Please provide numbers separated by commas to calculate the standard deviation, variance, mean, sum, & margin of error.

10, 12, 23, 23, 16, 23, 21, 16

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Standard deviation in statistics, typically denoted by σ, is a measure of variation or dispersion (refers to lớn a distribution"s extent of stretching or squeezing) between values in a mix of data. The lower the standard deviation, the closer the data points tend to be to lớn the mean (or expected value), μ. Conversely, a higher standard deviation indicates a wider range of values. Similar lớn other mathematical and statistical concepts, there are many different situations in which standard deviation can be used, và thus many different equations. In addition khổng lồ 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 lớn a mean. The above computes population standard deviation và 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 thành viên of a population can be sampled, the following equation can be used to find the standard deviation of the entire population:


xi is an individual valueμ is the mean/expected valueN 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 mix 1, 3, 4, 7, 8, i=1 would be 1, i=2 would be 3, & so on. Hence the summation notation simply means to lớn 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 thành viên 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 kích cỡ as the kích thước 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 the distribution. As such, the "corrected sample standard deviation" is the most commonly used estimator for population standard deviation, and is generally referred to lớn as simply the "sample standard deviation." It is a much better estimate than its uncorrected version, but still has a significant bias for small sample sizes (Ni is one sample value is the sample meanN is the sample size

Refer lớn the "Population Standard Deviation" section for an example of how to lớn 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 và industrial settings to test models against real-world data. An example of this in industrial applications is chất lượng control for some products. Standard deviation can be used khổng lồ calculate a minimum & maximum value within which some aspect of the hàng hóa should fall some high percentage of the time. In cases where values fall outside the calculated range, it may be necessary to lớn make changes to the production process to lớn ensure chất lượng control.

Standard deviation is also used in weather to determine differences in regional climate. Imagine two cities, one on the coast và 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 và the standard deviation ignored. Coastal cities tend to have far more stable temperatures due lớn 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, và coastal areas remain warmer in winter, và cooler in summer due to the amount of energy required lớn change the temperature of the water. Hence, while the coastal city may have temperature ranges between 60°F và 85°F over a given period of time khổng lồ result in a mean of 75°F, an inland city could have temperatures ranging from 30°F to lớn 110°F khổng lồ 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 the standard deviation of stock B is significantly larger, for the exact same return. That is not to lớn 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 khổng lồ 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 lớn know how far from the mean a typical value from a distribution can be.