• Random Variable: Process generating random outcomes.

• Sample: Outcomes from a random variable.

• Descriptive Statistics: Summary of a sample.

• Inferential Statistics: Educated guess about the random variable that generated a sample.

A random variable is a process that generates random outcomes.

• Example in Cognitive Neuroscience:

  • Measuring reaction times (RT) to visual stimuli. Each RT can be considered an outcome generated by a complex cognitive process.

A sample consists of measurements obtained from a random variable. It represents a subset of possible outcomes that we can analyze.

• Example in Cognitive Neuroscience:

  • Measuring reaction times (RT) to visual stimuli. The set of RTs collected from a group of participants is a sample.

Descriptive statistics summarize the characteristics of a sample, such as its central tendency and variability.

• Measures:

  • Mean, Median, Mode (for central tendency)
  • Variance, Standard Deviation (for variability)

• Example in Cognitive Neuroscience:

  • Calculating the average reaction time and its variability in the collected sample to understand the general responsiveness to the stimulus.

Inferential statistics involve making educated guesses about the population from which a sample was drawn. This involves hypothesis testing and estimation.

• Example in Cognitive Neuroscience:

  • Using a sample of reaction times to test a hypothesis about cognitive processing differences between conditions (e.g., with and without distraction).

• Random variables can be discrete or continuous.

• Discrete: Produces only outcomes on a finite or countably infinite number of values.

• Continuous: Produces outcomes on an interval of real numbers.

• Example:

  • Discrete: Number of correct responses on a memory test.

  • Continuous: Reaction time to a visual stimulus.

• Samples can also be discrete or continuous.

• Discrete: A sample with a finite or countably infinite number of measurements.

• Continuous: A sample with an interval of real numbers.

• Example:

  • Discrete: Number of correct responses on a memory test for a group of participants.

  • Continuous: Reaction times to a visual stimulus for a group of participants.

• A sample is discrete if it was generated from a discrete random variable.

• A sample is continuous if it was generated from a continuous random variable.

• Descriptive statistics can be calculated for both discrete and continuous samples.

• Example:

  • Mean, median, mode, variance, and standard deviation can be calculated for both discrete and continuous samples.

• Note:

  • The specific formulas for these measures may differ slightly for discrete and continuous samples.

• Different visuals will be better suited for different types of data.

• Example:

  • A histogram and density plot are good visuals for a continuous sample.

  • A bar plot is a good visual for a discrete sample.

Consider an fMRI experiemnt in which block of trials are performed under two conditions: A decision-making task and a motor task. The average BOLD signal is measured for each task and then substracted. The resulting difference scores – one for each participant – is a measure of the cerebellum’s role in non-motor cognitive processing. After the first 10 subjects, the difference scores you observe are as follows:

-0.2, 0.1, 0.3, -0.1, 0.2, 0.0, -0.3, 0.1, 0.2, -0.1

• What is the random variable in this experiment?

• Is it discrete or continuous?

• What is the sample in this experiment?

• How would you describe the sample using descriptive statistics?

• You need to make a decision regarding whether or not you think the cerebellum is involved in non-motor cognitive processing. What type of statistical tool or method would you use?