Please enable JavaScript.
Coggle requires JavaScript to display documents.
Probability - Coggle Diagram
Probability
Introduction to Probability
Defining Probability
If the possible outcomes are identified as A, B, C, D, and so on, then Probability of A = number of outcomes classified as A/total number of possible outcomes
For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes.
Random Sampling
Independent random sampling: requires that each individual has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next if more than one individual is selected.
Random sampling with replacement: a definition of random sampling that requires equal chance of selection and constant probabilities.
Random sampling: requires that each individual in the population has an equal chance of being selected. A sample obtained by this process is called a simple random sample.
A sample obtained by this process is called a simple random sample.
Random sampling without replacement: to define random sampling without the requirement of constant probabilities
Probability and Frequency Distributions
Frequency distributions allow us to visualize the population, and the area or proportion of the graph can be used to calculate probabilities for specific outcomes.
Percentiles and Percentile Ranks
Finding Percentiles
Step 2: Use the unit normal table to find the z-score that corresponds to that proportion. If the proportion is below the mean, assign a negative sign to the z-score.
Step 3: Convert the z-score into the X value (percentile) using the formula
X=μ+(z⋅σ)
Step 1: Convert the given percentile (percentage) into a proportion of the distribution.
Quartiles
Quartile: a value that divides a data set into four equal parts.
IQR: the range of the middle 50% of the data.
Percentile rank: a particular score is defined as the percentage of individuals in the distribution with scores at or below that particular score
Percentile: the particular score associated with a percentile rank
Probability and the Normal Distribution
The Unit Normal Table
Unit Normal Table: This table lists proportions of the normal distribution for a full range of possible z-score values.
Proportions on the right and left sides are equal. For negative z-scores, the body and tail positions swap but proportions remain the same.
The body is always the larger portion of the distribution, and the tail is always the smaller portion, regardless of side.
Although the z-score values change signs (+ and −) from one side to the other, the proportions are always positive.
Probabilities, Proportions, and z-Scores
Finding Proportions/Probabilities for Specific z-Score Values
The unit normal table lists relationships between z-score locations and proportions in a normal distribution.
For any z-score location, you can use the table to look up the corresponding proportions.
Similarly, if you know the proportions, you can use the table to find the specific z-score location.
Because we have defined probability as equivalent to proportion, you can also use the unit normal table to look up probabilities for normal distributions.
Probabilities and Proportions for Scores from a Normal Distribution
Calculate the probability for a specific X value.
Convert the X value into a z-score.
Use the unit normal table to find the proportion or probability corresponding to that z-score.
Calculate the score (X value) corresponding to a specific proportion in a distribution.
Use the unit normal table to find the z-score corresponding to the probability.
Convert the z-score into the X value using the mean and standard deviation.
Looking Ahead to Inferential Statistics
Probability provides the foundation for inferential statistics by allowing researchers to evaluate how likely certain sample outcomes are, given the population distribution.