Normal Distribution Explained: What Is Normal Distribution?

A normal distribution is a continuous probability distribution of data points that appears as a bell curve when graphed using a line graph or histogram. It also goes by the name Gauss distribution, Gaussian distribution, and Laplace-Gauss distribution. Compared to other probability distributions, a normal distribution has several notable properties.

• Its graph has a distinctive symmetrical shape. A normal distribution curve appears symmetrically in the shape of a bell.
• It is characterized by equal values. The mean, median, and mode of its data set will all be the same number.
• Data distribution is equally split. Regardless of sample size and the sample mean, half the data points will be below the mean, and half the data points will be above the mean.
• It divides its data using standard deviations. A standard deviation is a measurement of how far away a data point is from the mean. The skewness of a data point will depend on the number of standard deviations it falls from the sample mean.
• The distribution of data follows an empirical rule of normal probability. With a normal distribution of data, 99.7 percent of all data points fall within three standard deviations from the mean in either direction. The first standard deviation of the mean covers 34.1 percent of all data points above the mean and 34.1 percent of all data points below the mean. The second standard deviation encompasses an additional 13.6 percent of data points above the mean and another 13.6 percent below the mean. The third standard deviation includes 2.1 percent of data points in either direction—both at the very top of the scale and the very bottom of the scale. This means you could fall on the ninety-ninth percentile of a sample size and still be within three standard deviations of the mean. You have to be a true outlier, or special case, to go beyond three standard deviations.
• The total area under the curve has a value of one. If you were to calculate the total area under a normal distribution curve, its value would equal one unit of measure.

Professional statisticians use normal distributions for many applications.

• Curving tests: Standardized tests like the SAT and ACT use an imposed curve to make their test results fit a normal distribution curve. On the SAT, the population mean will always be 500 points on a given section of the test, like math or evidence-based reading and writing.
• Finding Z-scores for data points: A Z-score is a measurement between a data point and a sample mean, expressed in terms of standard deviations. You can use Z-scores to calculate probabilities and identify outliers in data sets. This comes up in the analysis of all sorts of population measurements. Height, weight, blood pressure, IQ, and test scores are a few of the things that get grouped using Z-scores and normal distributions.
• Establish confidence intervals: Confidence intervals come into play when statisticians make inferences about how likely it is that a data set will contain the real-world population mean (average). They use confidence intervals to measure their level of confidence that these data sets will contain the mean values of the population at large. When data falls in a normal distribution, it can be easier to establish a confidence interval.
• A gateway to high-level statistics: Those who work in stats for a living know that normal distributions can be used to calculate probability density functions, test the central limit theorem, and calculate linear regressions through the use of residuals. Professional statisticians often use statistical software to build off of normal distributions for more advanced statistical analysis.

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