Normal Distribution

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SPA5805NormalDISTRIBUTIONandZStatistic2020.ppt
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Many variables in communication disorders are continuously distributed as opposed to being placed into discrete categories

A distribution is a pattern of scores; distribution of a variable provides information about individual cases as well as information about the group of scores.

Distributions for categorical variables are shown in bar graphs; distributions for continuous variables are usually displayed in line graphs or histograms

Normal distribution is a theoretical distribution providing a model for evaluating distributions of many real-life variables. It has 3 important characteristics:

1) it is unimodal (one mode at center) and symmetrical

2) it is continuous

3) it is asymptotic: curved line gets closer to horizontal axis as it moves away from the center

Normal Distribution

Normal Distribution: Z Statistic

Think about the 3 characteristics of the normal distribution I just mentioned

Different variables have different means and standard deviations; thus, many distributions are possible. The distributions may approximate the normal distribution but they are not comparable because of their different units of measurement.

All normal distributions are transformed to fit the standard normal distribution

All normal distributions can be converted to a common distribution with the same mean and standard deviation (mean of zero and standard deviation of one). THIS IS THE Z STATISTIC.

The proportion of scores in a given area under the bell-shaped curve is always the same: area between -1 and +1 standard deviations always includes 68% of total distribution of scores. Area between -2 and +2 SD include 95% of scores. Area between -3 and +3 SD include 99% of scores. This information is important for determining the probability of an outcome.

Standard Normal Distribution

Normal Distribution: Score Proportions

Think about what I just said relative to score proportions in each area defined by standard deviations

Single score in a distribution of scores in standard normal distribution can be represented by standard unit of measure: the z score.

Z score = standard score. It is a measure of individual location, telling us where individual scores are located within a distribution of scores. Z score tells us how far the SD corresponding value of x (score) lies above/below the mean. This standard unit of measure allows us to make comparisons between different variables.

Standard Units

Z scores include negative values and decimals; thus, they are difficult to report and may be misleading.

So, z scores are transformed into a distribution of standard scores with a mean of 100 and SD equal to 15 units

Transformed Standard Scores

Not bell-shaped and not symmetrical.

Negatively skewed distribution has more scores with larger values toward the right tail, whereas a positively skewed distribution has more scores with larger values toward the left tail

Kurtosis: measure of peakedness for symmetrical distributions; measure of how fat or thin the tails of a distribution are relative to a normal distribution

Skewed Distributions

Skewed Distributions

Distribution: pattern of scores

  • All variables have distributions
  • Statistics such as mean, median, standard deviation have distributions. BECAUSE: The mean of one set of scores is not the same as the mean of another set of scores, etc.
  • The distribution of a sample statistic indicates how often different values of that statistic should occur if samples of the same size are collected repeatedly from the same populations
  • One example: distribution of sample means (for a particular variable)

Distribution of a Sample Statistic

Probability distributions: provide information about the chance occurrence of a particular outcome for a particular distribution of scores

When distribution of sample means is known, it is easy to compute probabilities.

  • Typically not practical because target population is large. So we usually do not know the DISTRIBUTION of sample means

Distribution of Sample Means

  • So: we use a theorem:
  • 1) mean of the distribution of sample means equals the mean of the population
  • 2) distribution of sample means is less variable than the population
  • -This means that the standard deviation of the sample means is smaller than the standard deviation of the population. WHY?
  • - standard deviation of sample means is known as the standard error of the mean or SEM
  • -to determine this, we need to have the parameters of the population which we typically do not have available. WHY?
  • - So: we use a solution, called the Central Limit Theorem

Distribution of Sample Means