How does normal distribution work

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

The standard normal distribution has two parameters: the mean and the standard deviation. The normal distribution model is motivated by the Central Limit Theorem.

This theory states that averages calculated from independent, identically distributed random variables have approximately normal distributions, regardless of the type of distribution from which the variables are sampled provided it has finite variance.

Normal distribution is sometimes confused with symmetrical distribution. Symmetrical distribution is one where a dividing line produces two mirror images, but the actual data could be two humps or a series of hills in addition to the bell curve that indicates a normal distribution.

Real life data rarely, if ever, follow a perfect normal distribution. The skewness and kurtosis coefficients measure how different a given distribution is from a normal distribution. The skewness measures the symmetry of a distribution.

The normal distribution is symmetric and has a skewness of zero. If the distribution of a data set has a skewness less than zero, or negative skewness, then the left tail of the distribution is longer than the right tail; positive skewness implies that the right tail of the distribution is longer than the left. The kurtosis statistic measures the thickness of the tail ends of a distribution in relation to the tails of the normal distribution.

Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution e. Distributions with low kurtosis exhibit tail data that is generally less extreme than the tails of the normal distribution. The normal distribution has a kurtosis of three, which indicates the distribution has neither fat nor thin tails. Therefore, if an observed distribution has a kurtosis greater than three, the distribution is said to have heavy tails when compared to the normal distribution.

If the distribution has a kurtosis of less than three, it is said to have thin tails when compared to the normal distribution. The assumption of a normal distribution is applied to asset prices as well as price action. Traders may plot price points over time to fit recent price action into a normal distribution. The further price action moves from the mean, in this case, the more likelihood that an asset is being over or undervalued.

Traders can use the standard deviations to suggest potential trades. This type of trading is generally done on very short time frames as larger timescales make it much harder to pick entry and exit points. Similarly, many statistical theories attempt to model asset prices under the assumption that they follow a normal distribution. In reality, price distributions tend to have fat tails and, therefore, have kurtosis greater than three.

What is the standard normal distribution? The standard normal distribution , also called the z -distribution , is a special normal distribution where the mean is 0 and the standard deviation is 1. While individual observations from normal distributions are referred to as x , they are referred to as z in the z -distribution. Every normal distribution can be converted to the standard normal distribution by turning the individual values into z -scores.

You only need to know the mean and standard deviation of your distribution to find the z -score of a value. Each z -score is associated with a probability, or p -value , that tells you the likelihood of values below that z -score occurring.

If you convert an individual value into a z -score, you can then find the probability of all values up to that value occurring in a normal distribution. The mean of our distribution is , and the standard deviation is The z -score tells you how many standard deviations away is from the mean.

For a z -score of 1. This is the probability of SAT scores being or less That means it is likely that only 6. Frequently asked questions about normal distributions What is a normal distribution? In a normal distribution , data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. The measures of central tendency mean, mode and median are exactly the same in a normal distribution.

The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1. Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies. The empirical rule, or the The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side.

It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown. The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation.

Have a language expert improve your writing. Check your paper for plagiarism in 10 minutes. Just so you get the intuition. If you want to know, what is the probability that I get a value less than 20? So I can get any value less than 20 given this distribution. The cumulative distribution right here, -- let me make it so you can see -- if you go to 20 you just go right to that point there and you say wow, the probability of getting 20 or less is pretty high. It's approaching percent. That makes sense because most of the area under this curve is less than Or if you said what's the probability of getting less than minus five.

Well minus 5 is the mean so half of your results should be above that and half should be below. And if you go to this point right here you could see that this right here is 50 percent.

So the probability of getting less than minus 5 is exactly 50 percent. So what you do is, if I wanted to know the probability of getting between negative 1 and 1 what I do is -- let me get back to my pen tool -- I figure out what is the probability of getting minus 1 or lower. So I figure out this whole area. And then I figure out the probability of getting one or lower which is this whole area -- well I'm going to give it a different color -- 1 or lower is everything there. And I subtract the yellow area from the magenta area and I'll just get what's ever left over here.

