Introduction to Statistics

Steven A. JonesBiomedical Engineering

Louisiana Tech University

(Created for our NSF-funded Research Experiences in Micro/Nano

Engineering Program)

Experimental Design

1.Develop a Hypothesis2.Design Statistical Analysis3.Set up Experiment4.Test Experiment (Positive Control)5.Collect Data6.Perform Statistical Analysis

Step 2 must not be done after the data have been collected!

Gaussian Distribution

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Uniform Distribution

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Rayleigh Distribution

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Some Common Probability Distributions

Gaussian: Sum of numbers.

Uniform: e.g. Dice throw.

Rayleigh: Square root of the sum of the squares of two Gaussians.

Introductory Question

There is a 60% chance of rain on Friday and a 45% chance of rain on Saturday.

What is the probability that it will rain on Friday and Saturday?

Introductory Question

There is a 60% chance of rain on Friday and a 45% chance of rain on Saturday.

What is the probability that it will rain on either Friday or Saturday?

Axioms of Probability

For independent outcomes A and B:

BPAPBPAP

BPAP

BPAPBAP

APAP

BPAPBAP

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or

If probabilities are small, then they can be added when “or” is used.

Relationships Among Probability Distributions

Assume that are uniformly distributed. Then:ix

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iiii xAy

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Is Gaussian (Normal) distributed for N sufficiently large.

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21 yyz Is distributed.

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zw Is Rayleigh distributed.

Simple Distribution

You expect that the correct value for a height measurement is 12 meters, and the standard deviation is 3 meters. One way to determine whether or not you are correct is to take some measurements.

What could you conclude if you made one measurement and the value fell as follows on the expected distribution?

Gaussian Distribution

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What if three values fell as follows on the expected distribution?

Gaussian Distribution

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Did the two data sets to the left come from

different distributions?9 9.5 10 10.5 11 11.5 12

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What about these two?

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Set 2:

Set 1:

Set 2:

Set 1:

How confident are you that the data sets in each plot below come from different

distributions?

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Lower standard deviation

Smaller difference in means

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Fewer data points

A Student’s T test measures the confidence you can have that two values are inherently different, based on three

parameters

1. Difference of the means

2. Standard deviations

3. Number of data points obtained

Particularly useful when there are multiple confounding variables.

E.g. Blood pressure drugs – are we, on average, lowering blood pressure?

Student’s T Test

A Student’s T test is used to answer the following question:

Given:

• Difference of the means

• Standard deviations

• Number of data points obtained

• That these data come from normal distributions

What is the probability (p) that they came from the same underlying distribution?

Example

Given the mean and standard deviation for pressure, along with the number of points measured from a clinical drug trial, what is the probability that the drug had an effect on the distribution (i.e. that it changed the blood pressure of these individuals on average).

Sample Mean: Mean from the sample that was taken (the 2000 people in the drug trial).

Distribution Mean: Mean that would occur if you could give the drug to everyone in the world and do the measurement.

Underlying and Sample Distributions

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Uniform Distribution

Uniform (Theory)

Gaussian Distribution

Gaussian (Theory)

Statistical Tests You Should Know

T-test: Are the means of two data sets the same?

F-test: Are the standard deviations of two data sets the same?

Chi-Squred Test: Does the distribution of a data set match a proposed distribution?

Anova: Like an F-test for multiple variables.

Pearson’s Correlation Coefficient: Does one variable depend on another?

To Run a T-Test

Calculate the mean of the data.

Calculate the standard deviation of the data.

Determine the T statistic (e.g. )

From T determine p.

p is “the probability that you would get a difference in means this large or smaller, given that the two measurement sets come from the same distribution.”

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Interpretation of T test

You set the value that you consider significant.

Medical applications: p < 0.05 is “significant.”

Since p < 0.05 is a 1/20 probability, you will typically be wrong once in every 20 T tests.

Hypothesis

Null hypothesis: statement that the two distributions are the same. i.e. “Altase causes no change in blood pressure.”

Alternative hypothesis: Can vary. Altase reduces the mean blood pressure. Altase changes the mean blood pressure.

One-Tailed vs Two-Tailed

Depends on “alternative hypothesis.” One tail: If alternative hypothesis is that one mean is

greater than the other. Two tail: If alternative hypothesis is that the means

are different.

Saying that one of the means is greater is more restrictive.

The confidence you have in your result depends on the prediction (1st law of the frisbee).

Example

A friend throws a frisbee. It bounces off a pole, goes to the roof of the house, rolls along an arc, flips off the gutter, and then lands in the fountain. Are you impressed?

A friend predicts that the frisbee will do the above, and then it happens. Are you impressed?

As with the frisbee, statistical analysis depends on how far you are willing to stick your neck out.

Confidence Interval

States the range of values that contains the true value within a given percent confidence.

Depends on number of samples, and desired confidence.

Not a statistical test of significance, but related to the T-test

,

Gaussian Distribution

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y• The more samples we have, the narrower the confidence interval.