Mean, variance, and histograms

By the end of this module, you should be able to analyze large data sets.

Key Concepts:

1. EEG
2. Mean
3. Variance
4. Standard Deviation
5. Histogram

Motivation from epilepsy

An EEG (electroencephalogram) is a tool that tracks and records electrical activity (voltage) in the brain. Data from an EEG can be used to diagnose and monitor seizure disorders.

Exercise 1.1

Go to the webpage EEG in Common Epilepsy Syndromes . This page shows the EEG signals recorded from patients with various types of epilepsy. What do you notice about these signals, particularly media files 3, 5, 6, 9, and 10? Can you spot the period in the data where the seizures occur?

While in some cases, it may be easy to recognize a seizure from a plot of EEG data, the EEG itself produces a large set of numbers corresponding to the voltage at discrete points in time. If the sampling rate were 250 Hz, then the EEG would record the voltage every 4 ms. This means that to record the activity in the brain for 10 minutes, the EEG would give 150,000 voltage readings. It would be very difficult to look at such a large set of data and make conclusions about what types of seizures, if any, are present. This module will give you the basic tools needed to analyze large sets of data, such as the data taken from an EEG. Below is a summary of the concepts you will learn to compute in this module.

Summary of Key Concepts

Mean

The mean of a set of numbers is the average. If x is a vector of n numbers, then the mean of x is given by

In Matlab, the mean of the vector x can be computed by typing mean(x).

Example 2.1

Let x = [1, 7, 2, 5, 9, 6]. Then, $\overline{x}=(1+7+2+5+9+6)/6=5.$

Example 2.2

Suppose you are interviewing for a job where the employees make the following daily salaries:

• 16 general employees: $100 each
• 3 managers: $900 each
• 1 owner: $1700

Then, the mean daily salary is given by

Notice that the mean is $300, which is 3 times the daily salary of 80% of the workers! Thus, if you are simply told the mean of the daily salaries, you will not have an accurate idea of how much money you would be making if you got the job. The variance and standard deviation are two measures that can give you an idea of how well the mean represents the data.

Exercise 2.1

Compute the mean of the vector y = [3, 8, 2, 5, 5, 7], both on paper and using Matlab.