Note that converting x scores to z scores does NOT change the shape of the distribution.For instance, on a scale which has a mean of 500 and a standard deviation of 100, a score of 450 would equal a z score of (450-500)/100 = -50/100 = -0.50, which indicates that the score is half a standard deviation below the mean. If we know the mean μ (“mu”), and standard deviation σ (“sigma”), of a set of scores which are normally distributed, we can standardized each “raw” score, x, by converting it into a z score by using the following formula on each individual score:Ī z score reflects how many standard deviations above or below the population mean a raw score is. of 1) is called the standard normal distribution, which represents a distribution of z-scores. Standard Normal Distributions and Z ScoresĪ normal distribution that is standardized (so that it has a mean of 0 and a S.D. This fact, as described in the Central Limit Theorem, is critical for many applications of statistical inference. Although a distribution of scores in a sample of N cases may be quite far from normal, the distribution of means for all possible samples of N cases (i.e., the sampling distribution of the mean) may be quite close to normal. Important note: Before we use the normal distribution to compute probabilities, we must verify that the distribution of interest is very close to normal. This allows us to compute the probability of obtaining a sample mean in any range of interest, given that the sample is drawn randomly from a population with a specified mean and standard deviation. However, natural distributions are rarely (never?) exactly normal, or even close enough to normal that we can apply the formula with confidence.Ī major reason why the normal distribution is so important in inferential statistics is because the distribution of possible sample means approaches normal as the sample size increases, as described by the Central Limit Theorem. Some measurements in the natural world may approximate normal distributions (e.g., perhaps the weights of hippopotamuses, heights of palm trees, students’ IQs, and people’s happiness). If we have a normal distribution, we can use the formula to calculate how likely it is to observe values in regions of interest. Three normal distributions whose means and standard deviations differ. The total area under the curve sums to 100%.įigure 1. Tails of a normal distribution are asymptotic, indefinitely decreasing but never touching the x-axis. Although normal distributions may have different means and standard deviations, all normal distributions are “bell-curve” shaped, symmetrical with the greatest height at the mean (see Figure 1 for examples).
The normal distribution is defined by a mathematical formula.