EXAMPLE 26Normal distribution mean and standard deviation
Figure 23 shows two normal distributions, with different means but the same standard deviation. Which distribution has mean μ=6 and which distribution has mean μ=2?
Figure 24 shows two normal distributions, with the same mean but different standard deviations. Which distribution has σ=1 and which distribution has σ=2?
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Solution
Note that the two distributions have precisely the same spread or shape because each distribution has the same standard deviation, σ=2. However, the yellow distribution is symmetric about an axis drawn at 2, and it is centered at 2. Therefore, it has mean μ=2. The green distribution is symmetrical about, and centered at μ=6. The two curves are essentially identical, with the green one shifted four units to the right.
Because σ is a measure of spread, the larger the value of σ, the more spread out the distribution of X will be. This is illustrated in Figure 24. The normal distribution with the smaller standard deviation (σ=1) has a curve with a higher peak in the center and thinner “tails” than the distribution with a larger standard deviation (σ=2). Thus, the green distribution has σ=1 and the yellow distribution has σ=2.