Estimating the 95% Confidence Interval Using a normal Distribution
Confidence is one of three fundamental confidence indicators. The other two are statistical confidence and business confidence. A confidence interval is a range that is used to calculate the deviation of an expected result from the actual or predicted value. This is an economic concept that is used in virtually every industry in order to determine what kind of risks may be involved in producing or carrying out a certain kind of product or service.
The standard deviation is a mathematical calculation that can be used as a measure of variability in the data being studied. The normal curve (a normal distribution) shows the range of estimates over time; therefore, a normal deviation will show the variation over time as well. The confidence level is the percentage difference between the actual value and the estimated one, expressed as a percentage. The standard deviation is equal to the mean square value of the deviation, divided by the number of points in the range.
A confidence interval can be calculated on any normal curve. If the result is negative (i.e., there is no range), the confidence level is zero. In order to calculate the confidence interval, take a large sample size and calculate the standard deviation. Once this is done, look up the value for your estimate against the normal curve; this will give a negative sign if the range is too small, and a positive sign if it is too large. You will then know how far the range deviates from the normal curve.
The standard deviation used is the square root of the probability that the actual value will be different from the predicted value. The normal curve depicts the normal distribution of the probability that the sample mean lies between zero and the mean of the normal distribution. There is actually a bit of luck involved with this probability, because a small sample mean lies close to the normal curve often. Therefore, it is possible that this small sample mean will have a high value when compared to the normal distribution. Therefore, the smaller the sample mean the larger the probability that the value lies close to the normal curve.
The standard error of difference uses the square root of the probability that the actual value will be different from the predicted value. It then calculates the standard deviation by taking the square root of the deviation. Standard errors are useful because they illustrate the range of possible deviations and how they can vary over time. Using this form of estimation allows one to easily compare estimates from different studies and come to a conclusion regarding whether the results are consistent and reliable. The standard error tells you the range of possible deviations but does not show you the range over time or how much or little variance exists between estimated parameters.
It is easier to measure an individual's confidence level than it is to estimate a population value or mean. It is easier because you don't have to deal with sample size and normal distribution considerations. You simply take a single random sample and calculate the standard deviation for it. When doing so, you will see that there is enough variability in the results to give you a range over the confidence interval where the true population value or mean can be found.
To calculate confidence intervals, it is important to remember that the sample mean and standard deviation are related. If one is larger than the other then the interval will be longer. In the 95%is, however, the confidence interval does not encompass the tails of the distribution. For instance, in a normal distribution with tails, the interval spans three-quarters of the range. In the interval just below the tails the range is longer than two thirds of the total range, which demonstrates that the confidence intervals do not encompass the tails of the normal distribution.
The range over the confidence interval can also be calculated by dividing the total range by the number of times the confidence interval crosses the tails of the distribution. This is done for different samples and different distributions. When doing this, it is important to remember that the calculation is usually round to the nearest whole number. For instance, if the number is three quarters of one hundred then the range over the interval would be six quarters of one hundred.