Question

Explain why the normal distribution is USEFUL. If we know that a quality has a normal distribution, list three things we can then predict about this quality.

Answer 1

The problem with collecting data is that you do not generally know what distribution the data follows. So you have a sample, but no distribution to help figure it out. The true distribution is generally not knowable, but you could often find something workable if you tried (which is why Stats I texts start with the binomial, which shows up a lot). The thing is, there are a ton of distributions (which is why many students start to get lost in Stats I as additional distributions are introduced). And quite possibly your sample might match up with none of them, and then you’d need to dream up a new one.

Then along comes the central limit theorem. It implies under fairly easy to satisfy conditions that some of the summary statistics you’d calculate from your sample do have a known distribution even if you do not know the distribution of your sample.

Think about what that means: you’ve gone from a situation of having little likelihood of success in understanding your sample, to one of having the great understanding of the summary statistics of your sample. That’s a ridiculous reduction in workload.

The normal distribution is important because of the Central limit theorem. In simple terms, if you have many independent variables that may be generated by all kinds of distributions, assuming that nothing too crazy happens, the aggregate of those variables will tend toward a normal distribution. This universality across different domains makes the normal distribution one of the centerpieces of applied mathematics and statistics.

Another corollary is that the normal distribution makes math easy - things like calculating moments, correlations between variables, and other calculations that are domain specific. For that reason, even if a distribution isn't actually normal, it is useful to assume that it is normal to get a good, first-order understanding of a set of data

if we talk about basic three advantages then,

It is very helpful in forecasting .We can calculate the estimated length of the bones of animals and woods and leaves . because if animals are one type their numbering is normally distributed .

Second reason the normal distribution is so important is that it is easy for mathematical statisticians to work with.Normal distribution is very useful for controlling the quality in business. With this we can fix the limit of quality . that will helpful for controlling the quality .

If we take one sample out of the universe and calculate the mean size of growing then it will normal distribution This means that many kinds of statistical tests can be derived for normal distributions. Almost all statistical tests discussed in this text assume normal distributions. Fortunately, these tests work very well even if the distribution is only approximately normally distributed. Some tests work well even with very wide deviations from normality.

Finally, if the mean and standard deviation of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles.