Glossary
Central Limit Theorem
A fundamental theorem stating that the sampling distribution of the sample mean (or sum) will be approximately normal, regardless of the population distribution, as long as the sample size is sufficiently large (typically n ≥ 30).
Example:
Even if individual student study times are skewed, the average study time from many large samples of students will tend to follow a normal distribution due to the Central Limit Theorem.
Empirical Rule (68-95-99.7 Rule)
A rule for normal distributions stating that approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.
Example:
If test scores are normally distributed with a mean of 75 and a standard deviation of 5, then about 95% of students scored between 65 and 85 according to the Empirical Rule.
Large Counts Condition
A condition for using a normal model for sample proportions, requiring that both the number of expected successes (np) and expected failures (n(1-p)) are at least 10.
Example:
To use a normal model for the proportion of students who prefer online learning, you'd need at least 10 students who prefer it and 10 who don't, satisfying the Large Counts Condition.
Mean (μ)
The arithmetic average of a dataset, calculated by summing all values and dividing by the number of values; it represents the center of a normal distribution.
Example:
If the average score on a pop quiz was 85, then 85 is the mean score for that quiz.
Median
The middle value in an ordered dataset, dividing the data into two equal halves; in a normal distribution, it is equal to the mean and mode.
Example:
In the ordered list of test scores {70, 75, 80, 85, 90}, the median is 80.
Mode
The value that appears most frequently in a dataset; in a normal distribution, it is equal to the mean and median.
Example:
If the most common shoe size sold at a store is size 9, then 9 is the mode of shoe sizes sold.
Normal Distribution
A continuous probability distribution characterized by its symmetrical, bell-shaped curve, where the mean, median, and mode are all equal.
Example:
The distribution of adult human heights often approximates a normal distribution, with most people clustered around the average height and fewer people at very short or very tall extremes.
Percentiles
Values below which a certain percentage of data falls in a distribution.
Example:
If a student scores in the 90th percentile on a standardized test, it means they scored better than 90% of the other test-takers.
Standard Deviation (σ)
A measure of the typical distance or spread of data points from the mean in a distribution.
Example:
A small standard deviation for test scores means most students scored very close to the average, indicating less variability.
Symmetrical
A property of a distribution where the left and right sides are mirror images of each other when folded along the center.
Example:
A perfectly symmetrical histogram would show the same frequency of data points on either side of its central peak.
Unimodal
A property of a distribution indicating that it has a single, distinct peak or mound.
Example:
A dataset showing the number of hours students sleep per night might be unimodal if most students sleep around 7-8 hours, with frequencies decreasing on either side.
Z-score
A standardized measure that indicates how many standard deviations a data point is away from the mean of its distribution.
Example:
A student with a z-score of 2.0 on a test scored two standard deviations above the average, indicating a very strong performance relative to their peers.