find the mean by all the three methods
Answers
In statistics, regression toward the mean (or regression to the mean) is the phenomenon that arises if a sample point of a random variable is extreme (nearly an outlier), a future point will be closer to the mean or average on further measurements.[1][2][3] To avoid making incorrect inferences, regression toward the mean must be considered when designing scientific experiments and interpreting data.[4] Historically, what is now called regression toward the mean was also called reversion to the mean and reversion to mediocrity.
Galton's experimental setup (Fig.8)
The conditions under which regression toward the mean occurs depend on the way the term is mathematically defined. The British polymath Sir Francis Galton first observed the phenomenon in the context of simple linear regression of data points. Galton[5] developed the following model: pellets fall through a quincunx to form a normal distribution centred directly under their entrance point. These pellets might then be released down into a second gallery corresponding to a second measurement. Galton then asked the reverse question: "From where did these pellets come?"
The answer was not 'on average directly above'. Rather it was 'on average, more towards the middle', for the simple reason that there were more pellets above it towards the middle that could wander left than there were in the left extreme that could wander to the right, inwards.[6]
Being a less restrictive approach, regression towards the mean can be defined for any bivariate distribution with identical marginal distributions. Two such definitions exist.[7] One definition accords closely with the common usage of the term "regression towards the mean". Not all such bivariate distributions show regression towards the mean under this definition. However, all such bivariate distributions show regression towards the mean under the other definition.
Jeremy Siegel uses the term "return to the mean" to describe a financial time series in which "returns can be very unstable in the short run but very stable in the long run." More quantitatively, it is one in which the standard deviation of average annual returns declines faster than the inverse of the holding period, implying that the process is not a random walk, but that periods of lower returns are systematically followed by compensating periods of higher returns, as is the case in many seasonal businesses, for example.[8]
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