2-2 Histograms 57 Remembering Skewness: Skewed Left: Resembles toes on left foot Skewed Right: Resembles toes on right foot FIGURE 2-5 Bell-Shaped Distribution of Arm Circumferences Because this histogram is roughly bell-shaped, we say that the data have a normal distribution. (A more rigorous definition will be given in Chapter 6.) Normal Distribution When graphed as a histogram, a normal distribution has a “bell” shape similar to the one superimposed in Figure 2-5. Many statistical methods require that sample data come from a population having a distribution that is approximately a normal distribution, and we can often use a histogram to judge whether this requirement is satisfied. There are more advanced and less subjective methods for determining whether the distribution is close to being a normal distribution. Normal quantile plots can be very helpful for assessing normality: see Part 2 of this section. Uniform Distribution With data having a uniform distribution, the different possible values occur with approximately the same frequency, so the heights of the bars in the histogram are approximately uniform, as in Figure 2-4(b). Figure 2-4(b) depicts outcomes of digits from state lotteries. Skewness A distribution of data is skewed if it is not symmetric and extends more to one side than to the other. Data skewed to the right (also called positively skewed) have a longer right tail, as in Figure 2-2 and Figure 2-4(c). Commute times in Los Angeles are skewed to the right. Annual incomes of adult Americans are also skewed to the right. Data skewed to the left (also called negatively skewed) have a longer left tail, as in Figure 2-4(d). Life span data in humans are skewed to the left. (Here’s a mnemonic for remembering skewness: A distribution skewed to the right resembles the toes on your right foot, and one skewed to the left resembles the toes on your left foot.) Distributions skewed to the right are more common than those skewed to the left because it’s often easier to get exceptionally large values than values that are exceptionally small. With annual incomes, for example, it’s impossible to get values below zero, but there are a few people who earn millions or billions of dollars in a year. Annual incomes therefore tend to be skewed to the right. PART 2 Assessing Normality with Normal Quantile Plots Some really important methods presented in later chapters have a requirement that sample data must be from a population having a normal distribution. Histograms can be helpful in determining whether the normality requirement is satisfied, but they are not very helpful with small data sets. Section 6-5 discusses methods for he ta uGo Figure 2.5 quintillion bytes: Amount of data generated each day. (A quintillion is 1 followed by 18 zeroes.)
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