Understanding Normal Distribution: Probability Density Curves & Continuous Variables, Slides of Probability and Statistics

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Chapter 6

The Normal Distribution

In this handout:

• Probability model for a continuous random variable

• Normal distribution

• The idea of a continuous probability distribution draws from the relative frequency histogram for a large number of measurements (recall chapter 2).
• Group data in class intervals, compute the relative frequencies of the intervals, and build a histogram (figure (a)).
• The total area under the histogram is 1.
• For two boundary points a and b of a class, the relative frequency of measurements in interval [a,b] is the area above this interval in the histogram.
• With increasing number of observations, refine the histogram by having more class intervals with smaller widths (fig. (b)).
• By proceeding in this manner, the jumps between consecutive rectangles tend to dampen out, and the top of the histogram approximates the shape of a smooth curve, as illustrated in figure (c).
• This curve is called probability density curve. Docsity.com

• For important distributions, areas have been extensively tabulated.
• In most tables, the entire area to the left of each point is tabulated.
• To obtain the probabilities of other intervals, we must apply the following rules: P[a < X < b] = (Area to left of b) – (Area to left of a)

Figure 6.2 (p. 225) Different shapes of probability density curves. (a) Symmetry and deviations from symmetry; (b) different peakedness.

Figure 6.4 (p. 226) Quartiles of two continuous distributions.

Normal distribution

A bell-shaped distribution has been found to provide a reasonable approximation in many situations.

The normal distribution with a mean of μ and a standard deviation of σ is denoted by N(μ, σ).

The curve never reaches 0 for any value of x, but because the tail areas outside (μ-3σ, μ+3σ) are very small, we usually terminate the graph at these points.

Figure 6.6 (p. 230) Two normal distributions with different means but the same standard deviation.