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Chapter 4: Variability. A quantitative measure of the differences in the…
Chapter 4: Variability. A quantitative measure of the differences in the scores of a distribution. Describes the degree to which scores are spread out or close together.
Distribution of scores. Ex. The differences in height of adult males. The average is 70 inches with most variation within 5 inches.
Measures how well a score(or scores) represent a distribution. When using a sample to represent a population, variability indicates how much error to expect. Ex. When selecting one male to represent the average height of an adult male, the likelihood of finding one closer to the mean height is higher than selecting one male to represent average weight (which has higher variability).
Range. Distance covered by the scores in a distribution. Ex. For the following scores: 3, 6, 7, 8, the range is 8 - 3 = 5
Range for continuous variables. The difference between the Upper Real Limit and Lower Real Limit. Ex. For continuous scores 1 to 5, the range is 5.5 - 0.5 = 5.
Range can also refer to the number of measurement categories. Ex. Suppose we measure the number of children in a family, including 0, 1, 2, 3, 4. The range is 5.
Range is not a good measure of variability, because it does not describe the typical distance among common scores.
Quartile. Represents one-fourth of the distribution of scores. Ex. Quartiles have percentile ranks of 25%, 50%, 75%, and 100%
Interquartile Range. Represents the distance between middle two quartiles, Q2 and Q3, or middle 50% of the distribution.
Deviation or deviation score. The difference between the score and the mean. Ex. If the score is 10 and the mean is 8, then the deviation is 2.
Variance. The average squared distance from the mean. It is squared to account for positive and negative deviations.
Sum of Squares (SS). The sum of squared deviations.
Ex. Step 1. Find deviation (X - µ).
Step 2. Square deviation (X - µ)^2
Step 3: Add squared deviations.
Population Variation. The Sum of Squares (SS) over N, the number of scores in the population. SS/N.
Standard deviation. Is the squared root of the Variance. It describes the average distance or deviation from the mean.
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Adding a constant to a scale (such as adding 2 years to all ages) does not change the standard deviation.
Multiplying each score by a constant (changing KG to pounds) requires the standard deviation to also be multiplied by that constant
Sample variability bias. A sample statistic is said to be biased if it consistently over or under estimates a population parameter.
Sum of Squares (SS) for Samples. Is the same symbol, SS, and similar calculation as for the population.
Sample variation and standard deviation. Compared to population variation and standard deviation, use (n-1) in the denominator. Why? This increases the variance and standard deviation, which more accurate and less biased estimate of the population variance and standard deviation.
Degrees of Freedom. For a sample of n scores, degrees of freedom is (n - 1). This represents the number of scores in a sample that are free to vary.
Unbiased statistic. When the average value of a sample statistic is the same as a population parameter.