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Understanding Linear Transformations and Z-scores in Psychology Data Analysis

Dive into the world of psychology data analysis, exploring linear transformations and Z-scores. Learn how to interpret and apply these concepts effectively to analyze psychological data with precision.

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Understanding Linear Transformations and Z-scores in Psychology Data Analysis

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  1. Psychology 10 Analysis of Psychological Data February 19, 2014

  2. The Plan for Today • Review of midterm exam • More on measures of variability • Linear transformations • Changes in measures of central tendency under linear transformation • Changes in measures of variability under linear transformation • Z scores

  3. Midterm Grades

  4. Midterm Distribution

  5. Review • IQR = 75th percentile – 25th percentile • Sample standard deviation = SS / (N – 1). • SS is defined as S(X - M)2. • Computationally, SS = SX2 – (SX)2 / N is more efficient and less likely to lead to errors. • Example.

  6. Linear transformations • Sometimes we might want to change the metric of a variable using multiplication and addition. • Example: Suppose an income data set is reported in hundreds of dollars. We might want to change it into dollars. We could accomplish that by multiplying by 100.

  7. Linear transformations (cont.) • If the transformation is of the form Y = a + b X, then we say that it is a linear transformation. • If the transformation is linear and we’ve already calculated the mean and sd, we don’t need to redo the tedious calculations.

  8. Rules for change under linear transformation • The mean of Y = a + b X is a + b MX. • The same rule works for the median. • The standard deviation of Y = a + b X is b sX. • The same rule works for the IQR. • The variance of Y = a + b X is b2s2X.

  9. Example • If I tell you that the mean income of a data set is 467 hundreds of dollars, and the standard deviation is 154.5 hundreds of dollars, what are the mean and sd in dollars? • Y = 0 + 100 X. • New mean = 0 + 100 * 467 = $46,700. • New sd = 100 * 154.5 = $15,450.

  10. Another example • In the Raven data set used on the midterm, the mean is 32.525, and the standard deviation is 11.94. • Suppose I want to change the metric using the transformation Y = -5 + 0.5 * X. • The new mean is -5 + 0.5 * 32.525 = 11.263. • The new sd is 0.5 * 11.94 = 5.97.

  11. Why are these transformations “linear?” • Equations of the form Y = a + b X define lines. • Consider this data set: X = (1, 2, 3). • Suppose I calculate Y = 5 – 10 X. • Then Y = (-5, -15, -25). • If I plot Y against X, the points form a line. • Hence, we say that the equation is the equation of a line (or “linear”).

  12. More detail on linear equations • If Y = a + b X, a is the height of the line when X = 0. • a is called the “Y intercept” or simply “intercept.” • b is called the “slope,” because it represents how steep the line is. • The slope is the number of units change in Y for each unit change in X.

  13. Z scores • Sometimes a special linear transformation of the form Z = (X – M) / s will be particularly useful. • Is that really a linear transformation? • Z = -M / s + (1 / s) X . • a = - M / s b = 1 / s • Yes, that’s linear.

  14. Apply the rules for change under linear transformation • Z = -M / s + (1 / s) X . • MZ = -M / s + (1 / s) M = 0. • sZ = (1 / s) s = 1. • So the Z transformation of any variable will have a mean of zero and a standard deviation of one. • This is sometimes called “putting the variable in standard form.”

  15. Characteristics of Z scores • The sign of the Z score tells us whether the score is above or below the mean of the distribution. • The magnitude of the Z score tells us how far above or below, in standard deviation units. • For example, a Z score of -0.4 represents an individual score that is four tenths of a standard deviation below the mean.

  16. Comparisons using Z scores • Because the Z score is scale free, it can help us compare variables that are reported in different metrics. • Tonight’s in-class exercise illustrates that process.

  17. Next time… • We will talk a little more about the utility of Z scores. • We will begin to discuss probability.

  18. Activity • 40 people took exam 1. Your score was 80. • S X = 3,160, and S X2 = 252,136. • 35 people took exam 2. Your score was 68. • S X = 2,240, and S X2 = 146,760. • Which of your exam scores represents better performance relative to the rest of the class?

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