![]() What's the chance of seeing someone with a height between between 5 feet 10 inches and 6 feet 2 inches? (That is, between 70 and 74 inches.) Now for the fun part: Let's apply what we've just learned. To continue our example, the average American male height is 5 feet 10 inches, with a standard deviation of 4 inches. It says: 68% of the population is within 1 standard deviation of the mean.ĩ5% of the population is within 2 standard deviation of the mean.ĩ9.7% of the population is within 3 standard deviation of the mean. The 68-95-99 rule is based on the mean and standard deviation. Together, the mean and the standard deviation make up everything you need to know about a distribution. And even fewer are three standard deviations away (or further). Fewer observations are two standard deviations from the mean. Most observations fall within one standard deviation of the mean. This tells you how rare an observation would be. For example, the average of these three numbers: 1, 2, 3 = (1 2 3) / 3 = 2 Most people just call this "the average." It's what you get if you add up the value of all your observations, then divide that number by the number of observations. There's equal mass before and after the peak.Īnother important property is that we don't need a lot of information to describe a normal distribution. You can reduce lots of complicated mathematics down to a few rules of thumb, because you don't need to worry about weird edge cases.įor example, the peak always divides the distribution in half. This distribution is exciting because it's symmetric – which makes it easy to work with. A lot of things follow this distribution, like your height, weight, and IQ. They are represented by a bell curve: they have a peak in the middle that tapers towards each edge. ![]() Today, we're interested in normal distributions. In some cases, 10x above average is common. Your answers to the two questions above are different, because the distribution of data is different. ![]() How often would you expect to meet someone who earns 10x as much as Mason?Īnd now, how often would you expect to meet someone who is 10x as tall as Mason? He's an average American 40-year-old: 5 foot 10 inches tall and earning $47,000 per year before tax. The larger variance and standard deviation in Dataset B further demonstrates that Dataset B is more dispersed than Dataset A.Meet Mason. The population variance \(\sigma^2\) (pronounced sigma squared) of a discrete set of numbers is expressed by the following formula: In a normal distribution, about 68% of the values are within one standard deviation either side of the mean and about 95% of the scores are within two standard deviations of the mean. The standard deviation of a normal distribution enables us to calculate confidence intervals. Therefore, if all values of a dataset are the same, the standard deviation and variance are zero. The smaller the variance and standard deviation, the more the mean value is indicative of the whole dataset. Where a dataset is more dispersed, values are spread further away from the mean, leading to a larger variance and standard deviation. In datasets with a small spread all values are very close to the mean, resulting in a small variance and standard deviation. They summarise how close each observed data value is to the mean value. The variance and the standard deviation are measures of the spread of the data around the mean.
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