EXAMPLE 4.9 Students who seek therapy at university counseling centers often do not attend many sessions. For example, in one study, the median number of therapy sessions was 3 and the mean was 4.6 (Hatchett, 2003). Let’s examine the spread of fictional scores for a sample of five students: 1, 2, 4, 4, and 10 therapy sessions, with a mean of 4.2. We find out how far each score deviates from the mean by subtracting the mean from every score. First, we label with an X the column that lists the scores. Here, the second column includes the results we get when we subtract the mean from each score, or X − M. We call each of these a deviation from the mean (or just a deviation)—the amount that a score in a sample differs from the mean of the sample. X X − M 1 − 3.2 2 − 2.2 4 − 0.2 4 − 0.2 10 5.8 But we can’t just take the mean of the deviations. If we do (and if you try this, don’t forget the signs—negative and positive), we get 0—every time. Are you surprised? Remember, the mean is the point at which all scores are perfectly balanced. Mathematically, the scores have to balance out. Yet we know that there is variability among these scores. The number representing the amount of variability is certainly not 0! When we ask students for ways to eliminate the negative signs, two suggestions typically come up: (1) Take the absolute value of the deviations, thus making them all positive, or (2) square all the scores, again making them all positive. It turns out that the latter, squaring all the deviations, is how statisticians solve this problem. Once we square the deviations, we can take their average and get a measure of variability. Later (using a beautifully descriptive term created by our students), we will “unsquare” those deviations in order to calculate the standard deviation
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
EXAMPLE 4.9
Students who seek therapy at university counseling centers often do not attend many sessions. For example, in one study, the median number of therapy sessions was 3 and the
| X | X − M |
|---|---|
| 1 | − 3.2 |
| 2 | − 2.2 |
| 4 | − 0.2 |
| 4 | − 0.2 |
| 10 | 5.8 |
But we can’t just take the mean of the deviations. If we do (and if you try this, don’t forget the signs—negative and positive), we get 0—every time. Are you surprised? Remember, the mean is the point at which all scores are perfectly balanced. Mathematically, the scores have to balance out. Yet we know that there is variability among these scores. The number representing the amount of variability is certainly not 0!
When we ask students for ways to eliminate the negative signs, two suggestions typically come up: (1) Take the absolute value of the deviations, thus making them all positive, or (2) square all the scores, again making them all positive. It turns out that the latter, squaring all the deviations, is how statisticians solve this problem. Once we square the deviations, we can take their average and get a measure of variability. Later (using a beautifully descriptive term created by our students), we will “unsquare” those deviations in order to calculate the standard deviation
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