Introduction, statistics review

Aaron Meyer

Class overview

Distribution moments

The moments of a distribution describe its shape: $\mu_{n}=\int_{-\infty}^{\infty}(x-c)^{n}\,f(x)\,\mathrm{d} x$

Mean ( $n=1$ ), variance ( $n=2$ ), skewness ( $n=3$ ), kertosis ( $n=4$ )
Essential properties to determining how data will behave during analysis:
- How might your measurements need to change with changes in variance?
- What are these values for a normal distribution?

https://gregorygundersen.com/blog/2020/04/11/moments/

Sample statistics

If we sampled a number of times ( $n=3$ , say) many times, we could build a sampling distribution of the statistics (e.g., one for the sample mean and one for the sample standard deviation).

General properties of sampling distributions:

The sampling distribution of a statistic often tends to be centered at the value of the population parameter estimated by the statistics
The spread of the sampling distributions of many statistics tends to grow smaller as sample size $n$ increases
As $n$ increases, sampling distributions tends towards normality. If a process has mean $\mu$ and standard deviation $\sigma$ , them the sample mean ( $\mu$ ) and the sample standard deviation ( $\sigma / \sqrt{n}$ )

Sample statistics

This means that as $n$ increases, the better estimate $\mu_x$ is of $\mu$ .
- Sample standard deviation is the standard deviation of the mean.
When a population distribution is normal, the sampling distribution of the sample statistic is also normal, regardless of $n$ .
And the central limit theorem states that the sampling distribution can be approximated by a normal distribution when the sample size, $n$ , is sufficiently large.
Rule of thumb is that $n=30$ is sufficiently large, but there are times when smaller $n$ will suffice. Greater $n$ is required with higher skew.

Hypothesis Testing

In hypothesis testing, we state a null hypothesis that we will test; if its likelihood is less than some value, then we reject it.

For example:

$H_0$ : A particular set of points come from a normal distribution with mean $\mu$ and variance $\sigma$ .
$H_0$ : Two sets of observations were sampled from distributions with different means.

When we do an experiment, we typically take a sampling of several observations, to be more certain of the result.

T-distribution

When $n$ is small use the t-distribution with $n-1$ degrees of freedom.

$H_0$ : Assume $\mu=\mu_0$ then calculate t. $t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}}$
- Can think of $t$ designed to be $z/s$ , where it’s sensitive to the magnitude of the difference to the alternate hypothesis and scaled to control for the spread.
When comparing the differences between two means: (null hypothesis the means are the same, variances/sizes assumed equal). $t=\frac{{\bar{X}}_1-{\bar{X}}_2}{\sqrt{\frac{s_{X_1}^2+s_{X_2}^2}{n}}}$

Testing errors

Type I error: error of rejecting $H_0$ when it is true (false positive)
Type II error: not rejecting $H_0$ when it is false (false negative)
Alpha: significance level in the long run $H_0$ would be rejected this amount of the time falsely (i.e., we are willing to accept $x$ fraction of false positives)

Beware of goodness-of-fit tests because they are unlikely to reject any distribution with little data, and are very sensitive to the smallest systematic error with lots of data.

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Introduction, statistics review Aaron Meyer

Introduction, statistics review
Outline
Class overview
Machine learning & data-driven modeling in bioengineering
Lecture Slides
Textbook / Other Course Materials
Support / Office Hours
Learning Goals:
Practical Learning Objectives
Grade Breakdown
Labs
Project
Exams
Keys to Success
Introduction
How do we need to learn about the world?
Three things we need to learn about the world
Area of Focus
Why do we need models to learn about the world?
Why we need models - Can a biologist fix a radio?
Why we need models - Can a biologist fix a radio?
Why we need models - Can a biologist fix a radio?
Comparisons
Machine learning outperforms humans for some tasks
Machine learning outperforms humans for some tasks
Machine learning outperforms humans for some tasks
Machine learning outperforms humans for some tasks
Basic Statistical Concepts
Data
Types of variables
Probability
Coin toss example
Distributions
Other Probabilities
Normal Distributions
Normal Distributions
Other Distributions
Distribution moments
Sample statistics
Sample statistics
Sample statistics
Hypothesis Testing
T-distribution
Effect size
Kolmogorov-Smirnov Test
Kolmogorov-Smirnov Test
Kolmogorov-Smirnov Test Statistic
Graphical Analysis
Testing errors
Multiple hypotheses
Reading & Resources
Review Questions