For example, the six-sigma method in quality management. The normal distribution describes phenomena with predictable outcomes that surround the mean value. Then the resulting distribution will often become skewed. When the conditions of the central limit theorem become invalid, i.e., when the factors are When there are a number of small factors or random events affecting a phenomenon, the sum of total effect often leads to the normal distribution, such as people’s height, weight, IQ, etc. Each individual random variable, however, can follow any distribution. Its popularity is confirmed by the central limit theorem, which shows that the sum of a sequence of independent random variables follows the normal distribution. From its name “normal”, we can guess it is deemed as the most common and well-studied form of distribution. When we collect and analyze data, that data can be distributed or spread out in different ways.The normal distribution has a symmetric bell-shaped probability density function. Histograms often give information about the general shape of a distribution.Ĭonsider the two histograms below, one showing the magnitude of earthquakes and the other the historical percent returns on stocks. Notice how the shape of each histogram tells us a lot about the data. The earthquake histogram has a large amount of data concentrated in just a few categories at the low magnitude values, and then the graph tapers off rather slowly as the magnitude increases. We say that the shape of this histogram is skewed to the right. In other words, "skewed to the right" means that most of the data points exist on the left side of the histogram, but we have some "outliers" on the right side of the histogram. An outlier is simply a numerical value that is far from the rest of the data. In our earthquake magnitude example, consider the number of values at 4. We'd consider these to be outliers since they are far from the rest of the data points. We see that it's much more unlikely to see an earthquake of magnitude 4 than of magnitude 2. The opposite situation occurs with the histogram showing the percent return on stocks. The percent return on stocks has more concentrated data in the positive returns, and the tapering is to the left. We say that this histogram is skewed to the left. In other words, a left skew means that most of the data points exist on the right side of the histogram, but we have some outliers on the left side of the histogram. For example, we see that a return in the range of 10 to 20 percent is much more likely than one in the range of -40 to -30 percent. Sometimes we illustrate right and left skewing with smooth curves rather than with histograms, as sketched here:Īgain, notice that in the right-skewed distribution most of the data points are on the left, with the long tail on the right. Similarly, notice that in a left-skewed distribution, most of the data points are on the right, with a long tail extending to the left. You've now seen the look of right- and left-skewed distributions. We don't always get a skewed distribution, however. In fact, many times, something very interesting occurs. When this occurs, we call this distribution of data the normal distribution, the normal curve, or sometimes the "bell curve" because of its resemblance to the shape of a bell.
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