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Key Concepts Study Tool: Chapter 05
Click on each concept below to check your understanding.
1. The History of the Normal Distribution
- Normal curve a.k.a. Bell Curve a.k.a. Gaussian Curve.
- Carl Friedrich Gauss argued that the distribution of errors in the equation is random, and that their shape therefore assumes a normal distribution.
- But ... Abraham de Moivre used the curve first by noticing that as the number of events increases, the distribution of outcomes approaches a predictable bell-like curve.
- Stephen Stigler: “law of eponymy” states that “no scientific discovery is named after its original discoverer.”
2. The Normal Curve: Central Limit Theorem, Outliers, and Asymptomatic Normality
- Central limit theorem: If any variable has a known range, then it will increasingly approximate the normal curve as the number of samples increases.
- Outliers: Sufficiently unique trials that will never fall within the parameters of the normal distribution.
- Asymptotic normality: The assumption that a distribution is normal, but only with an infinite amount of trials. This allows us to use statistical techniques and principles on our data that are designed for normal distributions.
3. Describing Distributions: Symmetrical, Skewness, Kurtosis, Uni/Bi/Multimodal, Bell Curve
- Symmetrical: Exactly half of the scores fall above the mean, and exactly half of them fall below the mean. Additionally, both sides of the mean have the same pattern of distribution.
- Skewness: Opposite of symmetrical. Skewness occurs when there are more scores on one side of the mean than on the other, making one of the tails of the histogram longer than the other.
- Positively skewed or skewed to the right: Higher values are more spread out than lower values, so the right tail is longer.
- Negatively skewed or skewed to the left Lower values are more spread out than higher values, so the left tail is longer.
- Kurtosis: Refers to how flat or peaked a distribution is.
- Negative kurtosis : If a distribution is flatter than usual.
- Positive kurtosis : If it is more peaked than normal.
- Unimodal: When a distribution has only one mode. Histograms will have only one major “hump” in them.
- Bimodal: A distribution with two modes is bimodal, and will have two major “humps.”
- Multimodal: Any distribution that has more than two modes.
- Bell Curve: The normal curve is shaped like a bell, so it is often called a bell curve.