{"id":641,"date":"2020-07-20T21:47:23","date_gmt":"2020-07-21T01:47:23","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=641"},"modified":"2025-06-19T16:58:38","modified_gmt":"2025-06-19T20:58:38","slug":"5-2-normal-density-curve","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/5-2-normal-density-curve\/","title":{"raw":"5.2 Normal Density Curve","rendered":"5.2 Normal Density Curve"},"content":{"raw":"The normal density curve characterizes the normal distribution, which is the most widely used probability distribution for continuous variables. The normal distribution is symmetric and bell-shaped (for this reason it is often referred to as the \u201cbell curve\u201d). The normal density function has two parameters: the mean [latex]\\mu[\/latex]\u00a0and the standard deviation [latex]\\sigma[\/latex].\u00a0The parameter [latex]\\mu[\/latex]\u00a0controls the centre (location) of the distribution and [latex]\\sigma[\/latex]\u00a0controls the shape of the distribution. When\u00a0[latex]\\sigma[\/latex]\u00a0is larger, the curve appears shorter and fatter; when [latex]\\sigma[\/latex]\u00a0is smaller, the curve appears taller and slimmer.\r\n\r\nFigure 5.2 shows three normal density curves--[latex]N(0, 2), N(0, 1)[\/latex] and [latex]N(4, 1)[\/latex]. [latex]N(0,1)[\/latex] and [latex]N(4, 1)[\/latex] have the same standard deviation; therefore, they have the same shape; if you shift the location of [latex]N(0, 1)[\/latex] to the right by 4, the two distributions are exactly the same. [latex]N(0,1)[\/latex] and [latex]N(0, 2)[\/latex] have the same mean; therefore, they center at the same location. [latex]N(0, 2)[\/latex] has a larger standard deviation; therefore, the density curve is shorter and fatter.<\/a>\r\n\r\n[caption id=\"attachment_2756\" align=\"aligncenter\" width=\"600\"]\"Three Figure 5.2<\/strong>: Three normal density curves. [Image Description (See Appendix D Figure 5.2)<\/a>][\/caption]If a random variable [latex]X[\/latex] follows a normal distribution with mean [latex]\\mu[\/latex]\u00a0and standard deviation [latex]\\sigma[\/latex], we write [latex]X \\sim N(\\mu, \\sigma)[\/latex]. The probability density function of a normal random variable [latex]X[\/latex]\u00a0is given by:\r\n

[latex]f(x) = \\frac{1}{\\sqrt{2 \\pi} \\sigma} e^{-\\frac{(x- \\mu)^2}{2\\sigma^2}}, -\\infty < x < \\infty \u00a0\\text { with } \\pi \\approx 3.142 \\text { and } e \\approx 2.718[\/latex].<\/p>\r\nThe normal density curve has the following properties:\r\n

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Key Facts: Properties of a Normal Density Curve<\/p>\r\n\r\n<\/header>\r\n

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