Standard deviation

1. Basic Concepts of Standard Deviation:

– Definition of Population Values
– Expected value of random variable X: μ = E[X]
– Standard deviation definition: σ = sqrt(E[(X-μ)^2])
– Relationship between standard deviation and variance
– Standard deviation of a probability distribution
– Cases where standard deviation might not exist

– Standard Deviation vs. Standard Error
– Difference between standard deviation and standard error
– Standard error calculation for sample mean
– Use of standard error in reporting findings
– Significance levels based on standard errors
– Importance of distinguishing between the two measures

– Notable Properties of Standard Deviation
– Standard deviation abbreviation: SD
– Representation in mathematical equations
– Relationship between standard deviation and variance
– Standard deviation’s unit consistency with data
– Usefulness in outlier detection and data interpretation

2. Calculation and Application of Standard Deviation:

– Discrete and Continuous Random Variables
– Standard deviation formula for finite data set
– Calculation using summation notation
– Standard deviation formula for different probabilities
– Application in scenarios with varying probabilities
– Importance of Bessel’s correction for bias correction

– Estimation and Uncertainty
– Estimation of standard deviation in population sampling
– Challenges in unbiased estimation of standard deviation
– Comparison of different estimators for standard deviation
– Significance of corrected sample standard deviation
– Uncorrected and corrected sample standard deviation

3. Interpretation and Significance of Standard Deviation:

– Interpretation and Application
– Large vs. Small standard deviation implications
– Measure of uncertainty and variability
– Precision in physical science and agreement with predictions
– Application examples in experiments and industry
– Statistical tests and data distribution

– Bounds and Identities
– Invariance under changes in location and scale
– Relationship with covariance and sum of squared deviations
– Sample standard deviation formula and population characteristics
– Variability measurement using coefficient of variation
– Standard deviation of the mean and its calculation

4. Practical Applications of Standard Deviation:

– Experimental Science
– Theoretical models and statistical certainty
– Particle physics discoveries and Higgs boson
– Gravitational waves and scientific research importance

– Weather and Finance
– Impact of location on temperature range
– Risk assessment and mean-variance optimization
– Standard deviation in weather analysis and investment decisions
– Relationship between risk and return in finance

5. Geometric Interpretation and Calculation Methods:

– Geometric Insights
– Relationship with distance and orthogonal movements
– Derivation of standard deviation formula
– Statistical analysis using geometric concepts

– Calculation Methods
– Rapid calculation methods and running summations
– Weighted calculations and incremental methods
– Efficiency in variance calculation and round-off considerations

Standard deviation (Wikipedia)

For other uses, see Standard deviation (disambiguation).

In statistics, the standard deviation is a measure of the amount of variation of a random variable expected about its mean. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not.

Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.

The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data.

The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the margin of error of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population.

In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant", a safeguard against spurious conclusion that is really due to random sampling error.

When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).