Normal Curve



The normal curve is important because many statistical procedures require that the population from which the sample is drawn is normally distributed, no one way more likely to be effected one way or the other. Complete random sampling. In normal distribution, the mean, median, and mode are all exactly the same. Because this is a rare event, researchers tend to be satisfied with an approximately normal distribution.

  • Mean
  • Median
  • Mode

In the population, the scores in a normal distribution tend to remain around the mean (average), with fewer scores in the tails. Specifically, 68.26% of scores are within one unit, 95.46% of scores fall within two units, and 99.76% fall within three units on both sides of the mean.

Each unit under the normal curve is called a standardized score. Therefore, about 95% of scores tend to be within 2 standard deviations around the mean.



normal2.gif







Z-Scores



A) Z – scores are also known as standardized scores. There are two important reasons to standardize scores.

1) Standardized scores allow for comparison of scores from data sets that have different means and standard deviations.

2) Standardized scores give the exact location of any score in a distribution in relation to the mean.

In a Z distribution (the normal distribution), the mean is always 0 and the standard deviation is always 1. Standard deviations below the mean are negative. Standard deviation results are not negative, this just indicates that the results are below the mean.

B) Computing Z-Scores
To compute a z-score, two things are needed: the mean and standard deviation of the data set. The formula is:
Z = X - M
s

Key: X = a score in the data
M = mean of the data
s = standard deviation of the data




Sources


Wikipedia: The Free Encyclopedia. (November 7, 2008). Standard Score. November 7, 2008, http://en.wikipedia.org/wiki/Z_scores.

Wikipedia: The Free Encyclopedia. (November 6, 2008). Normal Distribution. November 7, 2008, http://en.wikipedia.org/wiki/Normal_curve.

Tushar Mehta Consulting. (April 14, 2008). Drawing a Normal Curve. November 7, 2008, http://www.tushar-mehta.com/excel/charts/normal_distribution.