Understanding Z-Scores

• Here are study notes based on the images you sent

Z-Scores

• A Z-score, also known as a standard score, indicates the precise location of a score within a distribution.
• Useful for finding percentiles, which represent the percentage of people who scored lower than a given score.
• A positive Z-score (e.g., +1.0) indicates the score is above the mean.
• A negative Z-score (e.g., -1.0) indicates the score is below the mean.
• Population formula: z = (X - μ) / σ
• Sample formula: z = (X - M) / s
• Formula for finding a raw score (X): X = μ + (z × σ)
• Z-scores help determine proportions under the normal curve.
• The total area under the normal curve is 1.
• Z-tables help in finding the area corresponding to specific Z-scores.
• z=0 corresponds to the 50th percentile (mean).
• z=+1 corresponds to the 84th percentile.
• z=-1 corresponds to the 16th percentile.
• Curve proportion = 1 (100%).
• Normal distribution has a mean of 0 and a standard deviation of 1.
• Not for skewed distributions.
• Mean, median, and mode are at the center point in a normal distribution.

Probability

• 0.0 indicates an event won't occur.
• 1.0 indicates an event will occur.
• P(A) = (Number of outcomes in A) / (Total number of outcomes)
• In random sampling, each person has an equal chance, maintaining constant probability.

Multiplication Rule

• If two events are independent (the outcome of one does not affect the other), the probability is found by multiplying their individual probabilities.

Addition Rule

• For mutually exclusive events (only one can happen at a time), the probability of one event or another occurring is the sum of their individual probabilities.
• Use Z-scores and the area under the standard normal curve to determine percentile ranks.

Hypothesis Testing

• The null hypothesis (H₀) states no difference exists between the groups being compared.
• The alternative hypothesis represents what researchers want to support.
• Type I error: Null hypothesis is rejected when it's true.
• Type II error: Failure to reject a false null hypothesis.

Statistical Significance

• Determines if the result is meaningful or due to chance.
• A common alpha level is 0.05.
• If p is less than 5%, the result is less likely to be random.
• If p is higher than 5%, the result is more likely due to chance.

Inferential Statistics

• Inferential statistics involve drawing conclusions based on data collected from a sample.

Parametric Tests

• Parametric tests, such as t-tests and z-tests, require certain assumptions about estimates of parameters and are used with interval and ratio data.
• Ordinal and nominal data use nonparametric tests.

Simple Tests

• Allow testing the null hypothesis for a simple sample when the population variance is unknown.
• Standard error: Sx = S / √N
• Z test: z = (X - μ) / Sx

Central Limit Theorem

• When taking a large number of random samples, the distribution of the sample means will form a normal distribution.
• Standard Distribution: the distribution of sample means that is gotten when repeatedly taking random samples of a fixed size population