what is the law of large numbers 
the law of large numbers applies to probability and statistics it states that a sample size gets closer to the average of the whole population as it grows because the sample is more representative of the population as it becomes larger the law of large numbers indicates in a financial context that a large entity that s growing rapidly can t maintain that growth pace forever the biggest of the blue chips with market values in the hundreds of billions are frequently cited as examples of this phenomenon investopedia julie bangunderstanding the law of large numbersthe law of large numbers can be used in two ways the law of large numbers can be applied to a variety of subjects in statistical analysis it may not be feasible to poll every individual within a given population to collect the required amount of data but every additional data point gathered has the potential to increase the likelihood that the outcome is a true measure of the mean the law of large numbers doesn t mean that a given sample or group of successive samples will always reflect the true population characteristics if a given sample or series of samples deviates from the true population average especially for smaller samples nor does it guarantee that successive samples will move the observed average toward the population mean as suggested by the gambler s fallacy the term law of large numbers sometimes relates to growth rates in business it s stated as a percentage and suggests that the percentage rate of growth becomes increasingly difficult to maintain as a business expands this happens because the underlying dollar amount is increasing even if the growth rate as a percentage remains constant the law of large numbers isn t to be mistaken with the law of averages which states that the distribution of outcomes in a large or small sample reflects the distribution of outcomes of the population law of large numbers statistical analysis exampleyou re more likely to reach an accurate average in statistical analysis by choosing 20 data points instead of relying on just two if you want to determine the average value of a data set of 100 possible values there s a greater probability of the two data points being outliers or non representative of the average and there s a lower probability of all 20 data points being non representative you may determine the average to be approximately 67 5 if the data set includes all integers from one to 100 and you drew only two values such as 95 and 40 the average should shift toward the true average as you consider more data points if you continue to take random samplings up to 20 variables law of large numbers and central limit theoremthe law of large numbers is related to the central limit theorem in statistical analysis the central limit theorem states that the sample mean will be evenly distributed as the sample size increases this is often depicted as a bell shaped curve where the peak of the curve is the mean and even distributions of sample data fall to the left and right of the curve the law of large numbers also states that data is refined as the sample grows but the law of large numbers more closely relates to the center of the bell curve it indicates that the mean of the sample will more closely resemble the mean of the population as the sample size increases the law of large numbers therefore relates to the peak or mean of a curve the central limit theorem relates to the distribution of a curve law of large numbers and business growththe term law of large numbers is sometimes used colloquially in business and finance to refer to the observation that exponential growth rates often don t scale this isn t related to the law of large numbers but may be a result of the law of diminishing marginal returns or diseconomies of scale the same principles can be applied to other metrics such as market capitalization or net profit investing decisions can be guided based on the associated difficulties that companies with very high market capitalization can experience as they relate to stock appreciation this concept is somewhat central to growth versus value stocks because a company may use it to maintain its business strategy of rapid growth after it achieves market success law of large numbers business exampletesla reported automotive sales not gross sales of 24 604 million in fiscal year 2020 the company reported 44 125 million the next year an increase of roughly 79 1 electric vehicles are an emerging market and tesla was beginning to finally experience economies of scale experiencing success very quickly the law of large numbers indicates that it will become harder for tesla to maintain this level of productivity as the company continues to grow it becomes apparent that tesla simply can t maintain its current growth trajectory due to the underlying dollar values becoming unreasonable assuming a steady growth rate over several years law of large numbers and insurancethe law of large numbers is also prominent in the insurance industry it s used to calculate and refine projected risk imagine a situation where an insurance company is assessing how much to charge customers for car insurance the company won t be able to adequately determine appropriate risk profiles should it have a small data set the company experiences the law of large numbers as it collects more data it may soon find that young male drivers are most likely to cause an accident this larger sample becomes more representative of driving incidents and the insurance company can arrive at more accurate conclusions about the appropriate insurance premiums to charge the law of large numbers also allows insurance companies to deeply refine the criteria in which to assess premiums by analyzing what traits cause higher risk 