That's exactly what I did in the spread sheet. Let me scroll down. This might be taxing my computer by taking the screen capture with it. So what I did is I evaluated the cumulative distribution function at one to be right there.

And I evaluate the cumulative distribution function at minus 1, which is right there. And the difference between these two, I subtract this number from this number and that tells me essentially the probability that I'm between those two numbers. Or another way to think about it, the area right here. I really encourage you to play with this and explore the excel formulas and everything.

This area right here would be minus 1 and 1. Just so you know this graph, the central line right here, this is the mean. And then these two lines I drew right here, these are 1 standard deviation below and 1 standard deviation above the mean. Some people think what's the probability that land within one standard deviation of the mean? Well that's easy to do. What I can do is I'll just click on this. And I'll just call this, what's the probability that I land between 1 standard deviation -- the mean is minus 5 -- one standard deviation below the mean is minus 15 and one standard deviation above the mean is 10 plus minus 5 is 5.

So that's between 5 and So That's actually always the case that you have a So once again, that number represents the area under the curve here, this area under the curve. And the way you get it is with the cumulative distribution function.

I'll go down here. Every time I move this I have to get rid of the pen tool. You evaluate it at plus 5, which is right here. This is 1 standard deviation above the mean, that's a number right around there.

It looks like it's like 80 something percent, maybe 90 percent roughly. And then you evaluate it at 1 standard deviation below the mean which is minus And this one looks like roughly 15 percent or so, 15, 16, maybe 17 percent, I'll say 18 percent.

But the big picture is when you subtract this value from this value you get the probability that you land between those two. And that's because this value tells the probability that you're less. So when you go to the cumulative distribution function you get that right there. It keeps crawling back and forth. So when you go to five -- and you just go right over here -- this is essentially tells you this area under the curve, the probability that you're less than or equal to 5, everything up there.

And then when you evaluate it at minus 15 down here, it tells you the probability that you're back here. So when you subtract this from the larger thing you're just left with what's under the curve right there. Just to understand this spreadsheet a little bit better because I really want you to play with it and see what happens if I make this distribution, the mean was minus 5 now let me make it 5.

It just shifted to the right. It just moved over to the right by 5. Let me use the pen tool. If I were to try to make the standard deviation smaller we'll see that the whole thing just gets a little bit tighter. Let's make it 6. And all of a sudden this looks a like a tighter curve, we make it 2, it becomes even tighter. I really want you to play with this and play with the formula and get an intuitive feeling for this, the cumulative distribution function and think a lot about how it relates to the binomial distribution and I cover that in the last video.

To plot this I just took each of these points. I went to plot the points between minus 20 and 20 and I just incremented by 1. I just decided to increment by 1. So this isn't a continues curve, it's actually just plotting a point at each point and connecting it with the line. Then I did the distance between each of those points and the mean. So I just took 0 minus 5, this is this distance. So this just tells you the point minus 20 is 25 less than the mean.

That's all I did there. Then I divided that by the standard deviation and this is the z score, the standard z score. This tells me how many standard deviations is minus 20 away from the mean. It's 12 and a half standard deviations below the mean. Then I use that and I just plugged it into essentially this formula to figure out the height of the function.

Let's say at minus 20 the height is very low. Let's say at minus 2 the heights a little bit better, the heights going to be some place, it's going to be right there. And so that gives me that value. But then to actually figure out the probability of that, what I do is I calculate the cumulative distribution function.

Well this is the probability that you're less than that. So the area under the curve below that which is very small. It's not zero, I know it looks like 0 here but that's only because I round it. It's going to be , it's going to be a really small number. There's some probability that we even get minus a thousand.

Another intuitive thing that you really should have a sense for is the integral over this or the entire area of the curve has to be 1 because that takes into account all possible circumstances. And that should happen if we put a suitably smaller number here and a suitably large number here.

There you go, we get percent. Although, this isn't a percent. We'd have to go from minus infinity to plus infinity to really get a percent.

It's just rounding to percent. It's probably