What is a normal distribution? (Definition and examples)
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Analysing a large amount of data sometimes becomes necessary, and statistical tools enable an individual or company to do so successfully. Gaussian distribution makes it possible to understand and analyse how a single variable presents itself in a large population. Understanding what this term means is beneficial for industries or organisations to make strategic decisions. In this article, we review the meaning of normal distribution, provide some examples and explain how it can help an organisation's decision-making process.
What is a normal distribution?
A normal distribution, which you can also refer to as Gaussian distribution, is a continuous probability distribution that describes a data set with values frequently occurring around the mean and appearing less as data gets further from the mean. It's illustrated as a graph with a bell shape and comprises two primary parameters: the mean and standard deviation (SD). The data on the chart has a symmetrical arrangement to show most values cluster around the mean and the rest towards either end. This statistical analysis is the most common for technical stock market analysis, including other statistics research.
This distribution follows the central limit theory, which states that 'different independent factors may influence a single variable or trait'. The factor's result forms a sum that usually creates a normal or Gaussian distribution. In a single distribution curve, about 68% of values may fall within +/- one SD from the mean and about 95% of the values may be within +/- two SD from the mean.
Examples of a Gaussian distribution
Gaussian distribution is a tool that individuals and organisations may use for different reasons. To get a better understanding of how this statistical analysis works, here are some examples:
Height is a relatable example of a Gaussian distribution. For example, you may use it to analyse the height of a randomly selected population of 1,000 people. Usually, most people are within average height, with a smaller population of persons being taller or shorter than average. Therefore, you may expect to see more values towards the mean and fewer at the tail ends. Also, height is not an independent variable, as different genetic and environmental factors influence its outcome.
Intelligence quotient (IQ)
IQ is also another example of a Gaussian distribution variable. Suppose you want to analyse the IQ levels of a particular population, such as students in a classroom. The IQ of most persons is within the normal range and only a few may be within the deviate region with an IQ level that is above or below average.
Technical stock market
Most individuals are familiar with the volatility of the stock market with the falling and rising share values. When analysing stock returns, volatility is the term that describes SD. If the returns distribute typically, over 90% may fall with SD of the mean. The change in values of stock and price indices can create a bell-shaped curve. This characteristic of the bell-shaped curve allows investors to make inferences about the risk and return of stocks.
The income range depends on the government and how they distribute income amongst the rich and poor masses of the economy. Usually, the middle class forms a larger population than both the rich and poor, so their income forms the mean on a Gaussian distribution curve. Income distribution in an economy may create a bell-shaped graph because of the high population towards the expected value.
Weight of a newborn
The weight of healthy newborns usually ranges from 2.5kg to 3.5kg, which can form a normal curve. Most newborns weigh within the typical range and only a few may deviate or weigh below or above. You can examine a population of newly born to achieve this.
Average student performance
Schools may want to display their academic excellence by presenting the average result of their students via social media or TV ads to motivate other parents into enrolling their kids. Usually, the school management finds that the academic performance of its students follows a normal curve. The numbers of average performing students are often higher than other groups of students.
Size of clothing
Manufacturers of clothing may perform a test to see which sizes of clothing they sell the most. The size of clothes and shoes follow a normal curve, as more people are usually within the average range of size and fewer people are above or below average. This factor may influence the number of clothes made for a specific size range, which may make it more challenging for people outside this range to find clothes in their size.
Rolling a die
Rolling dice is another example of the Gaussian distribution. An experiment has shown that the chance of getting one when you throw a die 100 times is 15% to 18%. Even when you roll the die 1000 times, the probability of getting the same result still averages around the same value. Rolling two dice simultaneously with 36 combinations also produces a similar average as the probability of throwing one. Increasing the number of dice makes the distribution graph more elaborate.
Common industries that use a normal distribution
This type of distribution helps to analyse large data to determine a variable's expected values or occurrence. Here are some industries that use this tool:
science and research
sales and marketing
Skewness and kurtosis
Actual data doesn't usually provide a perfect Gaussian distribution curve. With skewness and kurtosis, you can measure how a particular distribution differs from normal. Skewness is a value that describes a distribution's symmetry. Normal or Gaussian distribution has zero skewness, indicating that both the left and right tails are equal. A positive value shows that the right tail is longer in the distribution and a negative value shows the left is longer.
The kurtosis coefficient measures how thick the tail ends are in contrast to the normal value. The ideal Gaussian distribution has a kurtosis value of three, which means there are no thin or fat tails. A large kurtosis value exceeds the normal, for example, above four SD from the mean value. Values below three show a distribution with low kurtosis or thin tails.
Characteristics of a normal curve
Here are a few characteristics to identify a normal curve:
The curve has a bell shape with one peak.
The mean of the curve is at its centre.
Both tails of a normal probability distribution never meet the horizontal axis and extend indefinitely.
A normal curve is unimodal because there's one maximum point.
The variable a normal curve distributes is continuous. This distinguishes it from Poisson and binomial distributions, which have discrete variables.
The mode, median and mean are on the same value in a normal curve.
Mean deviation about mean equals 4/5 or 0.7979 of SD.
Application of Gaussian distribution in business
You can use a Gaussian distribution for various purposes, and the business world utilises this tool to determine or identify salaries, profit, cost and other relevant values. For practical usage, it's beneficial to convert the normal curve into a standard one and its specific variable to an expected average value.
A phone company may wish to announce the minimum hours guarantee for its new model. The company can make calculations using the data available for the minimum hours to reveal to their audience. Performing tests may show the mean hours at 50,000 with an SD of 1,750 hours and that the distribution of hours follows a bell-shaped curve. This curve may help them decide on the ideal value for the official announcement and limit customer complaints by giving them what they promise.
How Gaussian distribution helps in decision-making processes
This distribution may help organisations make good decisions that involve:
Negotiation. Gaussian distribution help companies create beneficial contracts. The bell-shaped curve allows them to analyse prices to determine whether they're above or below the average range.
Product prices. Manufacturers may decide the cost of their products by using the bell-shaped curve. This action allows comparison with similar products in the industry and eventually selects a reasonable price.
Financial trading. Traders usually compile prices to create a Gaussian distribution over a specific period. The SD from this distribution helps traders select possible trades that require a short time frame, as it's much easier to determine the entry or exit point than over a larger time frame.
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