In everyday usage, ‘normal’ means usual. Today, Wednesday, May 9, 2018, here in San Jose, California, it is a sunny day, 72 degrees Fahrenheit, with a light wind blowing, that entices me outdoors. It’s what I expect this day to be in San Jose at this time of the year, a typical day.
In statistics, on the other hand, ‘normal’ refers to a special type of distribution (also known as bell-shaped or Gaussian distribution) that describes how variables we are familiar with from everyday life such as height, weight, temperature, and salary vary by proportion or probability. The shape of the bell (symmetric or almost symmetric) suggests that most of the values are concentrated near the center while few values are at the edges, that is, near the two tails, relatively far from the center.
What makes statistical analysis of normal distributions powerful as well as convenient is the idea of standard deviation. Here, ‘deviation’ means how far the value is from the center of the distribution while ‘standard’ means that it is from the mean (or average, which is at the center of the distribution) from which the difference is calculated. Mean and standard deviation go hand-in-hand and the whole is more than the sum of its parts.
There is an empirical (by observation) rule that describes all normal distributions, the “68-95-99.7 Rule.” It says that for any variable that can be described by a normal, or approximately normal, distribution, about 68% of the data will be within -1 and +1 standard deviation around the mean, 95% within -2 and + 2 standard deviations around the mean, and about 99.7% (practically all) will be within -3 and +3 standard deviations around the mean.
Why is the idea of standard deviation so important? One reason is that random errors in measurements usually follow a normal distribution. Suppose I measure the weight of a mango 100 times on five different scales. Even though it is the same mango, the scales will not give me the same weight each time, which means my measurement will have a variation, that is, standard deviation. If I calculate the mean weight of my mango to be 1.2 pounds and the standard deviation 0.1 pound, then I can be 95% confident that the weight of my mango will be between 1.0 pound and 1.4 pounds.
When on July 4, 2012, the announcement was made at CERN that the Higgs boson was discovered with a 5-sigma result (the lower-case Greek letter sigma – ơ – is the conventional notation for standard deviation, so standard deviation is frequently referred to as sigma), it meant that the result was due to a random error only to the extent of 2.87 x 10– 7, that is, a 1-in-3 million chance that the result was a fluke.
Anyone can use the “68-95-99.7 Rule” to mentally calculate relevant percentages or probabilities of a normal distribution, if the mean and the standard deviation are given. Practice it and you too can experience the power of the normal!