The normal distribution or normal curve is a very common distribution. Many variables in real life, such as height, weight, and IQ, follow a normal distribution. As such, knowing how to work with the normal curve is important.
As mentioned before, the normal curve is characterized by two measures: the mean or average and the standard deviation. The mean is representative of what most people score or what the variable averages out to be.
The standard deviation is how far most scores are projected to be from the mean. A smaller standard deviation means that most scores are closer to the mean, while a larger standard deviation means that most scores are farther away from the mean.
This article will discuss what happens to the graph of the normal curve as the standard deviation decreases.
The curve becomes flatter
As the standard deviation of a normal distribution decreases, the graph of the normal distribution becomes flatter. This is because there is a greater range of values that are considered average.
There is also a greater range of values that are considered very low and very high compared to the average. When this range of values increases, the curve becomes flatter.
For example, consider two populations with the same average income. In population A, the average income is $50,000 per year and in population B, the average income is $25,000 per year. There is a much larger difference in income between these populations, which means there is a greater range of values that are considered low income.
Therefore, when comparing these two populations using the normal distribution curve, population B will have a higher proportion of individuals who are considered low income due to their lower average income.
The curve becomes steeper
As the standard deviation decreases, the curve becomes steeper. This is because as the standard deviation decreases, the average or mean value of the distribution of scores increases.
A smaller standard deviation means that most people score closer to the average, which makes the curve appear steeper. As the standard deviation decreases, the area between the curve and the lowest and highest scores increases as well.
The lower limit of normal drops and the upper limit of normal drops at similar rates, which creates a flatter upper and lower boundary on the graph. The curve appears more square-shaped as a result.
As you continue to decrease the standard deviation, eventually you will reach a point where there is no longer a normal distribution of scores. At this point, there will only be one score in the distribution and it will be average or mean. There will no longer be a Normal distribution.
The mean decreases
As the mean of the normal distribution decreases, or the average value below which most values in a population fall, the graph also shows a decrease in height.
This is because there are fewer and fewer values below the average value, so there are fewer and fewer small deviations from the normal distribution. These small deviations create bumps in the graph, or greater values than the average.
As more of these small deviations disappear, so does some of the height of the curve. The curve becomes flatter as there are less differences between values. The curve still maintains its shape, it just becomes less dramatic.
The lower mean value also means that there are less extreme values above the average value. These numbers pull down the overall mean, or average value. As this number decreases, so does the mean value of all of these numbers combined.
The standard deviation decreases
As the standard deviation of a normal distribution decreases, the shape of the graph of the distribution curve approaches a straight line. As the standard deviation approaches zero, the graph of the distribution curve becomes a straight line.
This is because as the standard deviation decreases, most values on the average are close to the middle value. For example, if you have a population with an average income of $50,000 and a standard deviation of $10,000, then most people in that population earn approximately $50,000 and very few earn less than that or more than that due to their being close to the middle value.
As more numbers on the average are close to the middle value, there are fewer variations between numbers. A decrease in variance is an increase in numerical closeness which results in a straight line.
The number of values on either end of the curve also decrease as the standard deviation decreases. When this happens, there are fewer high and low values which pull up or down on either end of the curve.
Examples of when this happens include:|endash|} With decreasing standard deviations, the graphs of normal distributions will look similar to the original graph, but will be shifted to the left or right. When plotted on a line, normal distribution curves with decreasing standard deviations will appear as a series of “V” shapes. Because these curves are all based on the same equation, they do not change shape; instead, they move along the horizontal axis. This movement is called translational shift. As you decrease your standard deviation even more, one side of your curve begins to dominate over the other side and it starts to become more vertical than horizontal shaped.{{cite book |author1=Jack K Gale |author2=Jeffrey S Hacker |title=Statistics for Social Science Research |publisher=Gale/Sage Publications |year=2001}} ===What Happens to the Graph of the Normal Curve as the Standard Deviation Decreases?=== For example, let’s say you have a normal distribution with μ = 20 and σ = 10. If you were to decrease σ by 1%, then μ would also decrease by 1%. However if you were to double σ (making μ = 40), then μ would only increase by 2%. So as σ gets smaller and smaller (more negative), its effect on μ gets smaller and smaller.{{cite web|url=https://mathworld.wolfram.com/NormalDistribution.html|title=Normal Distribution|access-date=2019-03-17}} ===The Shape Does Not Change=== As stated previously in this article, when calculating probability under a normal distribution function there are three parameters – mean (μ), variance (σ²), and sigma – that define its shape.
f(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) When calculating these parameters for an exponential distribution function however there are only two – mean (μ) and sigma (
In this article, we discussed what happens to the graph of the normal curve as the standard deviation decreases. We learned that as the standard deviation gets smaller and smaller, the effect on the mean (μ) gets smaller and smaller.
The shape does not change under a normal distribution function. Under an exponential distribution function, only the mean (μ) and sigma (n
ormal distribution functions have very similar graphs, so it is important to know which one you are dealing with.
