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Ch8_Statistics

# Ch8_Statistics - Statistics Point estimation Lets say you...

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Statistics Point estimation Let’s say you collected some data. What you really care about is the probability distribution that underlies your data. But all you can do is sample a finite amount of data from the distribution. So how do you estimate a parameter (e.g. mean, variance) of the underlying distribution based on your sampled data ? This process is known as calculating a point estimate . Although a point estimate might be an unfamiliar term, you are probably very used to calculating a point estimate for the mean or variance. Example : Estimating the mean of the underlying normal distribution based on 20 data points ( y 1 , y 2 ,… y 20 ). μ = mean of the underlying distribution μ ˆ = our estimate of μ (the point estimate) = i i y 20 ˆ μ Notice that the estimated mean (usually) differs from the true mean: Example : Estimating the variance of the underlying normal distribution based on 20 data points ( y 1 , y 2 ,… y 20 ). σ 2 = variance of the underlying distribution 2 ˆ σ = our estimate of σ 2 (the point estimate) - = i i y 20 ) ˆ ( ˆ 2 2 μ σ When plotting error bars, scientists often choose between plotting the standard deviation versus the standard error of the mean . What’s the difference? The standard deviation is your estimate of the standard deviation of the actual underlying distribution. The standard error of the mean, on the other hand, is your estimate of the standard deviation of your measurement of the mean. Thus, the more points that go into your μ μ

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measurement of the mean, the smaller the standard error of the mean should be. On the other hand, the standard deviation should not depend on your sample size. The first step of calculating either term is to calculate the point estimate of the variance: - = i i N y 2 2 ) ˆ ( ˆ μ σ N is the number of data points, y i are the data points themselves, and μ ˆ is the mean of the data points (i.e. point estimate of the mean). In order to estimate the standard deviation (STD) of the underlying distribution , take the square root of the estimate of the variance: 2 ˆ ˆ σ σ = In order to estimate the standard deviation in your estimate of the mean, take the standard error of the mean (SEM): N sem σ σ ˆ ˆ = The more data you have (the larger N ), the smaller the SEM. That’s because with more data, you have a better (less variable) estimate of the mean. The STD, on the other hand, does not change with the size of N. Confidence intervals Once you calculate a point estimate, how do you know how good an estimate of your underlying data it really is? If you calculated a point estimate of a mean, the SEM is one measure of how good your estimate is. But the confidence interval is a much more general and powerful approach. The confidence interval tells you the probability that the parameter of interest in the underlying distribution falls within some interval. More specifically, if you have taken a number of measurements of a parameter, the confidence interval will allow you to specify a range that with 95% probability contains the true value of the parameter.
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Ch8_Statistics - Statistics Point estimation Lets say you...

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