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# Importance of being uncertain
How samples are used to estimate population statistics and what this means in terms of uncertainty
### IBiG
### January 14th, 2020
Points of Significance - Nature Methods

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# Introduzione del ciclo di incontri

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## Modalita'
1. Cadenza mensile.
1. Durata di massimo 1 ora.
1. Gli incontri si svolgeranno come dei Journal clubs.
1. Verra' utilizzato come linea guida la rubrica di _Nature Methods_: .bold[Point of Significance].
1. I membri dell'IBiG si prendereanno la responsabilita' di preparare gli incontri.
1. Ogni suggerimento o proposta di tema dai colleghi e' il benvenuto.
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## Perche incontri sulla statistica?

- approximately half the articles published in medical journals that use statistical methods use them incorrectly

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## Obiettivi
1. Pianificazione degli esperimenti
1. Raccolta dei dati
1. Analisi dei dati
1. Interpretazione dei risultati
1. Visualizzazioni grafiche
]

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## Game show

- You are given the choice of three doors
 
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- Behind one door is a car; behind the others, goats.
  
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- You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat.
  
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- He then says to you, "Do you want to pick door No. 2?".
  
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### Is it to your advantage to switch your choice? 
]

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.content-box-grey[
.center[.black[To help you **move beyond** an intuitive understanding of fundamental statistics relevant to your work.

Our presentation will be practical and cogent, with focus on foundational concepts, practical tips and common misconceptions

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- andare oltre una comprensione intuitiva

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#### How samples are used to estimate population statistics and what this means in terms of uncertainty

It tells us the chances of being wrong.
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La statistica non da un numero che ci dice se abbiamo ragione o meno, la statistica da un range di probabilita. Da incertezza

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- summarizes the main features of a data set

- measures such as the mean and standard deviation
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- generalizes from observed data

- make inferences about the population
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Underpinning both are the concepts of sampling and estimation:

- collecting data

- quantifying the uncertainty

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# Population

The frequency histogram of all possible values is called the **population distribution**.

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Ciao

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### We are typically interested in inferring:

- The **mean** is calculated as the arithmetic average of values and can be unduly influenced by extreme values
  
- The **standard deviation** is calculated based on the square of the distance of each value from the mean

- The **median** is a more robust measure of location and more suitable for distributions that are skewed or otherwise irregularly shaped

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We cannot directly measure the mean and the s.d.

The best we can do is estimate them using **collected data through the process of sampling**
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## SAMPLING

- Samples are sets of data drawn from the population, characterized by the number of data points n, usually denoted by X and indexed by a numerical subscript.

- Larger samples approximate the population better.

- To maintain validity, the sample must be representative of the population.

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### Central event in the history of statistics and polling
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### Literary Digest

- Predict correctly the previous 4 elections

- 10M people sample

- Republican Alfred Mossman Landon wins by large majority
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- 0.05M people sample

- Democratic Franklin D. Roosevelt wins

- With 0.003M he replicated the results of Literary Digest
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]

1. Roosevelt won
 
1. Literary Digest ceased publishing within a few months of the election

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## SAMPLING

- where all values in the population have an equal chance of being selected at each stage of the sampling process.

- Representative does not mean that the sample is a miniature replica of the population.

- A sample will not resemble the population unless n is very large.

#### When constructing a sample, it is not always obvious whether it is free from bias.

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.black[Samples are our windows to the population, and their statistics are used to estimate those of the population.]
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## Central Limit Theorem

- Notice that the distribution of sample means looks quite different than the population one

- It appears similar in shape to a normal distribution.

- Also notice that its spread, is quite a bit smaller than that of the population
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Despite these differences, the population and sampling distributions are intimately related
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## Central Limit Theorem

- The CLT tells us that the distribution of sample means (Fig. 2c) will become increasingly close to a normal distribution as the sample size increases
  
- Regardless of the shape of the population distribution (Fig. 2a) as long as the frequency of extreme values drops off quickly.

- The CLT also relates population and sample distribution parameters

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delta X:  is the spread of sample means

delta: is the spread of the underlying population.

As we increase n: delta X will decrease (our samples will have more similar means) but delta will not change (sam-pling has no effect on the population)

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## A demonstration of the CLT for different population distributions

- Increase in precision of our estimate of the population mean with increase in sample size

- Notice that it is still possible for a sample mean to fall far from the population mean, especially for small n
 
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## A demonstration of the CLT for different population distributions

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For example, with samples of size n = 3 from the irregular distribution, the number of times the sample mean fell outside the standard deviation interval ranged from 7.6% to 8.6%.

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- Always keep in mind that your **measurements are estimates**, which you should not endow with an aura of exactitude and finality. 
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END
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