Psychology 121, Lecture 3

 Overview

Describing a group of scores

1. Measure of central tendency

• Mean: The mean is equal to the sum of all the scores divided by the number of scores.
• Median: The middle score of a ranked list of scores. E.g., of the list 2,4,5,6,7,8,13 6 is the median.
• Mode: The mode is the number with the highest frequency. E.g., of the list 2,4,3,6,4,9,5 4 is the mode because it occurs twice.

• Variance. Variance is a useful number because it is additive. For instance the total variance in a group of scores could be the sum of the valid variance, random variance, and bias variance.
• Standard deviation is useful because it is in the same units as the tests.

1. (Observed score - mean) = deviation score
2. square each deviation score (to render all deviations positive)
3. sum together all deviations scores, then divide by the number of scores

3. Skew

• A positive skew: The distribution is distorted by more people having higher scores than would be expected in a perfect distribution.
• A negative skew: The distribution is distorted by more people having lower scores than would be expected in a perfect distribution.
• to remember these remember things are skewed in the direction of "more tail"

Common Comparisons between a Score and Norms

• Percentile Rank = % of scores falling below a particular score
• Percentile = Score that is higher than a specific percent of scores related measures include quartiles, deciles, & stanines
• Z scores = a transformation to a distribution where mean=0, SD=1,

• T scores = a linear transformation to a distribution where mean=50, SD=10, typically rounded to an integer. T scores are easier to deal with than Z scores.

• Age, Grade-Equivalent , & Developmental Scales

• Age scale: Compares a particular score to the average score at a particular age. E.g., "Billy has the vocabulary of a twelve year old."
• Grade-equivalent and developmental scales are similar to age scales but use grade scales or developmental scales rather than ages as the basis for comparison. E.g., "You are at a ninth grade reading level."

Advantages & disadvantages of comparison methods

Percentile and percentile rank

The biggest problem is that they tell you had you did relative to other people rather than telling you how much you know (if knowledge is the thing being tested). They tend to overemphasize the importance of differences around the mean and underestimate the importance of differences at the extreme. Many people are concentrated around the mean so doing just a little bit better on a test will raise your percentile rank significantly.

Z and T scores

(You should know score distributions in a standard distribution in terms of Z scores: 34% between 0 and 1 in the Z score, 14% between 1 and 2 in the z score, and 2% between 2 and 3 in the z scores. The percentages are the same in the negatives.) The biggest problem is that nobody but statisticians usually understand them.

Age, grade-equivalent, and developmental scales

The drawbacks are (a) There is no idea of the variability of the scores, only what the average is; (b) There is no breakdown of the information known. For instance, you might be doing arithmetic at a grade level two beyond your own but you might only know some of the skills very well and others not at all. It would be unwise to skip the grades as might be suggested by the grad-equivalency scores; (c) The scores are only appropriate where there is a relatively linear relationship between the age (or grade or development) and the performance. If performance tends to come at uneven rates, you won't know how much better or worse you are doing by you age or grade or developmental scores.

Correlation and Regression

Regression Line = The best-fitting straight line summarizing a scatterplot

• Y' = a + bX,

• Slope = change in Y per unit of change in X (also called regression coefficient)
• Intercept = value of Y when X=0
• Residual = prediction error (Y' - Y)

Correlation and Regression

• Correlation coefficients summarize the strength and direction of a linear relationship between variable.
• Direction describes whether there is a positive or negative relationship between the variables. For instance, there might be a positive correlation between calorie intake and weight. And a negative correlation between the number of beers you have before a test and your score on the test.
• The strength of the correlation describes how well you can predict one variable knowing the other.
• The most common correlation coefficient is the Pearson Product Moment Correlation.
• This is often symbolized by an r.
• It has a scale of -1 to 1 with the negative sign indicating a negative correlation.
• The absolute value of the number indicates the strength with a 1 being the strongest correlation.
• There are many alternative coefficients, such as Spearman's rho (for rank order), Phi (for dichotomies), etc. See p.85 for more details.
• The significance of r (probability of finding that a correlation that large is due to chance alone). This is a joint function of association strength and sample size.
• Standard Error of Estimate measures residuals. Interpret as the standard deviation of prediction errors (p.87). Tells you how much unreliability there is in your estimate.
• Coefficient of Determination =r², indicating how much of the variability in measure is related to the other measure. Tells you how much of the variance of one set of scores can be predicted by known other set of scores.