Precision, Accuracy, Standard Error & the Central Limit Theorem - - PowerPoint PPT Presentation

Precision, Accuracy, Standard Error & the Central Limit Theorem

“He uses statistics as a drunken man uses lamp posts, for support rather than illumination.”

Andrew Lang (Scottish poet)

Statistical Vocabulary Review

• Descriptive statistics – numerical/graphical summary of data
• Inferential statistics – predict or control the values of variables

– make conclusions with

• Distribution – probability associated with each possible value of a variable

– a.k.a. probability function

• Parameter – population characteristic; unknown

– needs to be estimated for a sample (e.g. mean of a population)

• Statistic – estimation of parameter (e.g. mean of a sample)
• Error – difference between an observed value (or calculated) value and its true

(or expected) value

• Degrees of freedom – the number of values in the final calculation of a statistic that

are free to vary

More Terminology

• Precision – a measure of how close measured/estimated values are to each
• ther
• Accuracy – a measure of how close an estimator is expected to be to the

true value of a parameter

• Bias – how far the average statistic lies from the parameter it is estimating

– e.g. the error which arises when measuring or estimating a parameter

• Error – the difference between an observed value (or calculated) value and its

true (or expected) value

Errors from chance will cancel each other out in the long run, BUT those from bias will not.

ACCURACY ERROR

low high low high high low

PRECISION

high low

BIAS

“How confident are we in our statistic?” Standard error – standard deviation of a statistic Standard error of the mean - reflects the overall distribution of the means you would get from repeatedly resampling

n = sample size s = sample standard deviation

Standard error

𝑇𝐹𝑛𝑓𝑏𝑜 = 𝑡 𝑜

Small values = the more representative the sample will be of the overall population Large values = the less likely the sample adequately represents the overall population For parameter estimation - Less precision is reflected in a larger standard error

The Central Limit Theorem

“Sample means tend to cluster around the central population value.” Therefore:

• When sample size is large, you can assume that 𝑦 is close to the value of 𝜈
• With a small sample size you have a better chance to get a mean that is far off the true

population mean t-distribution

(sampling distribution)

Normal distribution

The Central Limit Theorem

Prove it to yourself: www.tinyurl.com/clt-simulator

• 1. Click Begin on left
• 2. Select Uniform distribution as our parent population
• 3. For the first measurement select Mean with N=2 (only use 2 samples to generate

mean)

• 4. For the first measurement select Mean with N=25 (use 25 samples to generate

mean)

• 5. Repeatedly click Animated to watch points be randomly selected, and the mean
• f the sample generated/plotted in the distributions below
• 6. Click 5 or 10,000 or 100,000 to generate multiple animations at once (number of

times you sample the population)

• 7. How does the resulting distribution of the means change?