Academic journal article Journal of Visual Impairment & Blindness

Distribution of Scores around the Mean and the Purpose of Z-Scores

Academic journal article Journal of Visual Impairment & Blindness

Distribution of Scores around the Mean and the Purpose of Z-Scores

Article excerpt

Many statistical tests are based on two fundamental measures derived from a set of data: a measure of the average score and a measure of what the spread of scores is around that average. With these two measures, a host of statistics make different kinds of comparisons between variables in the dataset or between subsets of the sample. In this issue's Statistical Sidebar, I will look a little closer at the second of those measures, the one that evaluates the spread of scores around a mean or average.

When calculating how a set of scores are distributed around a mean score, two of the more common ways of doing so are with a range or a standard deviation. The range is simply what the lowest and highest scores were that went into calculating the mean. The standard deviation is somewhat more useful in that it describes how individual scores would be expected to be arranged around the mean, as long as the set of data are "normally distributed." Normal distribution means that the majority of the individual scores are close to the mean and as scores get progressively higher or lower than the mean, there are fewer of them. If one plotted out how many individual instances there were of each possible score, the shape would come out looking like a bell, which is why normal distribution is called a bell curve. A standard deviation calculates where on that normal curve one would expect a given score to fall. For example, if a set of data had a mean of 100 and a standard deviation of 15 (as is often quoted for some IQ tests), it means that an individual score of 115 is 1 standard deviation above the mean. Since the properties of the normal distribution are known, it is possible to also know that one standard deviation equates to this score being higher than about 84. …

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