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Investment - Normal Distribution
The arithmetic mean and standard deviation are two useful techniques of representing numerous distributions of data. A distribution is just a group of values, representing their actual or hypothesized frequency of occurrence.
Analysing Data
Sometimes it is beneficial to look at a picture of the distribution to comprehend it. The form of the distribution has a bearing on how you perceive the summary measurements of the distribution. This data can be shown pictorially using a histogram — a bar chart with bars that are proportional to the frequency of occurrence of each group of observations — as illustrated in the following illustrations.
For a fully symmetrical distribution, such as a normal distribution the mean, median, and mode will be identical.
Normal Distribution Representation
A normal distribution is represented in a graph by a bell curve, an example of which is shown below. The shape of the curve is symmetrical, with a single central peak at the mean of the data and the graph falling off evenly on either side of the mean; 50% of the distribution is to the left of the mean, and 50% lies to the right of the mean. The shape of a normal distribution depends on the mean and the standard deviation.
The mean of the distribution dictates the placement of the centre of the curve, and the standard deviation determines the height and width of the curve. When the standard deviation is big, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow.
The arithmetic mean and standard deviation are two useful techniques of representing numerous distributions of data. A distribution is just a group of values, representing their actual or hypothesized frequency of occurrence.
Analysing Data
Sometimes it is beneficial to look at a picture of the distribution to comprehend it. The form of the distribution has a bearing on how you perceive the summary measurements of the distribution. This data can be shown pictorially using a histogram — a bar chart with bars that are proportional to the frequency of occurrence of each group of observations — as illustrated in the following illustrations.
For a fully symmetrical distribution, such as a normal distribution the mean, median, and mode will be identical.
Normal Distribution Representation
A normal distribution is represented in a graph by a bell curve, an example of which is shown below. The shape of the curve is symmetrical, with a single central peak at the mean of the data and the graph falling off evenly on either side of the mean; 50% of the distribution is to the left of the mean, and 50% lies to the right of the mean. The shape of a normal distribution depends on the mean and the standard deviation.
The mean of the distribution dictates the placement of the centre of the curve, and the standard deviation determines the height and width of the curve. When the standard deviation is big, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow.
A normal distribution has special importance in statistics because many variables have the approximate shape of a normal distribution — for example, height, blood pressure, and lengths of items created by machines. This distribution is often useful as a description of data when there are a large number of observations.
Observation of a Normal Distribution
A normal distribution is a distribution of a continuous random variable (i.e., a variable that can take on an unlimited number of values). The vertical axis for the normal distribution represents the probability or likelihood of occurrence. By contrast, on the histograms for the companies showed before, the vertical axis was frequency of occurrence.
The mean (and median) is the centre of the distribution and has the highest likelihood of occurring. Half of the observations are on one side of the mean and half on the other. Approximately two-thirds of the observations are within one standard deviation of the mean, and 95% of observations are within two standard deviations of the mean.
Standard Deviation and Normal Distribution
The whole area under the curve or bell is 100% of the distribution. The area under the curve that is within one standard deviation of the mean is around 68% of all the data. In other words, given a mean of 0 and a standard deviation of 1, around 68% of the observations lie between –1 and +1, and 32% of the observations are more than one standard deviation from the mean. The area under the curve that is within 2 standard deviations of the mean is around 95% of the data.
Given a mean of 0 and a standard deviation of 1, around 95% of the observations lie between –2 and +2, and 5% of the observations are more than two standard deviations from the mean. The area under the curve that is within three standard deviations of the mean represents around 99% of the observations. Given a mean of 0 and a standard deviation of 1, nearly 99% of the observations fall between –3 and +3, and less than 1% of the observations occur more than three standard deviations away from the mean.
The observations that are more than a specific number of standard deviations from the mean can be regarded as residing in the tails of the distribution. Assuming that returns on a portfolio of stocks are normally distributed, the chance of significant losses (a return more than three standard deviations lower than the mean return) is quite small. The chance of the return being in the left tail more than two standard deviations from the mean (which would be a significant loss under usual circumstances) is just 2.5%.
In other words, out of 200 days, 5 days are projected to contain observations that are greater than two standard deviations from the mean. But during financial crises, losses made by banks and other financial organizations over short periods have been many standard deviations below the mean.
Bell-Shaped Distributions with Fat and Thin Tails
Observation of a Normal Distribution
A normal distribution is a distribution of a continuous random variable (i.e., a variable that can take on an unlimited number of values). The vertical axis for the normal distribution represents the probability or likelihood of occurrence. By contrast, on the histograms for the companies showed before, the vertical axis was frequency of occurrence.
The mean (and median) is the centre of the distribution and has the highest likelihood of occurring. Half of the observations are on one side of the mean and half on the other. Approximately two-thirds of the observations are within one standard deviation of the mean, and 95% of observations are within two standard deviations of the mean.
Standard Deviation and Normal Distribution
The whole area under the curve or bell is 100% of the distribution. The area under the curve that is within one standard deviation of the mean is around 68% of all the data. In other words, given a mean of 0 and a standard deviation of 1, around 68% of the observations lie between –1 and +1, and 32% of the observations are more than one standard deviation from the mean. The area under the curve that is within 2 standard deviations of the mean is around 95% of the data.
Given a mean of 0 and a standard deviation of 1, around 95% of the observations lie between –2 and +2, and 5% of the observations are more than two standard deviations from the mean. The area under the curve that is within three standard deviations of the mean represents around 99% of the observations. Given a mean of 0 and a standard deviation of 1, nearly 99% of the observations fall between –3 and +3, and less than 1% of the observations occur more than three standard deviations away from the mean.
The observations that are more than a specific number of standard deviations from the mean can be regarded as residing in the tails of the distribution. Assuming that returns on a portfolio of stocks are normally distributed, the chance of significant losses (a return more than three standard deviations lower than the mean return) is quite small. The chance of the return being in the left tail more than two standard deviations from the mean (which would be a significant loss under usual circumstances) is just 2.5%.
In other words, out of 200 days, 5 days are projected to contain observations that are greater than two standard deviations from the mean. But during financial crises, losses made by banks and other financial organizations over short periods have been many standard deviations below the mean.
Bell-Shaped Distributions with Fat and Thin Tails
In the display, the curve with the solid line illustrates the normal distribution. The curve with the green dotted line is an example of a distribution with thinner tails than the normal distribution, indicating a lower risk of extreme outcomes. By contrast, the curve with the blue dotted line is an example of a distribution with fatter tails than the normal distribution, indicating increased possibility of extreme outcomes.
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