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Normal Distribution (Z-Score) Table


What is the Normal Distribution?
How to Read the Z-Table
Normal Distribution Properties
Common Applications
Z-Score Calculation







What is the Normal Distribution?

The normal distribution, also called the Gaussian distribution or bell curve, is a continuous probability distribution symmetric around its mean. It's characterized by two parameters: μ (mean) and σ (standard deviation). The standard normal distribution has μ = 0 and σ = 1, and any normal distribution can be standardized using the z-score transformation: z = (x - μ)/σ.
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How to Read the Z-Table

The z-table shows cumulative probabilities P(Z ≤ z) for the standard normal distribution. The left column shows the z-score to one decimal place, and the top row shows the second decimal place. To find P(Z ≤ 1.96), locate 1.9 in the left column and 0.06 across the top. For negative z-scores, use symmetry: P(Z ≤ -z) = 1 - P(Z ≤ z). For P(Z > z), calculate 1 - P(Z ≤ z).
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Normal Distribution Properties

The normal distribution is symmetric around the mean with 68% of values within ±1σ, 95% within ±2σ, and 99.7% within ±3σ (empirical rule). The total area under the curve equals 1. The mean, median, and mode are all equal. The probability density function is f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²)). The distribution extends from -∞ to +∞ but is nearly zero beyond ±3σ.
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Common Applications

Normal distributions model natural phenomena like human heights and weights, test scores, measurement errors, IQ scores, blood pressure readings, and manufacturing tolerances. In statistics, it's fundamental for hypothesis testing, confidence intervals, regression analysis, and the Central Limit Theorem. Many statistical tests assume normality, making z-tables essential for calculating p-values and critical values.
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Z-Score Calculation

A z-score indicates how many standard deviations a value is from the mean. Positive z-scores are above the mean, negative below. A z-score of 1.96 corresponds to the 97.5th percentile (used for 95% confidence intervals). Z-scores enable comparison across different normal distributions and conversion to percentiles. The table provides precise probabilities for standardized values essential in statistical inference.
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