Statistical data can be "distributed" (spread, dispersed, scattered) in a variety of ways. Bell-shaped Curve: There are certain sets of data where the data, when graphed, are symmetrical with a single central peak at the mean (average) of the data. The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean. Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean. Such graphs are called normal curves, and referred to as a normal distribution. The mean, median and mode are all the same in a normal distribution.
Notice how the histogram closely follows the form of the bell curve. A normal distribution is the most widely known and used of all distributions. It is an extremely important statistical data distribution pattern occurring in many natural phenomena, such a blood pressure, machined parts, human height, error in measurement, IQ scores, sizes of snowflakes, lifespans of light bulbs, weights of loaves of bread, test scores, milk production in cows, etc. When data pertaining to these phenomena are graphed as histograms with data on the horizontal axis and the amount of data on the vertical axis, a bell-shaped curve (normal curve) may be created. A normal distribution is actually a "family of distributions", since the mean and standard deviation, which determine the shape of the distribution, may differ from graph to graph.
Standard Deviation: See Refresher: Variance and Standard Deviation.
Percentages Under a Normal Curve: As seen in the previous section, the standard deviation can be used to sub-divide the space (the area) under a normal curve, starting from the mean. Each of these sub-divided sections can be used to represent a portion (a percentage) of the data falling into these sections of the graph. The normal curve actually shows how likely it is to find a value within a specific distance from the mean. Using 1 standard deviation to create the subdivisions: The most popular subdivision utilizes distances from the mean in increments of one standard deviation of that specific normal curve. When dealing with a normal curve:
NOTE: Normal distributions may also be referred to as normal probability distributions. Using ½ of 1 standard deviation to create the subdivisions: It is possible to subdivide the area under a normal curve into smaller intervals, such as widths of 0.5 standard deviations, as shown in the graph below. The addition of the percentages in this graph will be slightly different from the Empirical Rule values which are rounded approximations. These smaller subdivisions would be used when information presented in a question falls on the increments of one-half of one standard deviation from the mean.
Not all data is normally distributed. Statisticians use both simple and complex mathematical techniques to determine if a data set is distributed normally. The more data that is available, the more likely it can be determined if the population data is normal or not. The simplest test for normality is to make a histogram of the data. If the shape of the distribution resembles a bell curve, the data is likely normal. Further examinations such as whether the mean equals the median, and whether approximately 68% of the data is within one standard deviation of the mean, 95% within two standard deviations and 99.7% within three standard deviations may help verify if the data comes from a population that is normally distributed.
If a small sample set appears to be normal, it is dangerous to make the assumption that the population is also normal. Such an assumption may lead to an incorrect statistical analysis, and incorrect implications regarding the data. More sophisticated tests for normality require computer software packages and complex calculations. Examples: Directions: Look for the words "normally distributed" in the question. If you are NOT told that the data is normally distributed, do not use the percentages in the Empirical Rule (use your graphing calculator) to determine the information. Some examples may deal with graphical increments of width one standard deviation or one-half of one standard deviation, as seen in the charts, and some may need a graphing calculator.
Even though this problem is normally distributed, the most accurate answer cannot be obtained by using the Empirical Rule or the charts on this page. For this problem, one standard deviation above the mean will be located at 41.2 hours, two standard deviations will be at 42.4, and one and one-half standard deviations will be at 41.8. None of these locations corresponds exactly to the needed 42 hours. We need more power to find this solution. Let's use the graphing calculator! We will see in Understanding Z-Scores that this problem can also be accomplished without a graphing calculator. But if you know how to use your graphing calculator, you will find the process easier and faster.
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Normal Distribution - MathBitsNotebook(A2) (2024)
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