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Variability is to measure & describe the degree to which the scores in…
Variability is to measure & describe the degree to which the scores in a distribution are spread out of clustered together.
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Standard Deviation - is the square root of the variance & provides a measure of the standard distance from the mean.
Most commonly used and the most important measure of variability. It uses the MEAN as a reference point to find the distance b/w each score & the mean.
Deviation = X - mean (score minus the mean)
Step 1: Deviation score (represented by "x") is the result of the deviation. Deviation scores always equal 0.
Step 2: Calculate the mean of the deviation of scores: add up the deviation scores and then divide by N.
Step 3: Square each deviation score to get rid of -/+ signs --> mean squared deviation, which is called the VARIANCE
Step 4: Take the square root of the variance to obtain the Standard Deviation.
B/c the sum of the deviations is ALWAYS ZERO, the mean of the deviations is also zero and is of no value as a measure of variability.
SS, sum of squares - is the sum of the squared deviation scores.
- Find the deviation of each score (X-m).
- Square each deviation score (-m)^2
- Add the squared deviations.
Computational formula is used when there are not whole numbers (decimals) to prevent rounding errors. SS = standard deviation - (sum of scores)^2/N
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Is represented on a frequency distribution graph by a horizontal line or arrow extending from the mean for a distance equal to one standard deviation.
Usually variability is defined in terms of distance -- tells how much distance to expect b/w one score & another or how much distance to expect b/w an individual score & the mean.
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Sample Variability - sample statistic is biased if it consistently over estimates or underestimates corresponding population parameter.
Sample Variance & Standard Deviation - same formulas as variance and deviation. Sample Mean is M and n for sample scores.
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Sample standard deviation = s = square root of s^2 = square root of SS/n-1. Square root of the mean squared deviation from the sample mean.
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Transformations of Scale
- Adding a constant to each score does not change the standard deviation.
- Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant.
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Degrees of Freedom (df) for the sample variance is n-1. It indicates the # of scores in a sample that are independent and free to vary.
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