Statistics: Z-Scores, Normal Distribution, and Probability Concepts

Z-Score

The number of standard deviations a value is from the mean.

Positive Z-Score

The value is above the mean.

Negative Z-Score

The value is below the mean.

Z-Score Formula

z = (value - mean) / standard deviation.

Standard Deviation

Measures how spread out the data values are around the mean.

Larger Standard Deviation

Data is more spread out.

Smaller Standard Deviation

Data is more clustered near the mean.

Normal Distribution

A symmetric, bell-shaped curve centered at the mean.

Empirical Rule (68%)

About 68% of values lie within 1 standard deviation of the mean.

Empirical Rule (95%)

About 95% of values lie within 2 standard deviations of the mean.

Empirical Rule (99.7%)

About 99.7% of values lie within 3 standard deviations of the mean.

Probability Under a Normal Curve

Represents the proportion of values in the population.

Probability

A measure of how likely an event is to occur.

Complement Rule

P(A^c) = 1 - P(A).

General Addition Rule

P(A or B) = P(A) + P(B) - P(A and B).

Disjoint Events

Events that cannot happen at the same time.

Independent Events

One event does not affect the probability of another.

Multiplication Rule (Independent)

P(A and B) = P(A) × P(B).

Conditional Probability

P(B|A) = P(A and B) / P(A).

Sampling Distribution

The distribution of sample statistics (like sample means) from repeated sampling.

Central Limit Theorem

For large sample sizes, the distribution of sample means becomes approximately normal.

Standard Error (Mean)

SE = population standard deviation / √n.

Effect of Sample Size on Standard Error

Larger sample size → smaller standard error.

Stable Sample Means

Larger sample sizes produce more consistent sample means.

Standard Error (Proportion)

SE = √(p(1-p)/n).

Interpretation of Standard Error

Tells how far a sample statistic is expected to vary from the population value.

Mean

The arithmetic average of the data.

Median

The middle value when data is ordered.

Mode

The most frequently occurring value.

Mean vs Median (Skewed Data)

Use the median when the data is skewed or has outliers.

IQR (Interquartile Range)

Measures the spread of the middle 50% of data: Q3 - Q1.

Outlier Definition (Boxplot)

A value more than 1.5 × IQR above Q3 or below Q1.

Histogram

Used to display distributions of quantitative data.

Bar Chart

Used to display frequencies of categorical data.

Pie Chart

Shows parts of a whole, using percentages or proportions.

Misleading Graph

A graph that distorts or exaggerates the data.

Common Graph Misleading Trick

Manipulating the y-axis scale to exaggerate change.