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Normal Distribution
๐ The normal distribution shows that data near the mean are more frequent in occurrence than data far from the mean. It underpins much of probability theory and statistical inference.
๐ The Classical Bell-shaped Curve
๐ Overview
The Normal Distribution (Gaussian distribution) is a continuous probability distribution that is symmetrical around its mean. It is characterised by the iconic bell-shaped curve and is one of the most important distributions in medicine, biology, and social sciences.
โจ Key Characteristics
- โ๏ธ Symmetry: The curve is perfectly symmetrical around the mean.
- ๐ Bell-Shaped: Peak at the mean with tails extending indefinitely.
- ๐ Mean = Median = Mode: All measures of central tendency align at the centre.
- ๐ Standard Deviation (ฯ): Controls spread โ small ฯ = narrow & tall curve, large ฯ = wide & flat curve.
๐ Properties
- ๐ 68โ95โ99.7 Rule (Empirical Rule):
- ~68% of data within ยฑ1ฯ
- ~95% within ยฑ2ฯ
- ~99.7% within ยฑ3ฯ
- โ Total Probability = 1: Area under curve = 100% of all outcomes.
๐งฎ Standard Normal Distribution
- Defined with ฮผ = 0 and ฯ = 1.
- Values converted with Z-score: z = (X โ ฮผ) / ฯ
- ๐ Z-scores = "How many SDs away from the mean" โ allows comparison across datasets.
๐ Applications
- ๐ Statistical Inference: Confidence intervals, hypothesis testing.
- ๐ Central Limit Theorem: Sampling means approximate normality when n is large.
- โ๏ธ Quality Control: Control charts use normal distribution to detect abnormalities.
- ๐งฌ Natural & Social Sciences: Human height, IQ, lab measurements often approximate a normal curve.
๐ Checking for Normality
- ๐ Visual Methods: Histogram (bell curve shape), QโQ plots (linear trend).
- ๐ Statistical Tests: ShapiroโWilk test, KolmogorovโSmirnov test.
๐ Summary
โ The Normal Distribution is fundamental in statistics. It is bell-shaped, symmetrical, and defined by its mean (ฮผ) and standard deviation (ฯ). The 68โ95โ99.7 rule is a cornerstone concept, and Z-scores allow comparison across distributions. Understanding it is crucial for interpreting lab values, designing studies, and applying evidence-based medicine.
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