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64, 95, 99.7 The normal distribution is one of the most important distributions. The histogram has this general shape. When the histogram of the normal distribution is smoothed to form a curve, the curve is bell-shaped. This curve is called a normal curve and is used to model the normal distribution. The bell can vary in size but they all have the same basic properties. ◦ The curve is bell-shaped with the highest point at the mean µ. ◦ The curve is symmetrical about a vertical line 𝑥 = 𝜇. ◦ The mean, median, and mode are all equal. ◦ 50% of the data values of the distribution are to the right of the mean µ; 50% of the data values are to the left of the mean µ. Approximately 68% of the data values fall between 𝜇 − 𝜎 and 𝜇 + 𝜎; that is, 68% are between one standard deviation less and one standard deviation more than the mean. The normal curve model approaches the horizontal axis, but never touches or crosses the axis. Normal curves give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean. We can find these numbers precisely, but until then we will use a simple rule that tells us a lot about the normal curve. About 68% of the values fall within one standard deviation ơ of the mean µ. About 95% of the values fall within two standard deviations ơ of the mean µ. About 99.7% (almost all!) of the values fall within three standard deviations ơ of the mean µ. The following shows what the 68-95-99.7 rule looks like on the graph. Data is considered “normal” if it falls within two standard deviations ơ of the mean µ, or within the central 95% of the curve. Data is considered “unusual” if it falls outside two standard deviations ơ of the mean µ, or within the upper 2.5% and the lower 2.5% of the curve. pg 866 #3 & 4