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Published byErik Fox Modified about 1 year ago

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Measures of Dispersion and Standard Scores

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Deviation Review Deviation is the difference from a standard or reference value (usually the mean). This is the starting point for determining both the variance and the standard deviation of a set of scores. We want to measure the dispersion of the scores around the mean, so it makes sense to use the deviation scores.

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Activity #1 (Part 1) Measure to the nearest millimeter the writing utensil you are using and write the measurement on a piece of paper.

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Sum of Squares (SS) Sum of squares = sum of squared deviations SS is used when calculating the variance and the standard deviation If you are using a frequency distribution table:

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Sum of Squares (SS) The population formulas are as follows:

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Calculating SS Step 1: Find the mean. Step 2: Find the deviation scores. Step 3: Square the deviation scores. Step 4: Sum the squared deviation scores.

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Sum of Squares Computational Formulas When you are trying to understand SS, use the previous formulas, but when you are computing SS, use these formulas (it’s faster).

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Watch Out Know the difference between and. means that you square Xs, then sum the squared Xs. means that you sum the Xs, then square the sum of the Xs.

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Calculating SS (the easy way) Step 1: Sum the X column and square the sum. Step 2: Square each score (X) Step 3: Sum the X 2 column. Step 4: Plug the values into the formula and solve.

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Calculating SS with Frequency Distribution Tables Step 1: Create a fX column and sum the values, then square the sum. Step 2: Create a fX 2 column by multiplying fX(X) (do not square fX). Step 3: Sum the fX 2 column. Step 4: Plug the values into the equation and solve (remember N is the sum of f).

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Population Variance (σ 2 ) Variance = average of squared deviations Recall that SS is the sum of the squared deviations, the numerator in the above equation. So we can rewrite the equation as:

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Sample Variance (s 2 ) If we use the population formula for sample data, we will probably underestimate the variance (i.e., s 2 will be smaller than σ 2 ). To correct for this and get a better estimate of the population variance, we change the denominator to N-1.

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Sample Variance (s 2 ) When you use N-1 in the denominator, the result is a larger estimate of the population variance. Why do we need to make our estimate larger? (Unadjusted) sample variance is always less than or equal to population variance.

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Standard Deviation The standard deviation is the square root of the variance.

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Standard Deviation (Sample)

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Calculation: Break it Down Sum of Squares (SS) Variance (s 2 )

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Know These Equations With these three equations you can understand and calculate sample variance and standard deviation.

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What Does it Look Like? Let’s look at an example (also see p. 103): The standard deviation is another unit of measurement on the X axis.

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What Should We Expect? For a small sample, we should expect the standard deviation units to divide the sample distribution into about 4 parts. For a large sample, we should expect closer to six parts.

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Activity #1 (Part 2) Given our writing utensil data, calculate (in the following order): – SS – s 2 – s Use either of the sample formulas and show all of your work. Write your name on the paper.

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Standard Scores (z scores) z score = the deviation of a raw score from the mean in standard deviation units. The closer a score is to the mean, the smaller its z score will be. A positive z-score indicates that the score is above the mean, negative indicates that it is below the mean. A z score of zero will always be at the mean.

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You Try What raw score would have a z score of -1?

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You Try What raw score would have a z score of 2?

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Calculating z scores Population: Sample: So you have to know the standard deviation and the mean before you can find the z score.

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Calculating Raw Scores If I know the z score but I don’t know the corresponding raw score, using basic algebra I can change the equation to solve for X.

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Activity #2 Given the above information, calculate the z score values for the following raw scores: 75, 95, 100, 50, 125, and 80. Using the same sample information, calculate the raw scores for the following z scores: 3.25, -.25, 2, -1.75, and 2.5

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Homework Study for Chapter 6 Quiz (know the equations in red). Read Chapter 7 Do Chapter 6 Homework

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