Relationship between confidence level, confidence interval

Suppose a particular phenomenon follows a normal distribution, ~(mu, sigma), two points on the horizontal axis of the normal distribution curve define the ends of the confidence interval, and the area under the curve, between the two points, measures the confidence.

For normal distribution ~(mu, sigma), about 95.5% fall within two standard deviation (sigma * 2). By tradition, statistical significance is often taken at the 95% level, and this occurs at the (1.96 * sigma) deviations from the average.

The distance from a value to the average, measured in standard deviation, is called the z-score. 

z-score=(value - mu) / sigma

using z-score, the confidence level can be computed by:

<confidence level>=NORMSDIST(<z-score>) - NORMSDIST(-<z-score>)

where NORMSDIST(<z-score>) measures the area from negative inifiity to <z-score> under the normal distribution curve.

The 95% confidence level occurs at confidence interval (z-score=-1.96, z-score=1.96). The confidence of 95% implies that the confidence interval ranges 1.96 times the standard deviaion on either side of the average. In other words, the confidence interval for the phenomenon (e.g. a poll) with 95% confidence ranges from (mu - 1.96 * sigma) to (mu + 1.96 * sigma)