a switch, our standard deviation calculator supports both formulas.
Because you’ll almost always be sampling from a population and won’t have access to data on the entire population, you’ll most often use the sample standard deviation formula. The “adjusted sample standard deviation” is the formula our calculator employs in this case, and it is not unique since, unlike the sample mean and variance, there is no one formula that provides an efficient estimator across all distributions. For n <10, this formula, for example, can be severely skewed.
In other circumstances, you will have information about the entire population; for example, if the population of interest includes pupils in a class or school, you may be able to obtain grades for all of them at any one time. However, most of the time, the population of interest spans too much time and includes too many people to be assessed practically.
The standard deviation for proportional data, event rates, and other types of data can be calculated using the following formula:
where p denotes the percentage of the population who is exposed to the event of interest or has a feature of interest. Because a proportion is merely a subset of a mean, this standard deviation formula is generated by transforming the previous ones. Our standard deviation calculator works with proportions where all you need to know is the sample size and the event rate to estimate the difference between the observed and expected outcome.
A lower standard deviation in a data collection suggests less dispersion – the values are more grouped around the mean, as illustrated in the example above. In physics, medicine, biology, physiology, chemistry, and other fields, standard deviation measures and estimations can be used to signify the precision of measuring techniques, instruments, or methods. It can be viewed as a measure of data uncertainty – expected, known, or accepted, depending on the situation.
A statistical cut-off point in standard deviations will be offered to you in many instances. These can be equivalent to percentiles, or how many cases are x standard deviations away from the predicted value. The following are some important levels and percentile cut-offs:
Standard deviation cut-offs for normally distributed variables are listed in the table below:
So, if an observation deviates from the predicted value by 1.645 standard deviations, it is in the top 10% of the population of interest. The direction of the effect you’re interested in is referred to as 2-sided. In most cases, the 1-sided number is the one that matters. In population studies, the 2-sided percentile corresponds to the proportion inside the standard deviation’s bounds.
The standard deviation, in a geometrical sense, represents the percentage of a distribution’s area that is included or omitted.
Finance
Financial managers and academic papers frequently utilise the standard deviation of price changes of a financial asset (stock, bond, property, etc.) to measure the degree of risk of single assets or asset portfolios. This is, however, a contentious issue, with many notable financial practitioners criticising the risk-standard-deviation equation. The Bollinger Bands are a common technical analysis tool that plots lines calculated to be two standard deviations in each direction from the mean price of a rolling period.
Because standard deviation and other statistical tools are only applicable to stationary series, and financial data is non-stationary, it must be transformed by removing trend, seasonality, and auto-correlation from the dataset, which is usually done by differencing using complex regressions such as ARIMA (Auto Regressive Integrated Moving Average) and exponential smoothing models.
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